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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|---|---|---|---|---|---|---|
AlgebraicGeometry.Scheme.residueFieldCongr_trans_hom
|
Mathlib/AlgebraicGeometry/ResidueField.lean
|
@[reassoc (attr := simp)]
lemma residueFieldCongr_trans_hom (X : Scheme) {x y z : X} (e : x = y) (e' : y = z) :
(X.residueFieldCongr e).hom ≫ (X.residueFieldCongr e').hom =
(X.residueFieldCongr (e.trans e')).hom
|
X : Scheme
x y z : ↑↑X.toPresheafedSpace
e : x = y
e' : y = z
⊢ (residueFieldCongr e).hom ≫ (residueFieldCongr e').hom = (residueFieldCongr ⋯).hom
|
subst e e'
|
X : Scheme
x : ↑↑X.toPresheafedSpace
⊢ (residueFieldCongr ⋯).hom ≫ (residueFieldCongr ⋯).hom = (residueFieldCongr ⋯).hom
|
677b1f74c913e216
|
MultilinearMap.dfinsuppFamily_single_left_apply
|
Mathlib/LinearAlgebra/Multilinear/DFinsupp.lean
|
theorem dfinsuppFamily_single_left_apply [∀ i, DecidableEq (κ i)]
(p : Π i, κ i) (f : MultilinearMap R (fun i ↦ M i (p i)) (N p)) (x : Π i, Π₀ j, M i j) :
dfinsuppFamily (Pi.single p f) x = DFinsupp.single p (f fun i => x _ (p i))
|
case h.inl
ι : Type uι
κ : ι → Type uκ
R : Type uR
M : (i : ι) → κ i → Type uM
N : ((i : ι) → κ i) → Type uN
inst✝⁷ : DecidableEq ι
inst✝⁶ : Fintype ι
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)
inst✝³ : (p : (i : ι) → κ i) → AddCommMonoid (N p)
inst✝² : (i : ι) → (k : κ i) → Module R (M i k)
inst✝¹ : (p : (i : ι) → κ i) → Module R (N p)
inst✝ : (i : ι) → DecidableEq (κ i)
p : (i : ι) → κ i
f : MultilinearMap R (fun i => M i (p i)) (N p)
x : (i : ι) → Π₀ (j : κ i), M i j
⊢ ((dfinsuppFamily (Pi.single p f)) x) p = (DFinsupp.single p (f fun i => (x i) (p i))) p
|
simp
|
no goals
|
a3acc84950fc829b
|
IsPreconnected.preperfect_of_nontrivial
|
Mathlib/Topology/Perfect.lean
|
lemma IsPreconnected.preperfect_of_nontrivial [T1Space α] {U : Set α} (hu : U.Nontrivial)
(h : IsPreconnected U) : Preperfect U
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
h : IsPreconnected U
⊢ Preperfect U
|
intro x hx
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
h : IsPreconnected U
x : α
hx : x ∈ U
⊢ AccPt x (𝓟 U)
|
30f2e088a928e593
|
AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem
|
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
|
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q
|
case refine_2.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ HomogeneousLocalization.mk
{ deg := m * i, num := ⟨(if n ≤ i then a * ↑(((decompose 𝒜) x) (i - n)) else 0) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩,
den_mem := ⋯ } ∈
q.asIdeal
|
let product : A⁰_ f :=
(HomogeneousLocalization.mk
⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f) *
(HomogeneousLocalization.mk
⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
|
case refine_2.mk.refine_3
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
HomogeneousLocalization.mk { deg := m • n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.mk.refine_1⟩, den_mem := ⋯ } *
HomogeneousLocalization.mk
{ deg := m • (i - n), num := ⟨(proj 𝒜 (i - n)) x ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.mk.refine_2⟩,
den_mem := ⋯ }
⊢ HomogeneousLocalization.mk
{ deg := m * i, num := ⟨(if n ≤ i then a * ↑(((decompose 𝒜) x) (i - n)) else 0) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩,
den_mem := ⋯ } ∈
q.asIdeal
case refine_2.mk.refine_1
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ n ∈ 𝒜 (m • n)
case refine_2.mk.refine_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ f ^ (i - n) ∈ 𝒜 (m • (i - n))
|
a6e1542c5c7977fc
|
Real.IsConjExponent.inv_add_inv_conj_ennreal
|
Mathlib/Data/Real/ConjExponents.lean
|
theorem inv_add_inv_conj_ennreal : (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1
|
p q : ℝ
h : p.IsConjExponent q
⊢ (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1
|
rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_inv_of_pos h.pos,
← ENNReal.ofReal_inv_of_pos h.symm.pos, ← ENNReal.ofReal_add h.inv_nonneg h.symm.inv_nonneg,
h.inv_add_inv_conj]
|
no goals
|
46c010c443c7a174
|
NormedField.completeSpace_iff_isComplete_closedBall
|
Mathlib/Analysis/Normed/Field/Lemmas.lean
|
lemma NormedField.completeSpace_iff_isComplete_closedBall {K : Type*} [NormedField K] :
CompleteSpace K ↔ IsComplete (Metric.closedBall 0 1 : Set K)
|
K : Type u_4
inst✝ : NormedField K
⊢ CompleteSpace K ↔ IsComplete (Metric.closedBall 0 1)
|
constructor <;> intro h
|
case mp
K : Type u_4
inst✝ : NormedField K
h : CompleteSpace K
⊢ IsComplete (Metric.closedBall 0 1)
case mpr
K : Type u_4
inst✝ : NormedField K
h : IsComplete (Metric.closedBall 0 1)
⊢ CompleteSpace K
|
a36770b11a5c8086
|
Turing.BlankExtends.above_of_le
|
Mathlib/Computability/Tape.lean
|
theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂
|
case h
Γ : Type u_1
inst✝ : Inhabited Γ
l₁ l₂ : List Γ
i j : ℕ
e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default
h : l₁.length ≤ l₂.length
⊢ l₂ = l₁ ++ List.replicate (i - j) default
|
refine List.append_cancel_right (e.symm.trans ?_)
|
case h
Γ : Type u_1
inst✝ : Inhabited Γ
l₁ l₂ : List Γ
i j : ℕ
e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default
h : l₁.length ≤ l₂.length
⊢ l₁ ++ List.replicate i default = l₁ ++ List.replicate (i - j) default ++ List.replicate j default
|
5c9b524679a1747d
|
Dynamics.netMaxcard_le_coverMincard
|
Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean
|
lemma netMaxcard_le_coverMincard (T : X → X) (F : Set X) {U : Set (X × X)} (U_symm : SymmetricRel U)
(n : ℕ) :
netMaxcard T F U n ≤ coverMincard T F U n
|
X : Type u_1
T : X → X
F : Set X
U : Set (X × X)
U_symm : SymmetricRel U
n : ℕ
⊢ netMaxcard T F U n ≤ coverMincard T F U n
|
rcases eq_top_or_lt_top (coverMincard T F U n) with h | h
|
case inl
X : Type u_1
T : X → X
F : Set X
U : Set (X × X)
U_symm : SymmetricRel U
n : ℕ
h : coverMincard T F U n = ⊤
⊢ netMaxcard T F U n ≤ coverMincard T F U n
case inr
X : Type u_1
T : X → X
F : Set X
U : Set (X × X)
U_symm : SymmetricRel U
n : ℕ
h : coverMincard T F U n < ⊤
⊢ netMaxcard T F U n ≤ coverMincard T F U n
|
77281fcc433e4e06
|
VitaliFamily.withDensity_le_mul
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem withDensity_le_mul {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
μ.withDensity (v.limRatioMeas hρ) s ≤ (t : ℝ≥0∞) ^ 2 * ρ s
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ν (s ∩ f ⁻¹' {⊤}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ∫⁻ (a : α) in s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), v.limRatioMeas hρ a ∂μ ≤
(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
|
calc
(∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ) ≤ ∫⁻ _ in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ :=
lintegral_mono_ae ((ae_restrict_iff' M).2 (Eventually.of_forall fun x hx => hx.2.2.le))
_ = (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
_ = (t : ℝ≥0∞) ^ (2 : ℤ) * ((t : ℝ≥0∞) ^ (n - 1) * μ (s ∩ f ⁻¹' I)) := by
rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]
congr 2
abel
_ ≤ (t : ℝ≥0∞) ^ (2 : ℤ) * ρ (s ∩ f ⁻¹' I) := by
gcongr
rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']
apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
intro x hx
apply lt_of_lt_of_le _ hx.2.1
rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,
zpow_add₀ t_ne_zero']
conv_rhs => rw [← mul_one (t ^ n)]
gcongr
rw [zpow_neg_one]
exact inv_lt_one_of_one_lt₀ ht
|
no goals
|
5a46bd1d651b9448
|
MeasureTheory.Measure.measure_preimage_of_map_eq_self
|
Mathlib/MeasureTheory/Measure/Map.lean
|
/-- If `map f μ = μ`, then the measure of the preimage of any null measurable set `s`
is equal to the measure of `s`.
