name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
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|---|---|---|---|---|---|---|
CategoryTheory.Functor.PreservesRightHomologyOf.mk'
|
Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean
|
/-- If a functor preserves a certain right homology data of a short complex `S`, then it
preserves the right homology of `S`. -/
lemma PreservesRightHomologyOf.mk' (h : S.RightHomologyData) [h.IsPreservedBy F] :
F.PreservesRightHomologyOf S where
isPreservedBy h' :=
{ f := ShortComplex.RightHomologyData.IsPreservedBy.hf h F
g'
|
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{u_4, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
S : ShortComplex C
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
⊢ PreservesLimit (parallelPair h'.g' 0) F
|
have := ShortComplex.RightHomologyData.IsPreservedBy.hg' h F
|
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{u_4, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
S : ShortComplex C
h : S.RightHomologyData
inst✝ : h.IsPreservedBy F
h' : S.RightHomologyData
this : PreservesLimit (parallelPair h.g' 0) F
⊢ PreservesLimit (parallelPair h'.g' 0) F
|
39fd2293ef3ce48f
|
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet
|
Mathlib/Analysis/Complex/AbelLimit.lean
|
theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
|
case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (B₁ ⊔ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (B₁ ⊔ B₂)
this : ‖l - ∑ i ∈ range (i + 1), f i‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M
|
exact this.le
|
no goals
|
e1f8d437480849f9
|
Filter.tendsto_atTop_finset_of_monotone
|
Mathlib/Order/Filter/AtTopBot/Finset.lean
|
theorem tendsto_atTop_finset_of_monotone [Preorder β] {f : β → Finset α} (h : Monotone f)
(h' : ∀ x : α, ∃ n, x ∈ f n) : Tendsto f atTop atTop
|
case intro
α : Type u_3
β : Type u_4
inst✝ : Preorder β
f : β → Finset α
h : Monotone f
h' : ∀ (x : α), ∃ n, x ∈ f n
a : α
b : β
hb : a ∈ f b
⊢ ∀ᶠ (a_1 : β) in atTop, f a_1 ∈ Ici {a}
|
exact (eventually_ge_atTop b).mono fun b' hb' => (Finset.singleton_subset_iff.2 hb).trans (h hb')
|
no goals
|
7efa857265d50b55
|
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three
|
Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean
|
private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E'
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : E.IsElliptic
inst✝¹ : E'.IsElliptic
p : ℕ
inst✝ : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝⁴ : NeZero 2
this✝³ : NeZero 4
this✝² : NeZero 6
this✝¹ : Invertible 2 := invertibleOfNonzero hchar2
this✝ : Invertible 3 := invertibleOfNonzero hchar3
this : ∀ (E : WeierstrassCurve F) [inst : E.IsElliptic], E.j = E'.j → E.IsShortNF → ∃ C, E.variableChange C = E'
h✝ : ¬E.IsShortNF
C : VariableChange F
heq : (E.variableChange C).j = E'.j
hE : (E.variableChange C).IsShortNF
C' : VariableChange F
hC : (E.variableChange C).variableChange C' = E'
⊢ E.variableChange (C'.comp C) = E'
|
rwa [variableChange_comp]
|
no goals
|
5599f8482dc6841e
|
Finsupp.support_single_add
|
Mathlib/Data/Finsupp/Single.lean
|
theorem support_single_add {a : α} {b : M} {f : α →₀ M} (ha : a ∉ f.support) (hb : b ≠ 0) :
support (single a b + f) = cons a f.support ha
|
α : Type u_1
M : Type u_5
inst✝ : AddZeroClass M
a : α
b : M
f : α →₀ M
ha : a ∉ f.support
hb : b ≠ 0
⊢ (single a b + f).support = cons a f.support ha
|
have H := support_single_ne_zero a hb
|
α : Type u_1
M : Type u_5
inst✝ : AddZeroClass M
a : α
b : M
f : α →₀ M
ha : a ∉ f.support
hb : b ≠ 0
H : (single a b).support = {a}
⊢ (single a b + f).support = cons a f.support ha
|
e3a6e48c80b3900b
|
IsLocalizedModule.mk'_eq_zero
|
Mathlib/Algebra/Module/LocalizedModule/Basic.lean
|
theorem mk'_eq_zero {m : M} (s : S) : mk' f m s = 0 ↔ f m = 0
|
R : Type u_1
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type u_2
M' : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M'
inst✝² : Module R M
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
m : M
s : ↥S
⊢ mk' f m s = 0 ↔ f m = 0
|
rw [mk'_eq_iff, smul_zero]
|
no goals
|
db62fb65c91114a2
|
PartENat.lt_def
|
Mathlib/Data/Nat/PartENat.lean
|
theorem lt_def (x y : PartENat) : x < y ↔ ∃ hx : x.Dom, ∀ hy : y.Dom, x.get hx < y.get hy
|
case h
x y : PartENat
h : ∀ (x_1 : x.Dom → y.Dom), ¬∀ (hy : x.Dom), y.get ⋯ ≤ x.get hy
hyx : y.Dom → x.Dom
H : ∀ (hy : y.Dom), x.get ⋯ ≤ y.get hy
hx : x.Dom
hy : y.Dom
⊢ x.get hx < y.get hy
|
specialize H hy
|
case h
x y : PartENat
h : ∀ (x_1 : x.Dom → y.Dom), ¬∀ (hy : x.Dom), y.get ⋯ ≤ x.get hy
hyx : y.Dom → x.Dom
hx : x.Dom
hy : y.Dom
H : x.get ⋯ ≤ y.get hy
⊢ x.get hx < y.get hy
|
d9afc030dbc26ee8
|
Finset.Nonempty.zero_smul
|
Mathlib/Algebra/GroupWithZero/Pointwise/Finset.lean
|
lemma Nonempty.zero_smul (ht : t.Nonempty) : (0 : Finset α) • t = 0 :=
t.zero_smul_subset.antisymm <| by simpa [mem_smul] using ht
|
α : Type u_1
β : Type u_2
inst✝³ : DecidableEq β
inst✝² : Zero α
inst✝¹ : Zero β
inst✝ : SMulWithZero α β
t : Finset β
ht : t.Nonempty