Note that this lemma does not assume (a.e.) measurability of `f`. -/
lemma measure_preimage_of_map_eq_self {f : α → α} (hf : map f μ = μ)
{s : Set α} (hs : NullMeasurableSet s μ) : μ (f ⁻¹' s) = μ s
|
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → α
hf : map f μ = μ
s : Set α
hs : NullMeasurableSet s μ
hfm : AEMeasurable f μ
⊢ μ (f ⁻¹' s) = μ s
|
rw [← map_apply₀ hfm, hf]
|
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → α
hf : map f μ = μ
s : Set α
hs : NullMeasurableSet s μ
hfm : AEMeasurable f μ
⊢ NullMeasurableSet s (map f μ)
|
1342c2772e8df2cf
|
AlgebraicGeometry.Scheme.Cover.fromGlued_injective
|
Mathlib/AlgebraicGeometry/Gluing.lean
|
theorem fromGlued_injective : Function.Injective 𝒰.fromGlued.base
|
case intro.intro.intro.intro
X : Scheme
𝒰 : X.OpenCover
i : (gluedCover 𝒰).J
x : ↑↑((gluedCover 𝒰).U i).toPresheafedSpace
j : (gluedCover 𝒰).J
y : ↑↑((gluedCover 𝒰).U j).toPresheafedSpace
h : (ConcreteCategory.hom (𝒰.map i).base) x = (ConcreteCategory.hom (𝒰.map j).base) y
e : (TopCat.pullbackCone (forgetToTop.map (𝒰.map i)) (forgetToTop.map (𝒰.map j))).pt ≅
(PullbackCone.mk (forgetToTop.map (pullback.fst (𝒰.map i) (𝒰.map j)))
(forgetToTop.map (pullback.snd (𝒰.map i) (𝒰.map j))) ⋯).pt :=
(TopCat.pullbackConeIsLimit (forgetToTop.map (𝒰.map i)) (forgetToTop.map (𝒰.map j))).conePointUniqueUpToIso
(isLimitOfHasPullbackOfPreservesLimit forgetToTop (𝒰.map i) (𝒰.map j))
⊢ (ConcreteCategory.hom ((gluedCover 𝒰).ι i).base) x = (ConcreteCategory.hom ((gluedCover 𝒰).ι j).base) y
|
rw [𝒰.gluedCover.ι_eq_iff]
|
case intro.intro.intro.intro
X : Scheme
𝒰 : X.OpenCover
i : (gluedCover 𝒰).J
x : ↑↑((gluedCover 𝒰).U i).toPresheafedSpace
j : (gluedCover 𝒰).J
y : ↑↑((gluedCover 𝒰).U j).toPresheafedSpace
h : (ConcreteCategory.hom (𝒰.map i).base) x = (ConcreteCategory.hom (𝒰.map j).base) y
e : (TopCat.pullbackCone (forgetToTop.map (𝒰.map i)) (forgetToTop.map (𝒰.map j))).pt ≅
(PullbackCone.mk (forgetToTop.map (pullback.fst (𝒰.map i) (𝒰.map j)))
(forgetToTop.map (pullback.snd (𝒰.map i) (𝒰.map j))) ⋯).pt :=
(TopCat.pullbackConeIsLimit (forgetToTop.map (𝒰.map i)) (forgetToTop.map (𝒰.map j))).conePointUniqueUpToIso
(isLimitOfHasPullbackOfPreservesLimit forgetToTop (𝒰.map i) (𝒰.map j))
⊢ (gluedCover 𝒰).Rel ⟨i, x⟩ ⟨j, y⟩
|
e5353fbe9b3f7ecf
|
Urysohns.CU.continuous_lim
|
Mathlib/Topology/UrysohnsLemma.lean
|
theorem continuous_lim (c : CU P) : Continuous c.lim
|
case h
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ (dist (c.left.lim a✝) (c.left.lim x) + dist 0 0) / 2 ≤ 3 / 4 * (3 / 4) ^ n
|
rw [dist_self, add_zero, div_eq_inv_mul]
|
case h
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ 2⁻¹ * dist (c.left.lim a✝) (c.left.lim x) ≤ 3 / 4 * (3 / 4) ^ n
|
680ff52981894f6e
|
HomologicalComplex.homotopyCofiber.descSigma_ext_iff
|
Mathlib/Algebra/Homology/HomotopyCofiber.lean
|
lemma descSigma_ext_iff {φ : F ⟶ G} {K : HomologicalComplex C c}
(x y : Σ (α : G ⟶ K), Homotopy (φ ≫ α) 0) :
x = y ↔ x.1 = y.1 ∧ (∀ (i j : ι) (_ : c.Rel j i), x.2.hom i j = y.2.hom i j)
|
case mpr
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
inst✝ : DecidableRel c.Rel
φ : F ⟶ G
K : HomologicalComplex C c
x y : (α : G ⟶ K) × Homotopy (φ ≫ α) 0
⊢ (x.fst = y.fst ∧ ∀ (i j : ι), c.Rel j i → x.snd.hom i j = y.snd.hom i j) → x = y
|
obtain ⟨x₁, x₂⟩ := x
|
case mpr.mk
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
inst✝ : DecidableRel c.Rel
φ : F ⟶ G
K : HomologicalComplex C c
y : (α : G ⟶ K) × Homotopy (φ ≫ α) 0
x₁ : G ⟶ K
x₂ : Homotopy (φ ≫ x₁) 0
⊢ (⟨x₁, x₂⟩.fst = y.fst ∧ ∀ (i j : ι), c.Rel j i → ⟨x₁, x₂⟩.snd.hom i j = y.snd.hom i j) → ⟨x₁, x₂⟩ = y
|
515f5daed7a49791
|
TopCat.Sheaf.eq_of_locally_eq
|
Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean
|
theorem eq_of_locally_eq (s t : ToType (F.1.obj (op (iSup U))))
(h : ∀ i, F.1.map (Opens.leSupr U i).op s = F.1.map (Opens.leSupr U i).op t) : s = t
|
case a
C : Type u
inst✝⁵ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimits C
inst✝¹ : HasForget.forget.ReflectsIsomorphisms
inst✝ : PreservesLimits HasForget.forget
X : TopCat
F : Sheaf C X
ι : Type v
U : ι → Opens ↑X
s t : ToType (F.val.obj (op (iSup U)))
h :
∀ (i : ι), (ConcreteCategory.hom (F.val.map (leSupr U i).op)) s = (ConcreteCategory.hom (F.val.map (leSupr U i).op)) t
sf : (i : ι) → ToType (F.val.obj (op (U i))) := fun i => (ConcreteCategory.hom (F.val.map (leSupr U i).op)) s
sf_compatible : IsCompatible F.val U sf
gl : ToType (F.val.obj (op (iSup U)))
gl_uniq : ∀ (y : ToType (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl
⊢ IsGluing F.val U sf s
|
intro i
|
case a
C : Type u
inst✝⁵ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimits C
inst✝¹ : HasForget.forget.ReflectsIsomorphisms
inst✝ : PreservesLimits HasForget.forget
X : TopCat
F : Sheaf C X
ι : Type v
U : ι → Opens ↑X
s t : ToType (F.val.obj (op (iSup U)))
h :
∀ (i : ι), (ConcreteCategory.hom (F.val.map (leSupr U i).op)) s = (ConcreteCategory.hom (F.val.map (leSupr U i).op)) t
sf : (i : ι) → ToType (F.val.obj (op (U i))) := fun i => (ConcreteCategory.hom (F.val.map (leSupr U i).op)) s
sf_compatible : IsCompatible F.val U sf
gl : ToType (F.val.obj (op (iSup U)))
gl_uniq : ∀ (y : ToType (F.val.obj (op (iSup U)))), (fun s => IsGluing F.val U sf s) y → y = gl
i : ι
⊢ (ConcreteCategory.hom (F.val.map (leSupr U i).op)) s = sf i
|
8510ff05c576a104
|
Polynomial.Monic.natDegree_mul'
|
Mathlib/Algebra/Polynomial/Monic.lean
|
theorem natDegree_mul' (hp : p.Monic) (hq : q ≠ 0) :
(p * q).natDegree = p.natDegree + q.natDegree
|
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p.Monic
hq : q ≠ 0
⊢ p.leadingCoeff * q.leadingCoeff ≠ 0
|
simpa [hp.leadingCoeff, leadingCoeff_ne_zero]
|
no goals
|
20f0617818be8c81
|
Metric.uniformity_edist_aux
|
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε }
|
α : Type u_3
d : α → α → ℝ≥0
⊢ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε} = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε}
|
simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
|
α : Type u_3
d : α → α → ℝ≥0
⊢ (∀ i > 0, {p | ↑(d p.1 p.2) < i} ∈ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε}) ∧
∀ i > 0, {p | ↑(d p.1 p.2) < i} ∈ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | ↑(d p.1 p.2) < ε}
|
ab44719fcc963446
|
Nat.not_mem_of_lt_sInf
|
Mathlib/Data/Nat/Lattice.lean
|
theorem not_mem_of_lt_sInf {s : Set ℕ} {m : ℕ} (hm : m < sInf s) : m ∉ s
|
case inr
s : Set ℕ
m : ℕ
h : s.Nonempty
hm : m < Nat.find h
⊢ m ∉ s
|
exact Nat.find_min h hm
|
no goals
|
190805e255829153
|
InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
theorem angle_sub_eq_arcsin_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)
|
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x (-y) = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)
|
rw [or_comm, ← neg_ne_zero, or_comm] at h0
|
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x (-y) = 0
h0 : x ≠ 0 ∨ -y ≠ 0
⊢ angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)
|
621f12afbb27704c
|
IsFractional.sup
|
Mathlib/RingTheory/FractionalIdeal/Basic.lean
|
theorem _root_.IsFractional.sup {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
rw [smul_add]
apply isInteger_add
· rw [mul_smul, smul_comm]
exact isInteger_smul (hI bI hbI)
· rw [mul_smul]
exact isInteger_smul (hJ bJ hbJ)⟩
|
case intro.intro.intro.intro.ha
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R (aJ • aI • bI)
|
exact isInteger_smul (hI bI hbI)
|
no goals
|
5054dda448114ea2
|
Multiset.mem_powersetCardAux
|
Mathlib/Data/Multiset/Powerset.lean
|
theorem mem_powersetCardAux {n} {l : List α} {s} : s ∈ powersetCardAux n l ↔ s ≤ ↑l ∧ card s = n :=
Quotient.inductionOn s <| by
simp only [quot_mk_to_coe, powersetCardAux_eq_map_coe, List.mem_map, mem_sublistsLen,
coe_eq_coe, coe_le, Subperm, exists_prop, coe_card]
exact fun l₁ =>
⟨fun ⟨l₂, ⟨s, e⟩, p⟩ => ⟨⟨_, p, s⟩, p.symm.length_eq.trans e⟩,
fun ⟨⟨l₂, p, s⟩, e⟩ => ⟨_, ⟨s, p.length_eq.trans e⟩, p⟩⟩
|
α : Type u_1
n : ℕ
l : List α
s : Multiset α
⊢ ∀ (a : List α), (∃ a_1, (a_1 <+ l ∧ a_1.length = n) ∧ a_1 ~ a) ↔ (∃ l_1, l_1 ~ a ∧ l_1 <+ l) ∧ a.length = n
|
exact fun l₁ =>
⟨fun ⟨l₂, ⟨s, e⟩, p⟩ => ⟨⟨_, p, s⟩, p.symm.length_eq.trans e⟩,
fun ⟨⟨l₂, p, s⟩, e⟩ => ⟨_, ⟨s, p.length_eq.trans e⟩, p⟩⟩
|
no goals
|
305316f1982067af
|
Nat.div_eq_sub_mod_div
|
Mathlib/Data/Nat/Init.lean
|
lemma div_eq_sub_mod_div : m / n = (m - m % n) / n
|
m n : ℕ
⊢ m / n = (m - m % n) / n
|
obtain rfl | hn := n.eq_zero_or_pos
|
case inl
m : ℕ
⊢ m / 0 = (m - m % 0) / 0
case inr
m n : ℕ
hn : n > 0
⊢ m / n = (m - m % n) / n
|
946979a986a02452
|
PFunctor.M.iselect_eq_default
|
Mathlib/Data/PFunctor/Univariate/M.lean
|
theorem iselect_eq_default [DecidableEq F.A] [Inhabited (M F)] (ps : Path F) (x : M F)
(h : ¬IsPath ps x) : iselect ps x = head default
|
case pos.h
F : PFunctor.{u}
inst✝¹ : DecidableEq F.A
inst✝ : Inhabited F.M
ps_tail : List F.Idx
ps_ih : ∀ (x : F.M), ¬IsPath ps_tail x → (isubtree ps_tail x).head = default.head
a : F.A
i : F.B a
x_f : F.B a → F.M
h : ¬IsPath (⟨a, i⟩ :: ps_tail) (M.mk ⟨a, x_f⟩)
⊢ ¬IsPath ps_tail (x_f (cast ⋯ i))
|
intro h'
|
case pos.h
F : PFunctor.{u}
inst✝¹ : DecidableEq F.A
inst✝ : Inhabited F.M
ps_tail : List F.Idx
ps_ih : ∀ (x : F.M), ¬IsPath ps_tail x → (isubtree ps_tail x).head = default.head
a : F.A
i : F.B a
x_f : F.B a → F.M
h : ¬IsPath (⟨a, i⟩ :: ps_tail) (M.mk ⟨a, x_f⟩)
h' : IsPath ps_tail (x_f (cast ⋯ i))
⊢ False
|
6bd23acfd494ff8d
|
ProbabilityTheory.measure_limsup_eq_one
|
Mathlib/Probability/BorelCantelli.lean
|
theorem measure_limsup_eq_one {s : ℕ → Set Ω} (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ)
(hs' : (∑' n, μ (s n)) = ∞) : μ (limsup s atTop) = 1
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ⊤
this : IsProbabilityMeasure μ
⊢ μ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (μ[(s (k + 1)).indicator 1|↑(filtrationOfSet hsm) k]) ω) atTop atTop} =
1
|
suffices {ω | Tendsto (fun n => ∑ k ∈ Finset.range n,
(μ[(s (k + 1)).indicator (1 : Ω → ℝ)|filtrationOfSet hsm k]) ω) atTop atTop} =ᵐ[μ] Set.univ by
rw [measure_congr this, measure_univ]
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ⊤
this : IsProbabilityMeasure μ
⊢ {ω |
Tendsto (fun n => ∑ k ∈ Finset.range n, (μ[(s (k + 1)).indicator 1|↑(filtrationOfSet hsm) k]) ω) atTop
atTop} =ᶠ[ae μ]
Set.univ
|
b8c76ecaf1a3d1c2
|
LSeries_eventually_eq_zero_iff'
|
Mathlib/NumberTheory/LSeries/Injectivity.lean
|
/-- The `LSeries` of `f` is zero for large real arguments if and only if either `f n = 0`
for all `n ≠ 0` or the L-series converges nowhere. -/
lemma LSeries_eventually_eq_zero_iff' {f : ℕ → ℂ} :
(fun x : ℝ ↦ LSeries f x) =ᶠ[atTop] 0 ↔ (∀ n ≠ 0, f n = 0) ∨ abscissaOfAbsConv f = ⊤
|
case neg.refine_2
f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : ∀ (n : ℕ), ¬n = 0 → f n = 0
x : ℝ
⊢ (fun x => LSeries f ↑x) x = 0 x
|
simp [LSeries_congr x fun {n} ↦ H n, show (fun _ : ℕ ↦ (0 : ℂ)) = 0 from rfl]
|
no goals
|
7103bb783fd652c6
|
HasCompactSupport.contDiff_convolution_right
|
Mathlib/Analysis/Convolution.lean
|
theorem _root_.HasCompactSupport.contDiff_convolution_right {n : ℕ∞} (hcg : HasCompactSupport g)
(hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n (f ⋆[L, μ] g)
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace G
inst✝² : NormedAddCommGroup G
inst✝¹ : BorelSpace G
inst✝ : NormedSpace 𝕜 G
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
n : ℕ∞
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : ContDiff 𝕜 (↑n) g
⊢ ContDiff 𝕜 (↑n) (f ⋆[L, μ] g)
|
rcases exists_compact_iff_hasCompactSupport.2 hcg with ⟨k, hk, h'k⟩
|
case intro.intro
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace G
inst✝² : NormedAddCommGroup G
inst✝¹ : BorelSpace G
inst✝ : NormedSpace 𝕜 G
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
n : ℕ∞
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : ContDiff 𝕜 (↑n) g
k : Set G
hk : IsCompact k
h'k : ∀ x ∉ k, g x = 0
⊢ ContDiff 𝕜 (↑n) (f ⋆[L, μ] g)
|
e4b9034bdd5be77b
|
AlgebraicTopology.DoldKan.HigherFacesVanish.comp_Hσ_eq
|
Mathlib/AlgebraicTopology/DoldKan/Faces.lean
|
theorem comp_Hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ)
(hnaq : n = a + q) :
φ ≫ (Hσ q).f (n + 1) =
-φ ≫ X.δ ⟨a + 1, Nat.succ_lt_succ (Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm))⟩ ≫
X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm)⟩
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∑ x : Fin (n + 2), ((-1) ^ a * (-1) ^ ↑x) • (φ ≫ X.δ x) ≫ X.σ ⟨a, ⋯⟩ +
∑ x : Fin (n + 1 + 2), ((-1) ^ ↑x * (-1) ^ (a + 1)) • (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ x =
-(φ ≫ X.δ ⟨a + 1, ⋯⟩) ≫ X.σ ⟨a, ⋯⟩
|
rw [← Fin.sum_congr' _ (hnaq_shift 2).symm, Fin.sum_trunc]
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∑ i : Fin (a + 2),
((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i))) • (φ ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i))) ≫ X.σ ⟨a, ⋯⟩ +
∑ x : Fin (n + 1 + 2), ((-1) ^ ↑x * (-1) ^ (a + 1)) • (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ x =
-(φ ≫ X.δ ⟨a + 1, ⋯⟩) ≫ X.σ ⟨a, ⋯⟩
case hf
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∀ (j : Fin q),
((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.natAdd (a + 2) j))) •
(φ ≫ X.δ (Fin.cast ⋯ (Fin.natAdd (a + 2) j))) ≫ X.σ ⟨a, ⋯⟩ =
0
|
964779d74b03b839
|
Real.exists_isGLB
|
Mathlib/Data/Real/Archimedean.lean
|
theorem exists_isGLB (hne : s.Nonempty) (hbdd : BddBelow s) : ∃ x, IsGLB s x
|
s : Set ℝ
hne : s.Nonempty
hbdd : BddBelow s
hne' : (-s).Nonempty
⊢ ∃ x, IsGLB s x
|
have hbdd' : BddAbove (-s) := bddAbove_neg.mpr hbdd
|
s : Set ℝ
hne : s.Nonempty
hbdd : BddBelow s
hne' : (-s).Nonempty
hbdd' : BddAbove (-s)
⊢ ∃ x, IsGLB s x
|
85bc11b69c86eaa9
|
Matrix.det_updateCol_eq_zero
|
Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
|
theorem det_updateCol_eq_zero (h : i ≠ j) :
(M.updateCol j (fun k ↦ M k i)).det = 0 := det_zero_of_column_eq h (by simp [h])
|
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
M : Matrix n n R
i j : n
h : i ≠ j
⊢ ∀ (k : n), M.updateCol j (fun k => M k i) k i = M.updateCol j (fun k => M k i) k j
|
simp [h]
|
no goals
|
160cc2fe40b70d2b
|
AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen
|
Mathlib/Geometry/RingedSpace/Basic.lean
|
theorem isUnit_res_basicOpen {U : Opens X} (f : X.presheaf.obj (op U)) :
IsUnit (X.presheaf.map (@homOfLE (Opens X) _ _ _ (X.basicOpen_le f)).op f)
|
case h.intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↑↑X.toPresheafedSpace
hxU : x ∈ U
hx : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hxU)) f)
⊢ IsUnit
((ConcreteCategory.hom (X.presheaf.germ (X.basicOpen f) x ⋯))
((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f))
|
convert hx
|
case h.e'_3
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↑↑X.toPresheafedSpace
hxU : x ∈ U
hx : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hxU)) f)
⊢ (ConcreteCategory.hom (X.presheaf.germ (X.basicOpen f) x ⋯))
((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) =
(ConcreteCategory.hom (X.presheaf.germ U x hxU)) f
|
3112bb2ee47e3190
|
Multiset.toFinset_replicate
|
Mathlib/Data/Finset/Basic.lean
|
@[simp]
lemma toFinset_replicate (n : ℕ) (a : α) :
(replicate n a).toFinset = if n = 0 then ∅ else {a}
|
α : Type u_1
inst✝ : DecidableEq α
n : ℕ
a : α
⊢ (replicate n a).toFinset = if n = 0 then ∅ else {a}
|
ext x
|
case h
α : Type u_1
inst✝ : DecidableEq α
n : ℕ
a x : α
⊢ x ∈ (replicate n a).toFinset ↔ x ∈ if n = 0 then ∅ else {a}
|
aa7f47b181c09eaf
|
InnerProductGeometry.sin_angle_add_of_inner_eq_zero
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
theorem sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖
|
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖
|
rw [angle_add_eq_arcsin_of_inner_eq_zero h h0,
Real.sin_arcsin (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
(div_le_one_of_le₀ _ (norm_nonneg _))]
|
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : inner x y = 0
h0 : x ≠ 0 ∨ y ≠ 0
⊢ ‖y‖ ≤ ‖x + y‖
|
6afea2f3ae99dcc9
|
MeasureTheory.SimpleFunc.setToSimpleFunc_add
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem setToSimpleFunc_add (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
(hf : Integrable f μ) (hg : Integrable g μ) :
setToSimpleFunc T (f + g) = setToSimpleFunc T f + setToSimpleFunc T g :=
have hp_pair : Integrable (f.pair g) μ := integrable_pair hf hg
calc
setToSimpleFunc T (f + g) = ∑ x ∈ (pair f g).range, T (pair f g ⁻¹' {x}) (x.fst + x.snd)
|
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : Integrable (⇑g) μ
hp_pair : Integrable (⇑(f.pair g)) μ
⊢ ∑ x ∈ (f.pair g).range, (T (⇑(f.pair g) ⁻¹' {x})) x.1 + ∑ x ∈ (f.pair g).range, (T (⇑(f.pair g) ⁻¹' {x})) x.2 =
setToSimpleFunc T (map Prod.fst (f.pair g)) + setToSimpleFunc T (map Prod.snd (f.pair g))
|
rw [map_setToSimpleFunc T h_add hp_pair Prod.snd_zero,
map_setToSimpleFunc T h_add hp_pair Prod.fst_zero]
|
no goals
|
0c7344c6a33364aa
|
Fin.card_Ioc
|
Mathlib/Order/Interval/Finset/Fin.lean
|
@[simp]
lemma card_Ioc : #(Ioc a b) = b - a
|
n : ℕ
a b : Fin n
⊢ #(Ioc a b) = ↑b - ↑a
|
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
|
no goals
|
0d1d3ab707cebe97
|
Ideal.mem_map_C_iff
|
Mathlib/RingTheory/Polynomial/Basic.lean
|
theorem mem_map_C_iff {I : Ideal R} {f : R[X]} :
f ∈ (Ideal.map (C : R →+* R[X]) I : Ideal R[X]) ↔ ∀ n : ℕ, f.coeff n ∈ I
|
case mp.refine_1.intro
R : Type u
inst✝ : CommSemiring R
I : Ideal R
f✝ : R[X]
hf✝ : f✝ ∈ map C I
f : R[X]
hf : f ∈ ⇑C '' ↑I
n : ℕ
x : R
hx : x ∈ ↑I ∧ C x = f
⊢ (if n = 0 then x else 0) ∈ I
|
by_cases h : n = 0
|
case pos
R : Type u
inst✝ : CommSemiring R
I : Ideal R
f✝ : R[X]
hf✝ : f✝ ∈ map C I
f : R[X]
hf : f ∈ ⇑C '' ↑I
n : ℕ
x : R
hx : x ∈ ↑I ∧ C x = f
h : n = 0
⊢ (if n = 0 then x else 0) ∈ I
case neg
R : Type u
inst✝ : CommSemiring R
I : Ideal R
f✝ : R[X]
hf✝ : f✝ ∈ map C I
f : R[X]
hf : f ∈ ⇑C '' ↑I
n : ℕ
x : R
hx : x ∈ ↑I ∧ C x = f
h : ¬n = 0
⊢ (if n = 0 then x else 0) ∈ I
|
52a63b0c6c261831
|
basis_finite_of_finite_spans
|
Mathlib/LinearAlgebra/Basis/Cardinality.lean
|
/--
Over any nontrivial ring, the existence of a finite spanning set implies that any basis is finite.