⊢ 0 ⊆ 0 • t
|
simpa [mem_smul] using ht
|
no goals
|
68cccc7d617ad1f6
|
Path.Homotopy.continuous_transReflReparamAux
|
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
|
theorem continuous_transReflReparamAux : Continuous transReflReparamAux
|
case refine_5
x : ↑I
hx : ↑x = 1 / 2
⊢ 2 * ↑x = 1
|
simp [hx]
|
no goals
|
92d41a2e994f3e1d
|
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_1.ind
f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0
F : ℕ → ℂ := fun n => if n = 0 then 0 else f n
hF₀ : F 0 = 0
hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n
ha : ¬abscissaOfAbsConv F = ⊤
h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x
H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0
n : ℕ
ih : ∀ m < n, F m = 0
⊢ F n = 0
|
suffices Tendsto (fun x : ℝ ↦ n ^ (x : ℂ) * LSeries F x) atTop (nhds (F n)) by
replace this := this.congr' <| H' n
simp only [tendsto_const_nhds_iff] at this
exact this.symm
|
case neg.refine_1.ind
f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0
F : ℕ → ℂ := fun n => if n = 0 then 0 else f n
hF₀ : F 0 = 0
hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n
ha : ¬abscissaOfAbsConv F = ⊤
h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x
H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0
n : ℕ
ih : ∀ m < n, F m = 0
⊢ Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n))
|
7103bb783fd652c6
|
MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero
|
Mathlib/MeasureTheory/Function/Egorov.lean
|
theorem measure_notConvergentSeq_tendsto_zero [SemilatticeSup ι] [Countable ι]
(hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g) (hsm : MeasurableSet s)
(hs : μ s ≠ ∞) (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
Tendsto (fun j => μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0)
|
case inr
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : Nonempty ι
⊢ Tendsto (fun j => μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0)
|
rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]
|
case inr
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : Nonempty ι
⊢ Tendsto (fun j => μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 (μ (⋂ i, s ∩ notConvergentSeq f g n i)))
|
bd38541566755b41
|
List.getElem!_cons_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem getElem!_cons_succ [Inhabited α] {l : List α} : (a::l)[i+1]! = l[i]!
|
α : Type u_1
a : α
i : Nat
inst✝ : Inhabited α
l : List α
⊢ (a :: l)[i + 1]! = l[i]!
|
by_cases h : i < l.length
|
case pos
α : Type u_1
a : α
i : Nat
inst✝ : Inhabited α
l : List α
h : i < l.length
⊢ (a :: l)[i + 1]! = l[i]!
case neg
α : Type u_1
a : α
i : Nat
inst✝ : Inhabited α
l : List α
h : ¬i < l.length
⊢ (a :: l)[i + 1]! = l[i]!
|
5553a5f71d6d6c08
|
prod_generateFrom_generateFrom_eq
|
Mathlib/Topology/Constructions.lean
|
theorem prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : Set (Set Y)}
(hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
@instTopologicalSpaceProd X Y (generateFrom s) (generateFrom t) =
generateFrom (image2 (· ×ˢ ·) s t) :=
let G := generateFrom (image2 (· ×ˢ ·) s t)
le_antisymm
(le_generateFrom fun _ ⟨_, hu, _, hv, g_eq⟩ =>
g_eq.symm ▸
@IsOpen.prod _ _ (generateFrom s) (generateFrom t) _ _ (GenerateOpen.basic _ hu)
(GenerateOpen.basic _ hv))
(le_inf
(coinduced_le_iff_le_induced.mp <|
le_generateFrom fun u hu =>
have : ⋃ v ∈ t, u ×ˢ v = Prod.fst ⁻¹' u
|
X : Type u_5
Y : Type u_6
s : Set (Set X)
t : Set (Set Y)
hs : ⋃₀ s = univ
ht : ⋃₀ t = univ
G : TopologicalSpace (X × Y) := generateFrom (image2 (fun x1 x2 => x1 ×ˢ x2) s t)
v : Set Y
hv : v ∈ t
this : ⋃ u ∈ s, u ×ˢ v = Prod.snd ⁻¹' v
⊢ TopologicalSpace.IsOpen (⋃ u ∈ s, u ×ˢ v)
|
exact
isOpen_iUnion fun u =>
isOpen_iUnion fun hu => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩
|
no goals
|
6bf5cf316a327555
|
MeasureTheory.eLpNorm_sub_le_of_dist_bdd
|
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
theorem eLpNorm_sub_le_of_dist_bdd (μ : Measure α)
{p : ℝ≥0∞} (hp' : p ≠ ∞) {s : Set α} (hs : MeasurableSet[m] s)
{f g : α → β} {c : ℝ} (hc : 0 ≤ c) (hf : ∀ x ∈ s, dist (f x) (g x) ≤ c) :
eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)
|
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ x ∈ s, dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun x => c) x‖
⊢ eLpNorm (s.indicator fun x => c) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)
|
rw [eLpNorm_indicator_const hs hp hp']
|
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ x ∈ s, dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun x => c) x‖
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)
|
5a09cd678840d75b
|
Padic.exi_rat_seq_conv
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem exi_rat_seq_conv {ε : ℚ} (hε : 0 < ε) :
∃ N, ∀ i ≥ N, padicNormE (f i - (limSeq f i : ℚ_[p]) : ℚ_[p]) < ε
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ (↑i + 1) * ε
|
rw [right_distrib]
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ ↑i * ε + 1 * ε
|
a4713483bdf9656c
|
MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
(haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ 1 ∈ V₀
|
simp only [V₀, mem_iInter]
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ ∀ i ∈ t, 1 ∈ i.carrier
|
25c220e55fac0398
|
intervalIntegral.continuousAt_parametric_primitive_of_dominated
|
Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
|
theorem continuousAt_parametric_primitive_of_dominated [FirstCountableTopology X]
{F : X → ℝ → E} (bound : ℝ → ℝ) (a b : ℝ)
{a₀ b₀ : ℝ} {x₀ : X} (hF_meas : ∀ x, AEStronglyMeasurable (F x) (μ.restrict <| Ι a b))
(h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ.restrict <| Ι a b, ‖F x t‖ ≤ bound t)
(bound_integrable : IntervalIntegrable bound μ a b)
(h_cont : ∀ᵐ t ∂μ.restrict <| Ι a b, ContinuousAt (fun x ↦ F x t) x₀) (ha₀ : a₀ ∈ Ioo a b)
(hb₀ : b₀ ∈ Ioo a b) (hμb₀ : μ {b₀} = 0) :
ContinuousAt (fun p : X × ℝ ↦ ∫ t : ℝ in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)
|
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x => F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
⊢ ContinuousAt (fun p => ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)
|
have Ioo_nhds : Ioo a b ∈ 𝓝 b₀ := Ioo_mem_nhds hb₀.1 hb₀.2
|
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x => F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
⊢ ContinuousAt (fun p => ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)
|
700abcaa08c19cdb
|
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod
|
Mathlib/LinearAlgebra/FreeModule/Int.lean
|
/-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
(snf : Basis.SmithNormalForm N ι n) :
N.toAddSubgroup.index = Nat.card R ^ (Fintype.card ι - n) *
∏ i : Fin n, (Ideal.span {snf.a i}).toAddSubgroup.index
|
case h.refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN' : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
g : ι → R
h : ∀ (i : ι), (if h : ∃ j, f j = i then a h.choose else 0) ∣ g i
⊢ g = ∑ x : Fin n, (fun j => Exists.choose ⋯) x • Pi.single (f x) (a x)
|
ext i
|
case h.refine_2.h
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN' : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
g : ι → R
h : ∀ (i : ι), (if h : ∃ j, f j = i then a h.choose else 0) ∣ g i
i : ι
⊢ g i = (∑ x : Fin n, (fun j => Exists.choose ⋯) x • Pi.single (f x) (a x)) i
|
3d7a9cb7455a5c0f
|
bind₁_rename_expand_wittPolynomial
|
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
theorem bind₁_rename_expand_wittPolynomial (Φ : MvPolynomial idx ℤ) (n : ℕ)
(IH :
∀ m : ℕ,
m < n + 1 →
map (Int.castRingHom ℚ) (wittStructureInt p Φ m) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) m) :
bind₁ (fun b => rename (fun i => (b, i)) (expand p (W_ ℤ n))) Φ =
bind₁ (fun i => expand p (wittStructureInt p Φ i)) (W_ ℤ n)
|
case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
|
have key := (wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n).symm
|
case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
(bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n)
⊢ (bind₁ fun i => (expand p) ((rename fun i_1 => (i, i_1)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ fun i => (expand p) ((map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)
|
227cfd398809aa75
|
Set.le_einfsep_pair
|
Mathlib/Topology/MetricSpace/Infsep.lean
|
theorem le_einfsep_pair : edist x y ⊓ edist y x ≤ ({x, y} : Set α).einfsep
|
case inr.inl
α : Type u_1
inst✝ : EDist α
a b : α
hab : a ≠ b
⊢ edist b a ≤ edist a b ∨ edist a b ≤ edist a b
|
simp only [le_refl, true_or, or_true]
|
no goals
|
a64ce198c4883813
|
CategoryTheory.Adjunction.leftAdjointUniq_hom_counit
|
Mathlib/CategoryTheory/Adjunction/Unique.lean
|
theorem leftAdjointUniq_hom_counit {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
whiskerLeft G (leftAdjointUniq adj1 adj2).hom ≫ adj2.counit = adj1.counit
|
case w.h
C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F F' : C ⥤ D
G : D ⥤ C
adj1 : F ⊣ G
adj2 : F' ⊣ G
x : D
⊢ F.map (adj2.unit.app (G.obj x)) ≫ adj1.counit.app (F'.obj (G.obj x)) ≫ adj2.counit.app x = adj1.counit.app x
|
rw [← adj1.counit_naturality, ← Category.assoc, ← F.map_comp]
|
case w.h
C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F F' : C ⥤ D
G : D ⥤ C
adj1 : F ⊣ G
adj2 : F' ⊣ G
x : D
⊢ F.map (adj2.unit.app (G.obj x) ≫ G.map (adj2.counit.app x)) ≫ adj1.counit.app x = adj1.counit.app x
|
9669f69f339158ab
|
finprod_mem_inter_mul_diff'
|
Mathlib/Algebra/BigOperators/Finprod.lean
|
theorem finprod_mem_inter_mul_diff' (t : Set α) (h : (s ∩ mulSupport f).Finite) :
((∏ᶠ i ∈ s ∩ t, f i) * ∏ᶠ i ∈ s \ t, f i) = ∏ᶠ i ∈ s, f i
|
case hs
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Set α
h : (s ∩ mulSupport f).Finite
⊢ (s ∩ t ∩ mulSupport f).Finite
case ht
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Set α
h : (s ∩ mulSupport f).Finite
⊢ (s \ t ∩ mulSupport f).Finite
|
exacts [h.subset fun x hx => ⟨hx.1.1, hx.2⟩, h.subset fun x hx => ⟨hx.1.1, hx.2⟩]
|
no goals
|
b5ec9483672c1fac
|
PrimeSpectrum.isTopologicalBasis_basic_opens
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