-/
lemma basis_finite_of_finite_spans (w : Set M) (hw : w.Finite) (s : span R w = ⊤) {ι : Type w}
(b : Basis ι R M) : Finite ι
|
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
w : Set M
hw : w.Finite
s : span R w = ⊤
ι : Type w
b : Basis ι R M
this : Finite ↑w
val✝ : Fintype ↑w
i : Infinite ι
S : Finset ι := Finset.univ.sup fun x => (b.repr ↑x).support
bS : Set M := ⇑b '' ↑S
h : ∀ x ∈ w, x ∈ span R bS
k : span R bS = ⊤
x : ι
nm : x ∉ S
⊢ b x ∈ span R bS
|
rw [k]
|
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
w : Set M
hw : w.Finite
s : span R w = ⊤
ι : Type w
b : Basis ι R M
this : Finite ↑w
val✝ : Fintype ↑w
i : Infinite ι
S : Finset ι := Finset.univ.sup fun x => (b.repr ↑x).support
bS : Set M := ⇑b '' ↑S
h : ∀ x ∈ w, x ∈ span R bS
k : span R bS = ⊤
x : ι
nm : x ∉ S
⊢ b x ∈ ⊤
|
27ec1fa4ada49c51
|
AddMonoidAlgebra.freeAlgebra_lift_of_surjective_of_closure
|
Mathlib/RingTheory/FiniteType.lean
|
theorem freeAlgebra_lift_of_surjective_of_closure [CommSemiring R] {S : Set M}
(hS : closure S = ⊤) :
Function.Surjective
(FreeAlgebra.lift R fun s : S => of' R M ↑s : FreeAlgebra R S → R[M])
|
R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
f : R[M]
⊢ ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = f
|
induction' f using induction_on with m f g ihf ihg r f ih
|
case hM
R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
m : M
⊢ ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = (of R M) (Multiplicative.ofAdd m)
case hadd
R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
f g : R[M]
ihf : ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = f
ihg : ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = g
⊢ ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = f + g
case hsmul
R : Type u_1
M : Type u_2
inst✝¹ : AddMonoid M
inst✝ : CommSemiring R
S : Set M
hS : closure S = ⊤
r : R
f : R[M]
ih : ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = f
⊢ ∃ a, ((FreeAlgebra.lift R) fun s => of' R M ↑s) a = r • f
|
e03e61ce2ab85456
|
affineIndependent_of_ne
|
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
theorem affineIndependent_of_ne {p₁ p₂ : P} (h : p₁ ≠ p₂) : AffineIndependent k ![p₁, p₂]
|
case mk
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
p₁ p₂ : P
h : p₁ ≠ p₂
i₁ : { x // x ≠ 0 } := ⟨1, ⋯⟩
i : Fin 2
hi : i ≠ 0
⊢ ⟨i, hi⟩ = i₁
|
ext
|
case mk.a.h
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
p₁ p₂ : P
h : p₁ ≠ p₂
i₁ : { x // x ≠ 0 } := ⟨1, ⋯⟩
i : Fin 2
hi : i ≠ 0
⊢ ↑↑⟨i, hi⟩ = ↑↑i₁
|
2d25c971e4436589
|
uniformSpace_comap_id
|
Mathlib/Topology/UniformSpace/Basic.lean
|
theorem uniformSpace_comap_id {α : Type*} : UniformSpace.comap (id : α → α) = id
|
α : Type u_2
⊢ UniformSpace.comap id = id
|
ext : 2
|
case h.h
α : Type u_2
x✝ : UniformSpace α
⊢ 𝓤 α = 𝓤 α
|
352853a94e88d606
|
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.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✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))
this✝ : FiniteDimensional K L
this : IsGalois K L
H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac
⊢ k - s + 1 ≠ 0
|
exact Nat.succ_ne_zero _
|
no goals
|
460ee0dff43c909a
|
IsHomeomorphicTrivialFiberBundle.isOpenMap_proj
|
Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean
|
theorem isOpenMap_proj (h : IsHomeomorphicTrivialFiberBundle F proj) :
IsOpenMap proj
|
B : Type u_1
F : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalSpace Z
proj : Z → B
h : IsHomeomorphicTrivialFiberBundle F proj
⊢ IsOpenMap proj
|
obtain ⟨e, rfl⟩ := h.proj_eq
|
case intro
B : Type u_1
F : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalSpace Z
e : Z ≃ₜ B × F
h : IsHomeomorphicTrivialFiberBundle F (Prod.fst ∘ ⇑e)
⊢ IsOpenMap (Prod.fst ∘ ⇑e)
|
786cf80c321a1d1e
|
Matrix.invOf_eq
|
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
theorem invOf_eq [Invertible A.det] [Invertible A] : ⅟ A = ⅟ A.det • A.adjugate
|
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
A : Matrix n n α
inst✝¹ : Invertible A.det
inst✝ : Invertible A
this : Invertible A := A.invertibleOfDetInvertible
⊢ ⅟A = ⅟A.det • A.adjugate
|
convert (rfl : ⅟ A = _)
|
no goals
|
5729748099d6e6b6
|
cauchy_product
|
Mathlib/Algebra/Order/CauSeq/BigOperators.lean
|
theorem _root_.cauchy_product (ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n))
(hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ, ∀ j ≥ i,
abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k -
∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
|
case bc
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (N ⊔ M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k
h₃ :
∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : N ⊔ M + 1 < K
hKN : N < K
hsumlesum :
∑ i ∈ range (N ⊔ M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
∑ i ∈ range (N ⊔ M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (N ⊔ M + 1), abv (f n) < P
this : 0 < Q
⊢ ∑ k ∈ range K, abv (f k) - ∑ k ∈ range (N ⊔ M + 1), abv (f k) < ε / (4 * Q)
|
exact (le_abs_self _).trans_lt <|
hM _ ((Nat.le_succ_of_le (le_max_right _ _)).trans hNMK.le) _ <|
Nat.le_succ_of_le <| le_max_right _ _
|
no goals
|
b07014ed8fddf94c
|
CategoryTheory.Functor.preservesHomology_of_preservesEpis_and_kernels
|
Mathlib/CategoryTheory/Abelian/Exact.lean
|
/-- A functor preserving zero morphisms, epis, and kernels preserves homology. -/
lemma preservesHomology_of_preservesEpis_and_kernels [PreservesZeroMorphisms L]
[PreservesEpimorphisms L] [∀ {X Y} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) L] :
PreservesHomology L
|
A : Type u₁
B : Type u₂
inst✝⁶ : Category.{v₁, u₁} A
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Abelian A
inst✝³ : Abelian B
L : A ⥤ B
inst✝² : L.PreservesZeroMorphisms
inst✝¹ : L.PreservesEpimorphisms
inst✝ : ∀ {X Y : A} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) L
S : ShortComplex A
hS : S.Exact
⊢ L.map (Abelian.factorThruImage S.f) ≫ ((ShortComplex.mk (Abelian.image.ι S.f) S.g ⋯).map L).f =
(S.map L).f ≫ 𝟙 (S.map L).X₂
|
dsimp
|
A : Type u₁
B : Type u₂
inst✝⁶ : Category.{v₁, u₁} A
inst✝⁵ : Category.{v₂, u₂} B
inst✝⁴ : Abelian A
inst✝³ : Abelian B
L : A ⥤ B
inst✝² : L.PreservesZeroMorphisms
inst✝¹ : L.PreservesEpimorphisms
inst✝ : ∀ {X Y : A} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) L
S : ShortComplex A
hS : S.Exact
⊢ L.map (Abelian.factorThruImage S.f) ≫ L.map (kernel.ι (cokernel.π S.f)) = L.map S.f ≫ 𝟙 (L.obj S.X₂)
|
1450f99740d491ce
|
tendsto_integral_comp_smul_smul_of_integrable'
|
Mathlib/MeasureTheory/Integral/PeakFunction.lean
|
theorem tendsto_integral_comp_smul_smul_of_integrable'
{φ : F → ℝ} (hφ : ∀ x, 0 ≤ φ x) (h'φ : ∫ x, φ x ∂μ = 1)
(h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0))
{g : F → E} {x₀ : F} (hg : Integrable g μ) (h'g : ContinuousAt g x₀) :
Tendsto (fun (c : ℝ) ↦ ∫ x, (c ^ (finrank ℝ F) * φ (c • (x₀ - x))) • g x ∂μ)
atTop (𝓝 (g x₀))
|
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ : ∫ (x : F), φ x ∂μ = 1
h : Tendsto (fun x => ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)
g : F → E
x₀ : F
hg : Integrable g μ
h'g : ContinuousAt g x₀
f : F → E := fun x => g (x₀ - x)
If : Integrable f μ
this : Tendsto (fun c => ∫ (x : F), (c ^ finrank ℝ F * φ (c • x)) • f x ∂μ) atTop (𝓝 (f 0))
⊢ Tendsto (fun c => ∫ (x : F), (c ^ finrank ℝ F * φ (c • (x₀ - x))) • g x ∂μ) atTop (𝓝 (g x₀))
|
simp only [f, sub_zero] at this
|
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ : ∫ (x : F), φ x ∂μ = 1
h : Tendsto (fun x => ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)
g : F → E
x₀ : F
hg : Integrable g μ
h'g : ContinuousAt g x₀
f : F → E := fun x => g (x₀ - x)
If : Integrable f μ
this : Tendsto (fun c => ∫ (x : F), (c ^ finrank ℝ F * φ (c • x)) • g (x₀ - x) ∂μ) atTop (𝓝 (g x₀))
⊢ Tendsto (fun c => ∫ (x : F), (c ^ finrank ℝ F * φ (c • (x₀ - x))) • g x ∂μ) atTop (𝓝 (g x₀))
|
4121a55591318aa5
|
Matrix.mul_eq_one_comm
|
Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean
|
theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1
|
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
⊢ ∀ (A B : Matrix n n R), A * B = 1 → B * A = 1
|
intro A B hAB
|
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A✝ B✝ A B : Matrix n n R
hAB : A * B = 1
⊢ B * A = 1
|
7b3f481849b56cef
|
Function.updateFinset_updateFinset
|
Mathlib/Data/Finset/Update.lean
|
theorem updateFinset_updateFinset {s t : Finset ι} (hst : Disjoint s t)
{y : ∀ i : ↥s, π i} {z : ∀ i : ↥t, π i} :
updateFinset (updateFinset x s y) t z =
updateFinset x (s ∪ t) (Equiv.piFinsetUnion π hst ⟨y, z⟩)
|
case pos
ι : Type u_1
π : ι → Type u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
s t : Finset ι
hst : Disjoint s t
y : (i : { x // x ∈ s }) → π ↑i
z : (i : { x // x ∈ t }) → π ↑i
e : { x // x ∈ s } ⊕ { x // x ∈ t } ≃ { x // x ∈ s ∪ t } := Finset.union s t hst
i : ι
his : i ∈ s
hit : i ∈ t
⊢ z ⟨i, ⋯⟩ = if h : True ∨ True then (piFinsetUnion π hst) (y, z) ⟨i, ⋯⟩ else x i
|
exfalso
|
case pos
ι : Type u_1
π : ι → Type u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
s t : Finset ι
hst : Disjoint s t
y : (i : { x // x ∈ s }) → π ↑i
z : (i : { x // x ∈ t }) → π ↑i
e : { x // x ∈ s } ⊕ { x // x ∈ t } ≃ { x // x ∈ s ∪ t } := Finset.union s t hst
i : ι
his : i ∈ s
hit : i ∈ t
⊢ False
|
4ce6c3f1679cd67b
|
Polynomial.rootMultiplicity_eq_natTrailingDegree'
|
Mathlib/Algebra/Polynomial/Div.lean
|
/-- See `Polynomial.rootMultiplicity_eq_natTrailingDegree` for the general case. -/
lemma rootMultiplicity_eq_natTrailingDegree' : p.rootMultiplicity 0 = p.natTrailingDegree
|
case pos
R : Type u
inst✝ : CommRing R
p : R[X]
h : p = 0
⊢ rootMultiplicity 0 p = p.natTrailingDegree
|
simp only [h, rootMultiplicity_zero, natTrailingDegree_zero]
|
no goals
|
fb3fe56f8f18fafe
|
IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible
|
Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean
|
theorem _root_.IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible {K : Type*} [Field K]
{R : Type*} [CommRing R] [IsDomain R] {μ : R} {n : ℕ} [Algebra K R] (hμ : IsPrimitiveRoot μ n)
(h : Irreducible <| cyclotomic n K) [NeZero (n : K)] : cyclotomic n K = minpoly K μ
|
K : Type u_2
inst✝⁴ : Field K
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDomain R
μ : R
n : ℕ
inst✝¹ : Algebra K R
hμ : IsPrimitiveRoot μ n
h : Irreducible (cyclotomic n K)
inst✝ : NeZero ↑n
⊢ cyclotomic n K = minpoly K μ
|
haveI := NeZero.of_faithfulSMul K R n
|
K : Type u_2
inst✝⁴ : Field K
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDomain R
μ : R
n : ℕ
inst✝¹ : Algebra K R
hμ : IsPrimitiveRoot μ n
h : Irreducible (cyclotomic n K)
inst✝ : NeZero ↑n
this : NeZero ↑n
⊢ cyclotomic n K = minpoly K μ
|
9b8030f5fdd645f6
|
Nat.testBit_two_pow
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean
|
theorem testBit_two_pow {n m : Nat} : testBit (2 ^ n) m = decide (n = m)
|
n m : Nat
h : ¬n = m
h✝ : m < n
⊢ m ≤ n
|
omega
|
no goals
|
bbc66499b4dfb877
|
Finsupp.zipWith_single_single
|
Mathlib/Data/Finsupp/Single.lean
|
theorem zipWith_single_single (f : M → N → P) (hf : f 0 0 = 0) (a : α) (m : M) (n : N) :
zipWith f hf (single a m) (single a n) = single a (f m n)
|
α : Type u_1
M : Type u_5
N : Type u_7
P : Type u_8
inst✝² : Zero M
inst✝¹ : Zero N
inst✝ : Zero P
f : M → N → P
hf : f 0 0 = 0
a : α
m : M
n : N
⊢ zipWith f hf (single a m) (single a n) = single a (f m n)
|
ext a'
|
case h
α : Type u_1
M : Type u_5
N : Type u_7
P : Type u_8
inst✝² : Zero M
inst✝¹ : Zero N
inst✝ : Zero P
f : M → N → P
hf : f 0 0 = 0
a : α
m : M
n : N
a' : α
⊢ (zipWith f hf (single a m) (single a n)) a' = (single a (f m n)) a'
|
f8f9ada305dc4b27
|
CategoryTheory.Limits.limit_π_isIso_of_is_strict_terminal
|
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
|
theorem limit_π_isIso_of_is_strict_terminal (F : J ⥤ C) [HasLimit F] (i : J)
(H : ∀ (j) (_ : j ≠ i), IsTerminal (F.obj j)) [Subsingleton (i ⟶ i)] : IsIso (limit.π F i)
|
case pos.refl
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasStrictTerminalObjects C
J : Type v
inst✝² : SmallCategory J
F : J ⥤ C
inst✝¹ : HasLimit F
i : J
H : (j : J) → j ≠ i → IsTerminal (F.obj j)
inst✝ : Subsingleton (i ⟶ i)
k : J
h : ¬k = i
f : i ⟶ k
⊢ ((Functor.const J).obj (F.toPrefunctor.1 i)).map f ≫
(H k h).from (((Functor.const J).obj (F.toPrefunctor.1 i)).obj k) =
eqToHom ⋯ ≫ F.map f
|
apply (H _ h).hom_ext
|
no goals
|
464f78ec21e1a3ee
|
Module.reflection_mul_reflection_zpow_apply
|
Mathlib/LinearAlgebra/Reflection.lean
|
/-- A formula for $(r_1 r_2)^m z$, where $m$ is an integer and $z \in M$. -/
lemma reflection_mul_reflection_zpow_apply (m : ℤ) (z : M)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2
|
case neg
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
a✝ :
∀ (n : ℕ),
((reflection hf * reflection hg) ^ ↑n) z =
z +
(Polynomial.eval t (S R ((↑n - 2) / 2)) *
(Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
((g x * f z - g z) • y - f z • x) +
(Polynomial.eval t (S R ((↑n - 1) / 2)) *
(Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
((f y * g z - f z) • x - g z • y)
m : ℕ
ht' : t = g x * f y - 2
aux : ∀ (a b : ℤ), autoParam (a + b = -3) _auto✝ → a / 2 = -(b / 2) - 2
⊢ z +
(Polynomial.eval t (S R ((↑m - 1) / 2)) *
(Polynomial.eval t (S R (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
((g x * f z - g z) • y - f z • x) +
(Polynomial.eval t (S R ((↑m - 2) / 2)) *
(Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 2)))) •
((f y * g z - f z) • x - g z • y) =
z +
(Polynomial.eval t (S R ((-↑m - 2) / 2)) *
(Polynomial.eval t (S R ((-↑m - 1) / 2)) + Polynomial.eval t (S R ((-↑m - 3) / 2)))) •
((g x * f z - g z) • y - f z • x) +
(Polynomial.eval t (S R ((-↑m - 1) / 2)) *
(Polynomial.eval t (S R (-↑m / 2)) + Polynomial.eval t (S R ((-↑m - 2) / 2)))) •
((f y * g z - f z) • x - g z • y)
|
rw [aux (-m - 3) m, aux (-m - 2) (m - 1), aux (-m - 1) (m - 2), aux (-m) (m - 3)]
|
case neg
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
a✝ :
∀ (n : ℕ),
((reflection hf * reflection hg) ^ ↑n) z =
z +
(Polynomial.eval t (S R ((↑n - 2) / 2)) *
(Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
((g x * f z - g z) • y - f z • x) +
(Polynomial.eval t (S R ((↑n - 1) / 2)) *
(Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
((f y * g z - f z) • x - g z • y)
m : ℕ
ht' : t = g x * f y - 2
aux : ∀ (a b : ℤ), autoParam (a + b = -3) _auto✝ → a / 2 = -(b / 2) - 2
⊢ z +
(Polynomial.eval t (S R ((↑m - 1) / 2)) *
(Polynomial.eval t (S R (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
((g x * f z - g z) • y - f z • x) +
(Polynomial.eval t (S R ((↑m - 2) / 2)) *
(Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 2)))) •
((f y * g z - f z) • x - g z • y) =
z +
(Polynomial.eval t (S R (-((↑m - 1) / 2) - 2)) *
(Polynomial.eval t (S R (-((↑m - 2) / 2) - 2)) + Polynomial.eval t (S R (-(↑m / 2) - 2)))) •
((g x * f z - g z) • y - f z • x) +
(Polynomial.eval t (S R (-((↑m - 2) / 2) - 2)) *
(Polynomial.eval t (S R (-((↑m - 3) / 2) - 2)) + Polynomial.eval t (S R (-((↑m - 1) / 2) - 2)))) •
((f y * g z - f z) • x - g z • y)
|
ede6c0c006554b73
|
AlgebraicGeometry.Scheme.Pullback.range_fst
|
Mathlib/AlgebraicGeometry/PullbackCarrier.lean
|
lemma range_fst : Set.range (pullback.fst f g).base = f.base ⁻¹' Set.range g.base
|
case h.refine_1.intro
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
a : ↑↑(pullback f g).toPresheafedSpace
⊢ (ConcreteCategory.hom (pullback.fst f g).base) a ∈
⇑(ConcreteCategory.hom f.base) ⁻¹' Set.range ⇑(ConcreteCategory.hom g.base)
|
simp only [Set.mem_preimage, Set.mem_range, ← Scheme.comp_base_apply, pullback.condition]
|
case h.refine_1.intro
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
a : ↑↑(pullback f g).toPresheafedSpace
⊢ ∃ y, (ConcreteCategory.hom g.base) y = (ConcreteCategory.hom (pullback.snd f g ≫ g).base) a
|
11f78a61ac29370c
|
ProbabilityTheory.integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet
|
Mathlib/Probability/Moments/IntegrableExpMul.lean
|
lemma integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet
(hz : z.re ∈ interior (integrableExpSet X μ)) {p : ℝ} (hp : 0 ≤ p) :
Integrable (fun ω ↦ (X ω ^ p : ℝ) * cexp (z * X ω)) μ
|
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
p : ℝ
hp : 0 ≤ p
hX : AEMeasurable X μ
⊢ AEMeasurable (fun ω => ↑(X ω ^ p) * cexp (z * ↑(X ω))) μ
|
fun_prop
|
no goals
|
5757787a7a5feda6
|
SimpContFract.determinant_aux
|
Mathlib/Algebra/ContinuedFractions/Determinant.lean
|
theorem determinant_aux (hyp : n = 0 ∨ ¬(↑s : GenContFract K).TerminatedAt (n - 1)) :
((↑s : GenContFract K).contsAux n).a * ((↑s : GenContFract K).contsAux (n + 1)).b -
((↑s : GenContFract K).contsAux n).b * ((↑s : GenContFract K).contsAux (n + 1)).a =
(-1) ^ n
|
K : Type u_1
inst✝ : Field K
s : SimpContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬(↑s).TerminatedAt (n + 1 - 1)
g : GenContFract K := ↑s
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH : n = 0 ∨ ¬(↑s).TerminatedAt (n - 1) → ppred_conts.a * pred_conts.b - ppred_conts.b * pred_conts.a = (-1) ^ n
ppred_conts_eq : ppred_conts = g.contsAux n
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
not_terminated_at_n : ¬g.TerminatedAt n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
this : pA * ppB - pB * ppA = (-1) ^ (n + 1)
⊢ pA * (ppB + gp.b * pB) - pB * (ppA + gp.b * pA) = (-1) ^ (n + 1)
|
calc
pA * (ppB + gp.b * pB) - pB * (ppA + gp.b * pA) =
pA * ppB + pA * gp.b * pB - pB * ppA - pB * gp.b * pA := by ring
_ = pA * ppB - pB * ppA := by ring
_ = (-1) ^ (n + 1) := by assumption
|
no goals
|
5478ed7427a774c4
|
MeasureTheory.integrable_of_le_of_le
|
Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
|
lemma integrable_of_le_of_le {f g₁ g₂ : α → ℝ} (hf : AEStronglyMeasurable f μ)
(h_le₁ : g₁ ≤ᵐ[μ] f) (h_le₂ : f ≤ᵐ[μ] g₂)
(h_int₁ : Integrable g₁ μ) (h_int₂ : Integrable g₂ μ) :
Integrable f μ
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g₁ g₂ : α → ℝ
hf : AEStronglyMeasurable f μ
h_le₁ : g₁ ≤ᶠ[ae μ] f
h_le₂ : f ≤ᶠ[ae μ] g₂
h_int₁ : Integrable g₁ μ
h_int₂ : Integrable g₂ μ
⊢ Integrable f μ
|
have : ∀ᵐ x ∂μ, ‖f x‖ ≤ max ‖g₁ x‖ ‖g₂ x‖ := by
filter_upwards [h_le₁, h_le₂] with x hx1 hx2
simp only [Real.norm_eq_abs]
exact abs_le_max_abs_abs hx1 hx2
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g₁ g₂ : α → ℝ
hf : AEStronglyMeasurable f μ
h_le₁ : g₁ ≤ᶠ[ae μ] f
h_le₂ : f ≤ᶠ[ae μ] g₂
h_int₁ : Integrable g₁ μ
h_int₂ : Integrable g₂ μ
this : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ ‖g₁ x‖ ⊔ ‖g₂ x‖
⊢ Integrable f μ
|
b13e99f457d8487f
|
closure_diff
|
Mathlib/Topology/Basic.lean
|
theorem closure_diff : closure s \ closure t ⊆ closure (s \ t) :=
calc
closure s \ closure t = (closure t)ᶜ ∩ closure s
|
X : Type u
s t : Set X
inst✝ : TopologicalSpace X
⊢ closure ((closure t)ᶜ ∩ s) = closure (s \ closure t)
|
simp only [diff_eq, inter_comm]
|
no goals
|
40145cf1cb2fcf0d
|
CategoryTheory.kernelCokernelCompSequence.δ_fac
|
Mathlib/CategoryTheory/Abelian/DiagramLemmas/KernelCokernelComp.lean
|
lemma δ_fac : δ f g = - kernel.ι g ≫ cokernel.π f