theorem isTopologicalBasis_basic_opens :
TopologicalSpace.IsTopologicalBasis
(Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R)))
|
R : Type u
inst✝ : CommSemiring R
⊢ TopologicalSpace.IsTopologicalBasis (Set.range fun r => ↑(basicOpen r))
|
apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
|
case h_open
R : Type u
inst✝ : CommSemiring R
⊢ ∀ u ∈ Set.range fun r => ↑(basicOpen r), IsOpen u
case h_nhds
R : Type u
inst✝ : CommSemiring R
⊢ ∀ (a : PrimeSpectrum R) (u : Set (PrimeSpectrum R)),
a ∈ u → IsOpen u → ∃ v ∈ Set.range fun r => ↑(basicOpen r), a ∈ v ∧ v ⊆ u
|
38672fba2e8d5a02
|
MeasureTheory.Content.innerContent_iSup_nat
|
Mathlib/MeasureTheory/Measure/Content.lean
|
theorem innerContent_iSup_nat [R1Space G] (U : ℕ → Opens G) :
μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i)
|
case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ μ K ≤ ∑' (i : ℕ), μ.innerContent (U i)
|
convert le_trans (h3 t L) _
|
case h.e'_3.h.e'_6
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ K = t.sup L
case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ∑ i ∈ t, μ (L i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
|
47868df373390eca
|
MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_inner
|
Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
|
theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner {u : E → F'}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
(hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
eLpNorm u p' μ ≤ eLpNormLESNormFDerivOfEqInnerConst μ p * eLpNorm (fderiv ℝ u) p μ
|
case bc
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).IsConjExponent n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).IsConjExponent q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h0p' : p' ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ⊤
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤
h7u : Continuous u
h8u : Continuous (fderiv ℝ u)
v : E → ℝ := fun x => ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
⊢ ∫⁻ (x : E), ‖u x‖ₑ ^ (↑γ - 1) * ‖fderiv ℝ u x‖ₑ ∂μ ≤
(∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p)
|
convert ENNReal.lintegral_mul_le_Lp_mul_Lq μ
(.symm <| .conjExponent <| show 1 < (p : ℝ) from hp) ?_ ?_ using 5
|
case h.e'_4.h.e'_5.h.e'_5.h.e'_4.h
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).IsConjExponent n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).IsConjExponent q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h0p' : p' ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ⊤
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤
h7u : Continuous u
h8u : Continuous (fderiv ℝ u)
v : E → ℝ := fun x => ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
x✝ : E
⊢ ‖u x✝‖ₑ ^ ↑p' = (‖u x✝‖ₑ ^ (↑γ - 1)) ^ (↑p).conjExponent
case bc.convert_3
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).IsConjExponent n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).IsConjExponent q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h0p' : p' ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ⊤
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤
h7u : Continuous u
h8u : Continuous (fderiv ℝ u)
v : E → ℝ := fun x => ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
⊢ AEMeasurable (fun x => ‖u x‖ₑ ^ (↑γ - 1)) μ
case bc.convert_4
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).IsConjExponent n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).IsConjExponent q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h0p' : p' ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ⊤
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤
h7u : Continuous u
h8u : Continuous (fderiv ℝ u)
v : E → ℝ := fun x => ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
⊢ AEMeasurable (fun x => ‖fderiv ℝ u x‖ₑ) μ
|
728bbe5fb5258103
|
Std.Tactic.BVDecide.BVExpr.bitblast.mkOverflowBit.go_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Carry.lean
|
theorem go_decl_eq {aig : AIG α} {cin} {lhs rhs : AIG.RefVec aig w} :
∀ (idx : Nat) (h1) (h2),
(go aig lhs rhs curr cin).aig.decls[idx]'h2 = aig.decls[idx]'h1
|
case isTrue.h2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w curr : Nat
aig : AIG α
cin : aig.Ref
lhs rhs : aig.RefVec w
res : AIG.Entrypoint α
h✝ : curr < w
hgo :
go (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
(curr + 1) (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).ref =
res
idx✝ : Nat
h1✝ : idx✝ < aig.decls.size
h2✝ :
idx✝ <
(go (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig (lhs.cast ⋯)
(rhs.cast ⋯) (curr + 1)
(mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).ref).aig.decls.size
⊢ idx✝ < (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig.decls.size
|
apply AIG.LawfulOperator.lt_size_of_lt_aig_size (f := mkFullAdderCarry)
|
case isTrue.h2.h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w curr : Nat
aig : AIG α
cin : aig.Ref
lhs rhs : aig.RefVec w
res : AIG.Entrypoint α
h✝ : curr < w
hgo :
go (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig (lhs.cast ⋯) (rhs.cast ⋯)