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ (-kernel.ι g) ≫ (snakeInput f g).L₂.f = (kernel.ι g ≫ biprod.inr) ≫ (snakeInput f g).v₁₂.τ₂
|
aesop
|
no goals
|
74b74b8cd571ff07
|
Orthonormal.equiv_symm
|
Mathlib/Analysis/InnerProductSpace/Orthonormal.lean
|
theorem Orthonormal.equiv_symm {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) {v' : Basis ι' 𝕜 E'}
(hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') : (hv.equiv hv' e).symm = hv'.equiv hv e.symm :=
v'.ext_linearIsometryEquiv fun i =>
(hv.equiv hv' e).injective <| by
simp only [LinearIsometryEquiv.apply_symm_apply, Orthonormal.equiv_apply, e.apply_symm_apply]
|
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
ι' : Type u_5
E' : Type u_6
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
v' : Basis ι' 𝕜 E'
hv' : Orthonormal 𝕜 ⇑v'
e : ι ≃ ι'
i : ι'
⊢ (hv.equiv hv' e) ((hv.equiv hv' e).symm (v' i)) = (hv.equiv hv' e) ((hv'.equiv hv e.symm) (v' i))
|
simp only [LinearIsometryEquiv.apply_symm_apply, Orthonormal.equiv_apply, e.apply_symm_apply]
|
no goals
|
c4ffd895888577e5
|
Order.krullDim_eq_top_iff
|
Mathlib/Order/KrullDimension.lean
|
lemma krullDim_eq_top_iff : krullDim α = ⊤ ↔ InfiniteDimensionalOrder α
|
case inr.inr
α : Type u_1
inst✝ : Preorder α
h : krullDim α = ⊤
h✝¹ : Nonempty α
h✝ : InfiniteDimensionalOrder α
⊢ InfiniteDimensionalOrder α
|
infer_instance
|
no goals
|
02707beaa1c9c181
|
MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
(hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) :
Tendsto f atTop (𝓝 (limUnder atTop f))
|
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
ε : ℝ
εpos : ε > 0
L : Tendsto (fun n => ∫ (x : ℝ) in Ici ↑n, ‖f' x‖) atTop (𝓝 (∫ (x : ℝ) in ⋂ n, Ici ↑n, ‖f' x‖))
B : ⋂ n, Ici ↑n = ∅
⊢ ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε
|
simp only [B, Measure.restrict_empty, integral_zero_measure] at L
|
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
ε : ℝ
εpos : ε > 0
B : ⋂ n, Ici ↑n = ∅
L : Tendsto (fun n => ∫ (x : ℝ) in Ici ↑n, ‖f' x‖) atTop (𝓝 0)
⊢ ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε
|
baf6f2379f5dcbf2
|
tsirelson_inequality
|
Mathlib/Algebra/Star/CHSH.lean
|
theorem tsirelson_inequality [OrderedRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
[OrderedSMul ℝ R] [StarModule ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ √2 ^ 3 • (1 : R)
|
R : Type u
inst✝⁵ : OrderedRing R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : Algebra ℝ R
inst✝¹ : OrderedSMul ℝ R
inst✝ : StarModule ℝ R
A₀ A₁ B₀ B₁ : R
T : IsCHSHTuple A₀ A₁ B₀ B₁
M : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x
P : R := (√2)⁻¹ • (A₁ + A₀) - B₀
Q : R := (√2)⁻¹ • (A₁ - A₀) + B₁
w : √2 ^ 3 • 1 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁ = (√2)⁻¹ • (P ^ 2 + Q ^ 2)
⊢ 0 ≤ (√2)⁻¹ • (P ^ 2 + Q ^ 2)
|
have P_sa : star P = P := by
simp only [P, star_smul, star_add, star_sub, star_id_of_comm, T.A₀_sa, T.A₁_sa, T.B₀_sa,
T.B₁_sa]
|
R : Type u
inst✝⁵ : OrderedRing R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : Algebra ℝ R
inst✝¹ : OrderedSMul ℝ R
inst✝ : StarModule ℝ R
A₀ A₁ B₀ B₁ : R
T : IsCHSHTuple A₀ A₁ B₀ B₁
M : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x
P : R := (√2)⁻¹ • (A₁ + A₀) - B₀
Q : R := (√2)⁻¹ • (A₁ - A₀) + B₁
w : √2 ^ 3 • 1 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁ = (√2)⁻¹ • (P ^ 2 + Q ^ 2)
P_sa : star P = P
⊢ 0 ≤ (√2)⁻¹ • (P ^ 2 + Q ^ 2)
|
74b38187913f9314
|
AddMonoidAlgebra.mem_closure_of_mem_span_closure
|
Mathlib/RingTheory/FiniteType.lean
|
theorem mem_closure_of_mem_span_closure [Nontrivial R] {m : M} {S : Set M}
(h : of' R M m ∈ span R (Submonoid.closure (of' R M '' S) : Set R[M])) :
m ∈ closure S
|
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddMonoid M
inst✝ : Nontrivial R
m : M
S : Set M
S' : Submonoid (Multiplicative M) := Submonoid.closure S
h : of' R M m ∈ span R ↑(Submonoid.map (of R M) S')
h' : Submonoid.map (of R M) S' = Submonoid.closure ((fun x => (of R M) x) '' S)
⊢ Multiplicative.ofAdd m ∈ Submonoid.closure (⇑Multiplicative.toAdd ⁻¹' S)
|
simpa using of'_mem_span.1 h
|
no goals
|
70104ef5a58bb647
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.unsat_of_encounteredBoth
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem unsat_of_encounteredBoth {n : Nat} (c : DefaultClause n)
(assignment : Array Assignment) :
reduce c assignment = encounteredBoth → Unsatisfiable (PosFin n) assignment
|
case h_2.h_2
n : Nat
c : DefaultClause n
assignment : Array Assignment
hb : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
l : Literal (PosFin n)
x✝¹ : l ∈ c.clause
acc✝ : ReduceResult (PosFin n)
ih : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
x✝ : Assignment
heq✝ : assignment[l.fst.val]! = neg
h : (if (!l.snd) = true then reducedToUnit l else reducedToEmpty) = encounteredBoth
⊢ Unsatisfiable (PosFin n) assignment
|
split at h <;> simp at h
|
no goals
|
a1d564233a90b7cf
|
HomologicalComplex.acyclic_truncGE_iff_isSupportedOutside
|
Mathlib/Algebra/Homology/Embedding/TruncGEHomology.lean
|
lemma acyclic_truncGE_iff_isSupportedOutside :
(K.truncGE e).Acyclic ↔ K.IsSupportedOutside e
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ (K.truncGE e).Acyclic ↔ K.IsSupportedOutside e
|
constructor
|
case mp
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ (K.truncGE e).Acyclic → K.IsSupportedOutside e
case mpr
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ K.IsSupportedOutside e → (K.truncGE e).Acyclic
|
c22f4f01024eb4a0
|
HurwitzKernelBounds.isBigO_atTop_F_nat_one
|
Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
|
lemma isBigO_atTop_F_nat_one {a : ℝ} (ha : 0 ≤ a) : ∃ p, 0 < p ∧
F_nat 1 a =O[atTop] fun t ↦ exp (-p * t)
|
case inl
aux' : (fun t => ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x => 1
ha : 0 ≤ 0
⊢ ∃ p,
0 < p ∧
(fun t =>
rexp (-π * (0 ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ 2 +
0 * rexp (-π * 0 ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop]
fun t => rexp (-p * t)
|
exact ⟨_, pi_pos, by simpa only [zero_pow two_ne_zero, zero_add, mul_one, zero_mul, zero_div,
add_zero] using (isBigO_refl _ _).mul aux'⟩
|
no goals
|
5a5132f25c8455c4
|
IsSMulRegular.subsingleton
|
Mathlib/Algebra/Regular/SMul.lean
|
theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M :=
⟨fun a b => h (by dsimp only [Function.comp_def]; repeat' rw [MulActionWithZero.zero_smul])⟩
|
R : Type u_1
M : Type u_3
inst✝² : MonoidWithZero R
inst✝¹ : Zero M
inst✝ : MulActionWithZero R M
h : IsSMulRegular M 0
a b : M
⊢ 0 • a = 0 • b
|
repeat' rw [MulActionWithZero.zero_smul]
|
no goals
|
b20bfe08a69faefd
|
Matroid.map_dual
|
Mathlib/Data/Matroid/Map.lean
|
@[simp] lemma map_dual {hf} : (M.map f hf)✶ = M✶.map f hf
|
α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
⊢ ∀ a ⊆ M.E, ((M.map f hf).IsBase (f '' M.E \ f '' a) ∧ ∃ u ⊆ M.E, f '' u = f '' a) ↔ (M✶.map f hf).IsBase (f '' a)
|
intro B hB
|
α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
B : Set α
hB : B ⊆ M.E
⊢ ((M.map f hf).IsBase (f '' M.E \ f '' B) ∧ ∃ u ⊆ M.E, f '' u = f '' B) ↔ (M✶.map f hf).IsBase (f '' B)
|
29df05518640e03f
|
DirSupInacc.union
|
Mathlib/Topology/Order/ScottTopology.lean
|
lemma DirSupInacc.union (hs : DirSupInacc s) (ht : DirSupInacc t) : DirSupInacc (s ∪ t)
|
α : Type u_1
inst✝ : Preorder α
s t : Set α
hs : DirSupInacc s
ht : DirSupInacc t
⊢ DirSupInacc (s ∪ t)
|
rw [← dirSupClosed_compl, compl_union]
|
α : Type u_1
inst✝ : Preorder α
s t : Set α
hs : DirSupInacc s
ht : DirSupInacc t
⊢ DirSupClosed (sᶜ ∩ tᶜ)
|
7956754734308ef2
|
Fin.snoc_init_self
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem snoc_init_self : snoc (init q) (q (last n)) = q
|
case neg
n : ℕ
α : Fin (n + 1) → Sort u_1
q : (i : Fin (n + 1)) → α i
j : Fin (n + 1)
h : ¬↑j < n
⊢ snoc (init q) (q (last n)) (last n) = q (last n)
|
simp
|
no goals
|
e85188ef3c9ae2e3
|
Std.Tactic.BVDecide.BVExpr.bitblast.go_denote_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Expr.lean
|
theorem go_denote_eq (aig : AIG BVBit) (expr : BVExpr w) (assign : Assignment) :
∀ (idx : Nat) (hidx : idx < w),
⟦(go aig expr).val.aig, (go aig expr).val.vec.get idx hidx, assign.toAIGAssignment⟧
=
(expr.eval assign).getLsbD idx
|
case shiftLeft.hright
w : Nat
assign : Assignment
m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (idx : Nat) (hidx : idx < m✝),
⟦assign.toAIGAssignment, { aig := (go aig lhs).val.aig, ref := (go aig lhs).val.vec.get idx hidx }⟧ =
(eval assign lhs).getLsbD idx
rih :
∀ (aig : AIG BVBit) (idx : Nat) (hidx : idx < n✝),
⟦assign.toAIGAssignment, { aig := (go aig rhs).val.aig, ref := (go aig rhs).val.vec.get idx hidx }⟧ =
(eval assign rhs).getLsbD idx
aig : AIG BVBit
idx : Nat
hidx : idx < m✝
⊢ ∀ (idx : Nat)
(hidx : idx < { n := n✝, target := (go aig lhs).1.vec.cast ⋯, distance := (go (go aig lhs).1.aig rhs).1.vec }.n),
⟦assign.toAIGAssignment,
{ aig := (go (go aig lhs).1.aig rhs).1.aig,
ref :=
{ n := n✝, target := (go aig lhs).1.vec.cast ⋯, distance := (go (go aig lhs).1.aig rhs).1.vec }.distance.get
idx hidx }⟧ =
(eval assign rhs).getLsbD idx
|
intros
|
case shiftLeft.hright
w : Nat
assign : Assignment
m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (idx : Nat) (hidx : idx < m✝),
⟦assign.toAIGAssignment, { aig := (go aig lhs).val.aig, ref := (go aig lhs).val.vec.get idx hidx }⟧ =
(eval assign lhs).getLsbD idx
rih :
∀ (aig : AIG BVBit) (idx : Nat) (hidx : idx < n✝),
⟦assign.toAIGAssignment, { aig := (go aig rhs).val.aig, ref := (go aig rhs).val.vec.get idx hidx }⟧ =
(eval assign rhs).getLsbD idx
aig : AIG BVBit
idx : Nat
hidx : idx < m✝
idx✝ : Nat