(curr + 1) (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).ref =
res
idx✝ : Nat
h1✝ : idx✝ < aig.decls.size
h2✝ :
idx✝ <
(go (mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).aig (lhs.cast ⋯)
(rhs.cast ⋯) (curr + 1)
(mkFullAdderCarry aig { lhs := lhs.get curr h✝, rhs := rhs.get curr h✝, cin := cin }).ref).aig.decls.size
⊢ idx✝ < aig.decls.size
|
0e1562c4999c3b0c
|
BooleanSubalgebra.sdiff_mem
|
Mathlib/Order/BooleanSubalgebra.lean
|
lemma sdiff_mem (ha : a ∈ L) (hb : b ∈ L) : a \ b ∈ L
|
α : Type u_2
inst✝ : BooleanAlgebra α
L : BooleanSubalgebra α
a b : α
ha : a ∈ L
hb : b ∈ L
⊢ a \ b ∈ L
|
simpa [sdiff_eq] using L.infClosed ha (compl_mem hb)
|
no goals
|
0a19dc16f6028850
|
ExistsContDiffBumpBase.y_smooth
|
Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
|
theorem y_smooth : ContDiffOn ℝ ∞ (uncurry y) (Ioo (0 : ℝ) 1 ×ˢ (univ : Set E))
|
case refine_3
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
⊢ ContDiffOn ℝ ∞ (↿w) (Ioo 0 1 ×ˢ univ)
|
apply ContDiffOn.mul
|
case refine_3.hf
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
⊢ ContDiffOn ℝ ∞ (fun x => ((∫ (x : E), u x ∂μ) * |x.1| ^ finrank ℝ E)⁻¹) (Ioo 0 1 ×ˢ univ)
case refine_3.hg
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
⊢ ContDiffOn ℝ ∞ (fun x => u (x.1⁻¹ • x.2)) (Ioo 0 1 ×ˢ univ)
|
5d103a5b75b6c303
|
MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
|
Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean
|
theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1))
|
case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
one_lt_p : -1 < p - 1
g : ℝ → ℝ := fun t => t ^ (p - 1)
obs : ∀ (x : ℝ), intervalIntegral g 0 x volume = x ^ p / p
t : ℝ
t_pos : t ∈ Ioi 0
⊢ 0 ≤ g t
|
exact Real.rpow_nonneg (mem_Ioi.mp t_pos).le (p - 1)
|
no goals
|
720d77f4c064a30f
|
Module.End.exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow
|
Mathlib/LinearAlgebra/JordanChevalley.lean
|
theorem exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow {P : K[X]} {k : ℕ}
(sep : P.Separable) (nil : minpoly K f ∣ P ^ k) :
∃ᵉ (n ∈ adjoin K {f}) (s ∈ adjoin K {f}), IsNilpotent n ∧ IsSemisimple s ∧ f = n + s
|
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k
ff : ↥(adjoin K {f}) := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
⊢ ∃ n ∈ adjoin K {f}, ∃ s ∈ adjoin K {f}, IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s
|
obtain ⟨⟨s, mem⟩, ⟨⟨k, hk⟩, hss⟩, -⟩ := existsUnique_nilpotent_sub_and_aeval_eq_zero nil' sep'
|
case intro.mk.intro.intro.intro
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥(adjoin K {f}) := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End K V
mem : s ∈ adjoin K {f}
hss : (aeval ⟨s, mem⟩) P = 0
k : ℕ
hk : (ff - ⟨s, mem⟩) ^ k = 0
⊢ ∃ n ∈ adjoin K {f}, ∃ s ∈ adjoin K {f}, IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s
|
d5adbca57ec910fe
|
PNat.gcd_eq_left_iff_dvd
|
Mathlib/Data/PNat/Prime.lean
|
theorem gcd_eq_left_iff_dvd {m n : ℕ+} : m ∣ n ↔ m.gcd n = m
|
m n : ℕ+
⊢ (↑m).gcd ↑n = ↑m ↔ m.gcd n = m
|
rw [← coe_inj]
|
m n : ℕ+
⊢ (↑m).gcd ↑n = ↑m ↔ ↑(m.gcd n) = ↑m
|
2ad2bac44da65d21
|
AnalyticOnNhd.is_constant_or_isOpen
|
Mathlib/Analysis/Complex/OpenMapping.lean
|
theorem AnalyticOnNhd.is_constant_or_isOpen (hg : AnalyticOnNhd ℂ g U) (hU : IsPreconnected U) :
(∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s)
|
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
g : E → ℂ
hg : AnalyticOnNhd ℂ g U
hU : IsPreconnected U
h : ∀ z₀ ∈ U, ¬∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀
s : Set E
hs1 : s ⊆ U
hs2 : IsOpen s
⊢ ∀ x ∈ g '' s, g '' s ∈ 𝓝 x
|
rintro z ⟨w, hw1, rfl⟩
|
case neg.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
g : E → ℂ
hg : AnalyticOnNhd ℂ g U
hU : IsPreconnected U
h : ∀ z₀ ∈ U, ¬∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀
s : Set E
hs1 : s ⊆ U
hs2 : IsOpen s
w : E
hw1 : w ∈ s
⊢ g '' s ∈ 𝓝 (g w)
|
bcea65785f6ffd2a
|
intervalIntegral.continuousWithinAt_primitive
|
Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
|
theorem continuousWithinAt_primitive (hb₀ : μ {b₀} = 0)
(h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)) :
ContinuousWithinAt (fun b => ∫ x in a..b, f x ∂μ) (Icc b₁ b₂) b₀
|
case intro.h₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (a ⊓ b₁) (a ⊔ b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : b₁ ⊓ b₂ = b₁
x : ℝ
h₁ : b₁ ≤ x
h₂ : x ≤ b₂
⊢ x ∈ [[a ⊓ b₁, a ⊔ b₂]]
|
exact ⟨min_le_of_left_le <| (min_le_right _ _).trans h₁,
le_max_of_le_right <| h₂.trans <| le_max_right _ _⟩
|
no goals
|
9163a1d4de5d2966
|
Mathlib.Tactic.Bicategory.naturality_leftUnitor
|
Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.lean
|
theorem naturality_leftUnitor {p : a ⟶ b} {f : b ⟶ c} {pf : a ⟶ c} (η_f : p ≫ f ≅ pf) :
p ◁ (λ_ f) ≪≫ η_f = normalizeIsoComp (ρ_ p) η_f :=
Iso.ext (by simp)
|
B : Type u
inst✝ : Bicategory B
a b c : B
p : a ⟶ b
f : b ⟶ c
pf : a ⟶ c
η_f : p ≫ f ≅ pf
⊢ (p ◁ λ_ f ≪≫ η_f).hom = (normalizeIsoComp (ρ_ p) η_f).hom
|
simp
|
no goals
|
98745ba2aaaa51f8
|
eventually_homothety_mem_of_mem_interior
|
Mathlib/Analysis/Normed/Affine/AddTorsor.lean
|
theorem eventually_homothety_mem_of_mem_interior (x : Q) {s : Set Q} {y : Q} (hy : y ∈ interior s) :
∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s
|
case h
W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
x : Q
s : Set Q
y : Q
hy : y ∈ interior s
h : y = x
⊢ 0 < 1 ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < 1} → (homothety x x_1) y ∈ s
|
simp [h.symm, interior_subset hy]
|
no goals
|
1289875882d40d72
|
Real.continuousOn_tan_Ioo
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
|
theorem continuousOn_tan_Ioo : ContinuousOn tan (Ioo (-(π / 2)) (π / 2))
|
x : ℝ
⊢ x ∈ Ioo (-(π / 2)) (π / 2) → x ∈ {x | cos x ≠ 0}
|
simp only [and_imp, mem_Ioo, mem_setOf_eq, Ne]
|
x : ℝ
⊢ -(π / 2) < x → x < π / 2 → ¬cos x = 0
|
d91f87528b6a23d6
|
GromovHausdorff.hausdorffDist_optimal
|
Mathlib/Topology/MetricSpace/GromovHausdorff.lean
|
theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y
|
case intro.intro.intro.intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n => ℝ) ⊤) :=
fun x =>
match x with
| inl y => Φ y
| inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
|
have : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb :=
hausdorffDist_optimal_le_HD _ _ (candidatesBOfCandidates_mem F Fgood)
|
case intro.intro.intro.intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n => ℝ) ⊤) :=
fun x =>
match x with
| inl y => Φ y
| inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
|
e1b74305bb180259
|
Metric.Sum.mem_uniformity_iff_glueDist
|
Mathlib/Topology/MetricSpace/Gluing.lean
|
theorem Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s
|
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
⊢ s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ (a b : X ⊕ Y), glueDist Φ Ψ ε a b < δ → (a, b) ∈ s
|
simp only [Sum.uniformity, Filter.mem_sup, Filter.mem_map, mem_uniformity_dist, mem_preimage]
|
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hε : 0 < ε
s : Set ((X ⊕ Y) × (X ⊕ Y))
⊢ ((∃ ε > 0, ∀ ⦃a b : X⦄, dist a b < ε → Prod.map Sum.inl Sum.inl (a, b) ∈ s) ∧
∃ ε > 0, ∀ ⦃a b : Y⦄, dist a b < ε → Prod.map Sum.inr Sum.inr (a, b) ∈ s) ↔
∃ δ > 0, ∀ (a b : X ⊕ Y), glueDist Φ Ψ ε a b < δ → (a, b) ∈ s
|
fe4b5204c103372a
|
StieltjesFunction.outer_trim
|
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
theorem outer_trim : f.outer.trim = f.outer
|
f : StieltjesFunction
s : Set ℝ
t : ℕ → Set ℝ
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ⊤
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)
|
have hl :=
ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'
|
f : StieltjesFunction
s : Set ℝ
t : ℕ → Set ℝ
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ⊤
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hl : f.length (t i) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)
|
f9dd01cdb673361c
|
Real.cos_pi_div_five
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
theorem cos_pi_div_five : cos (π / 5) = (1 + √5) / 4
|
case inr
c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ c = (1 + √5) / 4
|
absurd (show 0 ≤ c from cos_nonneg_of_mem_Icc <| by constructor <;> linarith [pi_pos.le])
|
case inr
c : ℝ := cos (π / 5)
this : 4 * (c * c) + -2 * c + -1 = 0
hd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)
h : c = (- -2 - 2 * √5) / (2 * 4)
⊢ ¬0 ≤ c
|
386d26d34e57890c
|
Polynomial.cyclotomic_injective
|
Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean
|
theorem cyclotomic_injective [CharZero R] : Function.Injective fun n => cyclotomic n R
|
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : (cyclotomic m ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))
hmzero : NeZero m
⊢ n = m
|
rw [isRoot_cyclotomic_iff (R := ℂ)] at hroot
|
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n m : ℕ
hzero : n ≠ 0
this : NeZero n
hnm : cyclotomic n ℂ = cyclotomic m ℂ
hprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n
hroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m
hmzero : NeZero m
⊢ n = m
|
95f81674fb37b3a0
|
CategoryTheory.Comma.inv_right
|
Mathlib/CategoryTheory/Comma/Basic.lean
|
@[simp]
lemma inv_right [IsIso e] : (inv e).right = inv e.right
|
A : Type u₁
inst✝³ : Category.{v₁, u₁} A
B : Type u₂
inst✝² : Category.{v₂, u₂} B
T : Type u₃
inst✝¹ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
e : X ⟶ Y
inst✝ : IsIso e
⊢ (inv e).right = inv e.right
|
apply IsIso.eq_inv_of_hom_inv_id
|
case hom_inv_id
A : Type u₁
inst✝³ : Category.{v₁, u₁} A
B : Type u₂
inst✝² : Category.{v₂, u₂} B
T : Type u₃
inst✝¹ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
e : X ⟶ Y
inst✝ : IsIso e
⊢ e.right ≫ (inv e).right = 𝟙 X.right
|
c0e969a93d7f9594
|
Complex.continuousOn_norm_circleTransformBoundingFunction
|
Mathlib/MeasureTheory/Integral/CircleTransform.lean
|
theorem continuousOn_norm_circleTransformBoundingFunction {R r : ℝ} (hr : r < R) (z : ℂ) :
ContinuousOn ((‖·‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)
|
case hg.hf
R r : ℝ
hr : r < R