hidx✝ : idx✝ < { n := n✝, target := (go aig lhs).1.vec.cast ⋯, distance := (go (go aig lhs).1.aig rhs).1.vec }.n
⊢ ⟦assign.toAIGAssignment,
{ aig := (go (go aig lhs).1.aig rhs).1.aig,
ref :=
{ n := n✝, target := (go aig lhs).1.vec.cast ⋯, distance := (go (go aig lhs).1.aig rhs).1.vec }.distance.get
idx✝ hidx✝ }⟧ =
(eval assign rhs).getLsbD idx✝
|
30595031a1a8ac13
|
Int.subNatNat_add
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean
|
theorem subNatNat_add (m n k : Nat) : subNatNat (m + n) k = m + subNatNat n k
|
case inl
m n k : Nat
h' : n < k
⊢ subNatNat (m + n) k = ↑m + subNatNat n k
|
simp [subNatNat_of_lt h', sub_one_add_one_eq_of_pos (Nat.sub_pos_of_lt h')]
|
case inl
m n k : Nat
h' : n < k
⊢ subNatNat (m + n) k = subNatNat m (k - n)
|
9fc3dd7335f8add0
|
MvPolynomial.mul_X_mem_coeffsIn
|
Mathlib/Algebra/MvPolynomial/Basic.lean
|
@[simp]
lemma mul_X_mem_coeffsIn : p * X s ∈ coeffsIn σ M ↔ p ∈ coeffsIn σ M
|
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Module R S
M : Submodule R S
p : MvPolynomial σ S
s : σ
⊢ p * X s ∈ coeffsIn σ M ↔ p ∈ coeffsIn σ M
|
simpa [-mul_monomial_mem_coeffsIn] using mul_monomial_mem_coeffsIn (i := .single s 1)
|
no goals
|
6383bcfd7c3ed98f
|
Set.preimage_subset
|
Mathlib/Data/Set/Image.lean
|
lemma preimage_subset {s t} (hs : s ⊆ f '' t) (hf : Set.InjOn f (f ⁻¹' s)) : f ⁻¹' s ⊆ t
|
α : Type u_1
β : Type u_2
f : α → β
s : Set β
t : Set α
hs : s ⊆ f '' t
hf : InjOn f (f ⁻¹' s)
⊢ f ⁻¹' s ⊆ t
|
rintro a ha
|
α : Type u_1
β : Type u_2
f : α → β
s : Set β
t : Set α
hs : s ⊆ f '' t
hf : InjOn f (f ⁻¹' s)
a : α
ha : a ∈ f ⁻¹' s
⊢ a ∈ t
|
ee68442f946ac210
|
IsOpen.exists_smooth_support_eq
|
Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
|
theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ∞ f ∧ Set.range f ⊆ Set.Icc 0 1
|
case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : IsOpen s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n => ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ∞ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
n : ℕ
R : ℕ → ℝ
hR : ∀ (i : ℕ) (x : E), ‖iteratedFDeriv ℝ i (fun x => g n x) x‖ ≤ R i
M : ℝ := (Finset.image R (Finset.range (n + 1))).max' ⋯ ⊔ 1
δnpos : 0 < δ n
IR : ∀ i ≤ n, R i ≤ M
i : ℕ
hi : i ≤ n
x : E
⊢ ‖iteratedFDeriv ℝ i (g n) x‖ ≤ M
|
exact (hR i x).trans (IR i hi)
|
no goals
|
415898bf3b6091fb
|
PosNum.lt_to_nat
|
Mathlib/Data/Num/Lemmas.lean
|
theorem lt_to_nat {m n : PosNum} : (m : ℕ) < n ↔ m < n :=
show (m : ℕ) < n ↔ cmp m n = Ordering.lt from
match cmp m n, cmp_to_nat m n with
| Ordering.lt, h => by simp only at h; simp [h]
| Ordering.eq, h => by simp only at h; simp [h, lt_irrefl]
| Ordering.gt, h => by simp [not_lt_of_gt h]
|
m n : PosNum
h : Ordering.casesOn Ordering.gt (↑m < ↑n) (m = n) (↑n < ↑m)
⊢ ↑m < ↑n ↔ Ordering.gt = Ordering.lt
|
simp [not_lt_of_gt h]
|
no goals
|
3877950e8242553f
|
List.lookupAll_eq_nil
|
Mathlib/Data/List/Sigma.lean
|
theorem lookupAll_eq_nil {a : α} :
∀ {l : List (Sigma β)}, lookupAll a l = [] ↔ ∀ b : β a, Sigma.mk a b ∉ l
| [] => by simp
| ⟨a', b⟩ :: l => by
by_cases h : a = a'
· subst a'
simp only [lookupAll_cons_eq, mem_cons, Sigma.mk.inj_iff, heq_eq_eq, true_and, not_or,
false_iff, not_forall, not_and, not_not, reduceCtorEq]
use b
simp
· simp [h, lookupAll_eq_nil]
|
case neg
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a a' : α
b : β a'
l : List (Sigma β)
h : ¬a = a'
⊢ lookupAll a (⟨a', b⟩ :: l) = [] ↔ ∀ (b_1 : β a), ⟨a, b_1⟩ ∉ ⟨a', b⟩ :: l
|
simp [h, lookupAll_eq_nil]
|
no goals
|
8edbefa7d6f1441c
|
iSupIndep_sUnion_of_directed
|
Mathlib/Order/CompactlyGenerated/Basic.lean
|
theorem iSupIndep_sUnion_of_directed {s : Set (Set α)} (hs : DirectedOn (· ⊆ ·) s)
(h : ∀ a ∈ s, sSupIndep a) : sSupIndep (⋃₀ s)
|
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
s : Set (Set α)
hs : DirectedOn (fun x1 x2 => x1 ⊆ x2) s
h : ∀ a ∈ s, sSupIndep a
⊢ sSupIndep (⋃₀ s)
|
rw [Set.sUnion_eq_iUnion]
|
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
s : Set (Set α)
hs : DirectedOn (fun x1 x2 => x1 ⊆ x2) s
h : ∀ a ∈ s, sSupIndep a
⊢ sSupIndep (⋃ i, ↑i)
|
c76a851a46820f44
|
Fin.mem_find_iff
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem mem_find_iff {p : Fin n → Prop} [DecidablePred p] {i : Fin n} :
i ∈ Fin.find p ↔ p i ∧ ∀ j, p j → i ≤ j :=
⟨fun hi ↦ ⟨find_spec _ hi, fun _ ↦ find_min' hi⟩, by
rintro ⟨hpi, hj⟩
cases hfp : Fin.find p
· rw [find_eq_none_iff] at hfp
exact (hfp _ hpi).elim
· exact Option.some_inj.2 (Fin.le_antisymm (find_min' hfp hpi) (hj _ (find_spec _ hfp)))⟩
|
n : ℕ
p : Fin n → Prop
inst✝ : DecidablePred p
i : Fin n
⊢ (p i ∧ ∀ (j : Fin n), p j → i ≤ j) → i ∈ find p
|
rintro ⟨hpi, hj⟩
|
case intro
n : ℕ
p : Fin n → Prop
inst✝ : DecidablePred p
i : Fin n
hpi : p i
hj : ∀ (j : Fin n), p j → i ≤ j
⊢ i ∈ find p
|
bd23a709faee2e49
|
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)
⊢ ↑(Fin.cast ⋯ (↑(p.length + ↑i) + 1)) = ↑(Fin.natAdd (p.length + 1) i)
|
simp only [Fin.coe_cast, Fin.coe_natAdd]
|
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 + 1 + ↑i
|
81a20ecaf5e7e778
|
Mathlib.Tactic.Bicategory.evalWhiskerRight_cons_of_of
|
Mathlib/Tactic/CategoryTheory/Bicategory/Normalize.lean
|
theorem evalWhiskerRight_cons_of_of
{f g h i : a ⟶ b} {j : b ⟶ c}
{α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i} {ηs₁ : h ≫ j ⟶ i ≫ j}
{η₁ : g ≫ j ⟶ h ≫ j} {η₂ : g ≫ j ⟶ i ≫ j} {η₃ : f ≫ j ⟶ i ≫ j}
(e_ηs₁ : ηs ▷ j = ηs₁) (e_η₁ : η ▷ j = η₁)
(e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (whiskerRightIso α j).hom ≫ η₂ = η₃) :
(α.hom ≫ η ≫ ηs) ▷ j = η₃
|
B : Type u
inst✝ : Bicategory B
a b c : B
f g h i : a ⟶ b
j : b ⟶ c
α : f ≅ g
η : g ⟶ h
ηs : h ⟶ i
ηs₁ : h ≫ j ⟶ i ≫ j
η₁ : g ≫ j ⟶ h ≫ j
η₂ : g ≫ j ⟶ i ≫ j
η₃ : f ≫ j ⟶ i ≫ j
e_ηs₁ : ηs ▷ j = ηs₁
e_η₁ : η ▷ j = η₁
e_η₂ : η₁ ≫ ηs₁ = η₂
e_η₃ : (whiskerRightIso α j).hom ≫ η₂ = η₃
⊢ (α.hom ≫ η ≫ ηs) ▷ j = η₃
|
simp_all
|
no goals
|
7631aa8acffca5fd
|
MeasureTheory.continuousOn_convolution_right_with_param
|
Mathlib/Analysis/Convolution.lean
|
theorem continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ univ)) :
ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
H : ¬∀ p ∈ s, ∀ (x : G), g p x = 0
this : LocallyCompactSpace G
q₀ : P
x₀ : G
hq₀ : (q₀, x₀).1 ∈ s
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := -k +ᵥ t
k'_comp : IsCompact k'
g' : P × G → G → E' := fun p x => g p.1 (p.2 - x)
s' : Set (P × G) := s ×ˢ t
⊢ uncurry g' = uncurry g ∘ fun w => (w.1.1, w.1.2 - w.2)
|
ext y
|
case h
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
H : ¬∀ p ∈ s, ∀ (x : G), g p x = 0
this : LocallyCompactSpace G
q₀ : P
x₀ : G
hq₀ : (q₀, x₀).1 ∈ s
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := -k +ᵥ t
k'_comp : IsCompact k'
g' : P × G → G → E' := fun p x => g p.1 (p.2 - x)
s' : Set (P × G) := s ×ˢ t
y : (P × G) × G
⊢ uncurry g' y = (uncurry g ∘ fun w => (w.1.1, w.1.2 - w.2)) y
|
a579c7593509cfe8
|
MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ
|
Mathlib/MeasureTheory/Measure/Restrict.lean
|
theorem ext_iff_of_iUnion_eq_univ [Countable ι] {s : ι → Set α} (hs : ⋃ i, s i = univ) :
μ = ν ↔ ∀ i, μ.restrict (s i) = ν.restrict (s i)
|
α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable ι
s : ι → Set α
hs : ⋃ i, s i = univ
⊢ μ = ν ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
|
rw [← restrict_iUnion_congr, hs, restrict_univ, restrict_univ]
|
no goals
|
0ef26d718a599dda
|
RingHom.FormallyUnramified.holdsForLocalizationAway
|
Mathlib/RingTheory/RingHom/Unramified.lean
|
lemma holdsForLocalizationAway :
HoldsForLocalizationAway FormallyUnramified
|
R S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
⊢ (algebraMap R S).FormallyUnramified
|
rw [formallyUnramified_algebraMap]
|
R S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
⊢ Algebra.FormallyUnramified R S
|
ce99abdb1bed37fd
|
IsSepClosed.algebraMap_surjective
|
Mathlib/FieldTheory/IsSepClosed.lean
|
theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [Algebra.IsSeparable k K] :
Function.Surjective (algebraMap k K)
|
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : Algebra.IsSeparable k K
⊢ Function.Surjective ⇑(algebraMap k K)
|
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
|
k : Type u
inst✝⁴ : Field k
K : Type v
inst✝³ : Field K
inst✝² : IsSepClosed k
inst✝¹ : Algebra k K
inst✝ : Algebra.IsSeparable k K
x : K
⊢ (algebraMap k K) (-(minpoly k x).coeff 0) = x
|
c449367686c063ba
|
Module.exists_basis_of_basis_baseChange
|
Mathlib/RingTheory/LocalRing/Module.lean
|
/-- If `M` is of finite presentation over a local ring `(R, 𝔪, k)` such that
`𝔪 ⊗ M → M` is injective, then every family of elements that is a `k`-basis of
`k ⊗ M` is an `R`-basis of `M`. -/
lemma exists_basis_of_basis_baseChange [Module.FinitePresentation R M]
{ι : Type u} (v : ι → M) (hli : LinearIndependent k (TensorProduct.mk R k M 1 ∘ v))
(hsp : Submodule.span k (Set.range (TensorProduct.mk R k M 1 ∘ v)) = ⊤)
(H : Function.Injective ((𝔪).subtype.rTensor M)) :
∃ (b : Basis ι R M), ∀ i, b i = v i
|
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝¹ : Finite ι
this✝ : Fintype ι := Fintype.ofFinite ι
this : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
⊢ Surjective ⇑i
|
rw [← LinearMap.range_eq_top, Finsupp.range_linearCombination]
|
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this✝¹ : Finite ι
this✝ : Fintype ι := Fintype.ofFinite ι
this : IsNoetherian k (k ⊗[R] (ι →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (ι →₀ R))
i : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R v
⊢ Submodule.span R (Set.range v) = ⊤
|
59c6db943feaa269
|
WittVector.map_frobeniusPoly
|
Mathlib/RingTheory/WittVector/Frobenius.lean
|
theorem map_frobeniusPoly (n : ℕ) :
MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n
|
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
this :
↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))
aux : ∀ (k : ℕ), ↑p ^ k ≠ 0
⊢ ↑p ^ j * ↑p * (↑((p ^ (n - i)).choose (j + 1)) * (↑p ^ i * (↑p ^ n)⁻¹)) =
↑((p ^ (n - i)).choose (j + 1)) / ↑p ^ (n - i - v p (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p
|
simpa [aux, -one_div, -pow_eq_zero_iff', field_simps] using this.symm
|
no goals
|
1eb072545457e9c3
|
AlgebraicGeometry.stalkClosedPointIso_inv
|
Mathlib/AlgebraicGeometry/Stalk.lean
|
lemma stalkClosedPointIso_inv :
(stalkClosedPointIso R).inv = StructureSheaf.toStalk R _
|
case hf.a
R : CommRingCat
inst✝ : IsLocalRing ↑R
x : ↑R
⊢ (CommRingCat.Hom.hom (stalkClosedPointIso R).inv) x =
(CommRingCat.Hom.hom (StructureSheaf.toStalk (↑R) (closedPoint ↑R))) x
|
exact StructureSheaf.localizationToStalk_of _ _ _
|
no goals
|
32651b7dea07f1c7
|
LaurentSeries.algebraMap_C_mem_adicCompletionIntegers
|
Mathlib/RingTheory/LaurentSeries.lean
|
lemma algebraMap_C_mem_adicCompletionIntegers (x : K) :
((LaurentSeriesRingEquiv K).toRingHom.comp HahnSeries.C) x ∈
adicCompletionIntegers (RatFunc K) (idealX K)
|
K : Type u_2
inst✝ : Field K
x : K
⊢ HahnSeries.C x = (ofPowerSeries ℤ K) ((PowerSeries.C K) x)
|
simp [C_apply, ofPowerSeries_C]
|
no goals
|
8b9003a9548d41b1
|
HomologicalComplex.mapBifunctor₁₂.d_eq
|
Mathlib/Algebra/Homology/BifunctorAssociator.lean
|
lemma d_eq (j j' : ι₄) [HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄] :
(mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).d j j' =
D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' +
D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j'
|
case e_a
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{u_13, u_1} C₁
inst✝¹⁹ : Category.{u_14, u_2} C₂
inst✝¹⁸ : Category.{u_15, u_5} C₃
inst✝¹⁷ : Category.{u_16, u_6} C₄
inst✝¹⁶ : Category.{u_17, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
⊢ 0 = d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'
|
by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
|
case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{u_13, u_1} C₁
inst✝¹⁹ : Category.{u_14, u_2} C₂
inst✝¹⁸ : Category.{u_15, u_5} C₃
inst✝¹⁷ : Category.{u_16, u_6} C₄
inst✝¹⁶ : Category.{u_17, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₂.Rel i₂ (c₂.next i₂)
⊢ 0 = d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'
case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{u_13, u_1} C₁
inst✝¹⁹ : Category.{u_14, u_2} C₂
inst✝¹⁸ : Category.{u_15, u_5} C₃
inst✝¹⁷ : Category.{u_16, u_6} C₄
inst✝¹⁶ : Category.{u_17, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ 0 = d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'
|
bcc237c029dad5d6
|
EReal.add_le_of_forall_lt
|
Mathlib/Data/Real/EReal.lean
|
lemma add_le_of_forall_lt {a b c : EReal} (h : ∀ a' < a, ∀ b' < b, a' + b' ≤ c) : a + b ≤ c
|
a b c : EReal
h : ∀ a' < a, ∀ b' < b, a' + b' ≤ c
d : EReal
hd : d < a + b
⊢ d ≤ c
|
obtain ⟨a', ha', hd⟩ := exists_lt_add_left hd
|
case intro.intro
a b c : EReal
h : ∀ a' < a, ∀ b' < b, a' + b' ≤ c
d : EReal
hd✝ : d < a + b
a' : EReal
ha' : a' < a
hd : d < a' + b
⊢ d ≤ c
|
a698a52687e39bf3
|
Asymptotics.SuperpolynomialDecay.param_zpow_mul
|
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
theorem SuperpolynomialDecay.param_zpow_mul (hk : Tendsto k l atTop)
(hf : SuperpolynomialDecay l k f) (z : ℤ) :
SuperpolynomialDecay l k fun a => k a ^ z * f a
|
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : LinearOrderedField β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hf : ∀ (z : ℤ), Tendsto (fun a => k a ^ z * f a) l (𝓝 0)
z : ℤ
⊢ ∀ (z_1 : ℤ), Tendsto (fun a => k a ^ z_1 * (k a ^ z * f a)) l (𝓝 0)
|
refine fun z' => (hf <| z' + z).congr' ((hk.eventually_ne_atTop 0).mono fun x hx => ?_)
|
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : LinearOrderedField β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hf : ∀ (z : ℤ), Tendsto (fun a => k a ^ z * f a) l (𝓝 0)
z z' : ℤ
x : α
hx : k x ≠ 0
⊢ k x ^ (z' + z) * f x = (fun a => k a ^ z' * (k a ^ z * f a)) x
|
fe3d3cf9ecd4f068
|
MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
theorem denseRange_coeSimpleFuncNonnegToLpNonneg [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) :
DenseRange (coeSimpleFuncNonnegToLpNonneg p μ G) := fun g ↦ by
borelize G
rw [mem_closure_iff_seq_limit]
have hg_memLp : MemLp (g : α → G) p μ := Lp.memLp (g : Lp G p μ)
have zero_mem : (0 : G) ∈ (range (g : α → G) ∪ {0} : Set G) ∩ { y | 0 ≤ y }
|
α : Type u_1
inst✝¹ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝ : NormedLatticeAddCommGroup G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
g : { g // 0 ≤ g }
this✝² : MeasurableSpace G := borel G
this✝¹ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this✝ : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n => SimpleFunc.approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᶠ[ae μ] ⇑(x n)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ toLp (x n) ⋯
hx_tendsto : Tendsto (fun n => eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
this : Tendsto (fun b => dist ↑(toLp (x b) ⋯) ↑g) atTop (𝓝 0)
⊢ Tendsto (fun b => dist ↑(coeSimpleFuncNonnegToLpNonneg p μ G ⟨toLp (x b) ⋯, ⋯⟩) ↑g) atTop (𝓝 0)
|
exact this
|
no goals
|
1fc70ad144f2653f
|
Plausible.InjectiveFunction.applyId_mem_iff
|
Mathlib/Testing/Plausible/Functions.lean
|
theorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys)
(x : α) : List.applyId.{u} (xs.zip ys) x ∈ ys ↔ x ∈ xs
|
case some
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x val : α
h₃ : dlookup x (map Prod.toSigma (xs.zip ys)) = some val
h₂ : ys.Nodup
⊢ (some val).getD x ∈ ys ↔ x ∈ xs
|
replace h₁ : xs.length = ys.length := h₁.length_eq
|
case some
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
x val : α
h₃ : dlookup x (map Prod.toSigma (xs.zip ys)) = some val
h₂ : ys.Nodup
h₁ : xs.length = ys.length
⊢ (some val).getD x ∈ ys ↔ x ∈ xs
|
93266444922c9f67
|
Finsupp.supported_iUnion
|
Mathlib/LinearAlgebra/Finsupp/Supported.lean
|
theorem supported_iUnion {δ : Type*} (s : δ → Set α) :
supported M R (⋃ i, s i) = ⨆ i, supported M R (s i)
|
case intro.refine_1
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
δ : Type u_7
s : δ → Set α
this : DecidablePred fun x => x ∈ ⋃ i, s i
l : α →₀ M
⊢ 0 ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))
|
exact zero_mem _
|
no goals
|
831219d09d75faa3
|
MeasureTheory.setLaverage_eq'
|
Mathlib/MeasureTheory/Integral/Average.lean
|
theorem setLaverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
⊢ ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α), f x ∂(μ s)⁻¹ • μ.restrict s
|
simp only [laverage_eq', restrict_apply_univ]
|
no goals
|
5c8b945375da63d8
|
ZMod.dft_smul_const
|
Mathlib/Analysis/Fourier/ZMod.lean
|
lemma dft_smul_const {R : Type*} [Ring R] [Module ℂ R] [Module R E] [IsScalarTower ℂ R E]
(Φ : ZMod N → R) (e : E) :
𝓕 (fun j ↦ Φ j • e) = fun k ↦ 𝓕 Φ k • e
|
N : ℕ
inst✝⁶ : NeZero N
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
R : Type u_2
inst✝³ : Ring R
inst✝² : Module ℂ R
inst✝¹ : Module R E
inst✝ : IsScalarTower ℂ R E
Φ : ZMod N → R
e : E
⊢ (𝓕 fun j => Φ j • e) = fun k => 𝓕 Φ k • e
|
simp only [dft_def, sum_smul, smul_assoc]
|
no goals
|
5f42e264fba58c0c
|
List.mem_rtakeWhile_imp
|
Mathlib/Data/List/DropRight.lean
|
theorem mem_rtakeWhile_imp {x : α} (hx : x ∈ rtakeWhile p l) : p x
|
α : Type u_1
p : α → Bool
l : List α
x : α
hx : x ∈ rtakeWhile p l
⊢ p x = true
|
rw [rtakeWhile, mem_reverse] at hx
|
α : Type u_1
p : α → Bool
l : List α
x : α
hx : x ∈ takeWhile p l.reverse
⊢ p x = true
|
ea73bcbb2133334e
|
Vitali.exists_disjoint_covering_ae
|
Mathlib/MeasureTheory/Covering/Vitali.lean
|
theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ C * μ (B a))
(ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a))
(hf : ∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ t, r a ≤ ε ∧ c a = x) :
∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
|
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ⊤
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ⊤
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
k : Set α := ⋃ a ∈ w, B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ d ⊓ R (c a)
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : (B a ∩ B b).Nonempty
bdiam : r a ≤ 2 * r b
bv : b ∈ v
⊢ c a ∈ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)
|
let b' : v := ⟨b, bv⟩
|
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ⊤
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ⊤
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
k : Set α := ⋃ a ∈ w, B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ d ⊓ R (c a)
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : (B a ∩ B b).Nonempty
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := ⟨b, bv⟩
⊢ c a ∈ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)
|
4466d2fc3e80cb3f
|
Vector.toList_setIfInBounds
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean
|
theorem toList_setIfInBounds (a : Vector α n) (i x) :
(a.setIfInBounds i x).toList = a.toList.set i x
|
α : Type u_1
n : Nat
a : Vector α n
i : Nat
x : α
⊢ (a.setIfInBounds i x).toList = a.toList.set i x
|
simp [Vector.setIfInBounds]
|
no goals
|
3cc0ba914ba760b9
|
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