z : ℂ
⊢ ContinuousOn (fun x => deriv (circleMap z R) x.2) (closedBall z r ×ˢ univ)
|
simp only [deriv_circleMap]
|
case hg.hf
R r : ℝ
hr : r < R
z : ℂ
⊢ ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r ×ˢ univ)
|
eed8430be825edc0
|
NumberField.mixedEmbedding.negAt_symm
|
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
|
theorem negAt_symm :
(negAt s).symm = negAt s
|
case neg
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∉ s
⊢ ((negAt s).symm x).1 w = ((negAt s) x).1 w
|
simp_rw [negAt_apply_isReal_and_not_mem _ hw, negAt, prod_symm,
ContinuousLinearEquiv.prod_apply, piCongrRight_symm_apply, if_neg hw, refl_symm, refl_apply]
|
no goals
|
b65992f9db9e340a
|
Seminorm.ball_norm_mul_subset
|
Mathlib/Analysis/Seminorm.lean
|
theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r
|
case inl
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
⊢ p.ball 0 (‖0‖ * r) ⊆ 0 • p.ball 0 r
|
rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
|
case inl
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
⊢ ∅ ⊆ 0 • p.ball 0 r
|
b87a373bd1754990
|
cyclotomic_comp_X_add_one_isEisensteinAt
|
Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
|
theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟
|
case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < ((cyclotomic p ℤ).comp (X + 1)).natDegree
⊢ (∑ x ∈ range p, if x = i then ↑(p.choose (x + 1)) else 0) ∈ Submodule.span ℤ {↑p}
|
rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
natDegree_cyclotomic, Nat.totient_prime hp.out] at hi
|
case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < p - 1
⊢ (∑ x ∈ range p, if x = i then ↑(p.choose (x + 1)) else 0) ∈ Submodule.span ℤ {↑p}
|
bff95b337ffb159a
|
ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz
|
Mathlib/Analysis/Normed/Operator/Banach.lean
|
lemma bijective_iff_dense_range_and_antilipschitz (f : E →SL[σ] F) :
Bijective f ↔ (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c f
|
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
x✝ : (LinearMap.range f).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f
hd : (LinearMap.range f).topologicalClosure = ⊤
c : ℝ≥0
hf : AntilipschitzWith c ⇑f
⊢ Surjective ⇑f
|
rwa [← LinearMap.range_eq_top, ← closed_range_of_antilipschitz hf]
|
no goals
|
1255c3d2e1f157b7
|
OrderIso.strictConcaveOn_symm
|
Mathlib/Analysis/Convex/Function.lean
|
theorem OrderIso.strictConcaveOn_symm (f : α ≃o β) (hf : StrictConvexOn 𝕜 univ f) :
StrictConcaveOn 𝕜 univ f.symm
|
case intro.intro
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : StrictConvexOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' : α
hy'' : y = f y'
hxy' : x' ≠ y'
⊢ a • x' + b • y' < f.symm (a • f x' + b • f y')
|
rw [← f.lt_iff_lt, OrderIso.apply_symm_apply]
|
case intro.intro
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : StrictConvexOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' : α
hy'' : y = f y'
hxy' : x' ≠ y'
⊢ f (a • x' + b • y') < a • f x' + b • f y'
|
798e65840fb2b195
|
Set.inl_compl_union_inr_compl
|
Mathlib/Data/Set/Basic.lean
|
lemma inl_compl_union_inr_compl {α β : Type*} {s : Set α} {t : Set β} :
Sum.inl '' sᶜ ∪ Sum.inr '' tᶜ = (Sum.inl '' s ∪ Sum.inr '' t)ᶜ
|
α : Type u_1
β : Type u_2
s : Set α
t : Set β
⊢ Sum.inl '' sᶜ ∪ Sum.inr '' tᶜ = (Sum.inl '' s)ᶜ ∩ (Sum.inr '' t)ᶜ
|
aesop
|
no goals
|
912b559fa88b7d10
|
IsLindelof.disjoint_nhdsSet_left
|
Mathlib/Topology/Compactness/Lindelof.lean
|
theorem IsLindelof.disjoint_nhdsSet_left {l : Filter X} [CountableInterFilter l]
(hs : IsLindelof s) :
Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l
|
X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
l : Filter X
inst✝ : CountableInterFilter l
hs : IsLindelof s
⊢ Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l
|
refine ⟨fun h x hx ↦ h.mono_left <| nhds_le_nhdsSet hx, fun H ↦ ?_⟩
|
X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
l : Filter X
inst✝ : CountableInterFilter l
hs : IsLindelof s
H : ∀ x ∈ s, Disjoint (𝓝 x) l
⊢ Disjoint (𝓝ˢ s) l
|
f96889b611855166
|
ZMod.cast_sub_one
|
Mathlib/Data/ZMod/Basic.lean
|
theorem cast_sub_one {R : Type*} [Ring R] {n : ℕ} (k : ZMod n) :
(cast (k - 1 : ZMod n) : R) = (if k = 0 then (n : R) else cast k) - 1
|
case neg.zero
R : Type u_1
inst✝ : Ring R
k : ZMod 0
hk : ¬k = 0
⊢ (k - 1).cast = k.cast - 1
|
dsimp [ZMod, ZMod.cast]
|
case neg.zero
R : Type u_1
inst✝ : Ring R
k : ZMod 0
hk : ¬k = 0
⊢ ↑(k - 1) = ↑k - 1
|
4fdb3038bd174435
|
Set.Iic_union_Icc'
|
Mathlib/Order/Interval/Set/Basic.lean
|
theorem Iic_union_Icc' (h₁ : c ≤ b) : Iic b ∪ Icc c d = Iic (max b d)
|
case h
α : Type u_1
inst✝ : LinearOrder α
b c d : α
h₁ : c ≤ b
x : α
⊢ x ∈ Iic b ∪ Icc c d ↔ x ∈ Iic (b ⊔ d)
|
simp_rw [mem_union, mem_Iic, mem_Icc, le_max_iff]
|
case h
α : Type u_1
inst✝ : LinearOrder α
b c d : α
h₁ : c ≤ b
x : α
⊢ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ x ≤ b ∨ x ≤ d
|
cb2a065dca8b2fee
|
UV.shadow_compression_subset_compression_shadow
|
Mathlib/Combinatorics/SetFamily/Compression/UV.lean
|
theorem shadow_compression_subset_compression_shadow (u v : Finset α)
(huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜) :
∂ (𝓒 u v 𝒜) ⊆ 𝓒 u v (∂ 𝒜)
|
case intro.intro.refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ insert x ((s ∪ v) \ u) ∈ 𝒜
|
rwa [← insert_sdiff_of_not_mem _ ‹x ∉ u›, ← insert_union]
|
no goals
|
9666ab7e87eab58f
|
Commute.orderOf_mul_eq_right_of_forall_prime_mul_dvd
|
Mathlib/GroupTheory/OrderOfElement.lean
|
theorem orderOf_mul_eq_right_of_forall_prime_mul_dvd (h : Commute x y) (hy : IsOfFinOrder y)
(hdvd : ∀ p : ℕ, p.Prime → p ∣ orderOf x → p * orderOf x ∣ orderOf y) :
orderOf (x * y) = orderOf y
|
case hd
G : Type u_1
inst✝ : Monoid G
x y : G
h : Commute x y
hy : IsOfFinOrder y
hdvd : ∀ (p : ℕ), Nat.Prime p → p ∣ orderOf x → p * orderOf x ∣ orderOf y
hoy : 0 < orderOf y
hxy : orderOf x ∣ orderOf y
⊢ ∀ (p : ℕ), Nat.Prime p → p ∣ orderOf y → ¬orderOf (x * y) ∣ orderOf y / p
|
refine fun p hp hpy hd => hp.ne_one ?_
|
case hd
G : Type u_1
inst✝ : Monoid G
x y : G
h : Commute x y
hy : IsOfFinOrder y
hdvd : ∀ (p : ℕ), Nat.Prime p → p ∣ orderOf x → p * orderOf x ∣ orderOf y
hoy : 0 < orderOf y
hxy : orderOf x ∣ orderOf y
p : ℕ
hp : Nat.Prime p
hpy : p ∣ orderOf y
hd : orderOf (x * y) ∣ orderOf y / p
⊢ p = 1
|
ad488aedf0f1c9e4
|
Int.mul_le_mul_of_nonneg_left
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem mul_le_mul_of_nonneg_left {a b c : Int}
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
if hba : b ≤ a then by
rw [Int.le_antisymm hba h₁]; apply Int.le_refl
else if hc0 : c ≤ 0 then by
simp [Int.le_antisymm hc0 h₂, Int.zero_mul]
else by
exact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left
(Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩)
|
a b c : Int
h₁ : a ≤ b
h₂ : 0 ≤ c
hba : ¬b ≤ a
hc0 : c ≤ 0
⊢ c * a ≤ c * b
|
simp [Int.le_antisymm hc0 h₂, Int.zero_mul]
|
no goals
|
ebac8b4eee743453
|
clusterPt_iff_lift'_closure'
|
Mathlib/Topology/Basic.lean
|
theorem clusterPt_iff_lift'_closure' {F : Filter X} :
ClusterPt x F ↔ (F.lift' closure ⊓ pure x).NeBot
|
case mpr
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
⊢ (pure x ⊓ F.lift' closure).NeBot → pure x ≤ F.lift' closure
|
intro h U hU
|
case mpr
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
h : (pure x ⊓ F.lift' closure).NeBot
U : Set X
hU : U ∈ F.lift' closure
⊢ U ∈ pure x
|
70879bd9b2cfc6e6
|
Set.sInter_prod_sInter
|
Mathlib/Data/Set/Lattice.lean
|
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2
|
case intro.intro
α : Type u_1
β : Type u_2
S : Set (Set α)
T : Set (Set β)
s₁ : Set α
h₁ : s₁ ∈ S
s₂ : Set β
h₂ : s₂ ∈ T
x : α × β
hx : ∀ i ∈ S ×ˢ T, x ∈ i.1 ×ˢ i.2
⊢ x ∈ ⋂₀ S ×ˢ ⋂₀ T
|
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
|
no goals
|
5614e7d5edc46357
|
Ordinal.sup_mul_nat
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem sup_mul_nat (o : Ordinal) : (sup fun n : ℕ => o * n) = o * ω
|
case inr
o : Ordinal.{u_1}
ho : 0 < o
⊢ (sup fun n => o * ↑n) = o * ω
|
exact (mul_isNormal ho).apply_omega0
|
no goals
|
4ffb26912b5f26d1
|
MulAction.IsBlockSystem.of_normal
|
Mathlib/GroupTheory/GroupAction/Blocks.lean
|
theorem IsBlockSystem.of_normal {N : Subgroup G} [N.Normal] :
IsBlockSystem G (Set.range fun a : X => orbit N a)
|
case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
⊢ ∀ ⦃B : Set X⦄, (B ∈ range fun a => orbit (↥N) a) → IsBlock G B
|
intro b
|
case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
b : Set X
⊢ (b ∈ range fun a => orbit (↥N) a) → IsBlock G b
|
94f82df37ff2f4b9
|
MeasureTheory.volume_sum_rpow_le
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
theorem MeasureTheory.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ card ι *
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1))
|
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
⊢ volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
|
have nm_neg := fun x : ι → ℝ => norm_neg (E := PiLp (.ofReal p) (fun _ : ι => ℝ)) x
|
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-x‖ = ‖x‖
⊢ volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
|
b43945fead980666
|
ProbabilityTheory.Kernel.tendsto_integral_density_of_monotone
|
Mathlib/Probability/Kernel/Disintegration/Density.lean
|
lemma tendsto_integral_density_of_monotone (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ ∫ x, density κ ν a x (seq m) ∂(ν a)) atTop (𝓝 (κ a univ).toReal)
|
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ Tendsto (fun m => ∫ (x : γ), κ.density ν a x (seq m) ∂ν a) atTop (𝓝 ((κ a) univ).toReal)
|
simp_rw [integral_density hκν a (hseq_meas _)]
|
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ Tendsto (fun m => ((κ a) (univ ×ˢ seq m)).toReal) atTop (𝓝 ((κ a) univ).toReal)
|
e97057e55029f4e1
|
SimpleGraph.chromaticNumber_sum
|
Mathlib/Combinatorics/SimpleGraph/Sum.lean
|
theorem chromaticNumber_sum :
(G ⊕g H).chromaticNumber = max G.chromaticNumber H.chromaticNumber
|
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
d : ℕ∞
hG : G.chromaticNumber ≤ d
hH : H.chromaticNumber ≤ d
⊢ (G ⊕g H).chromaticNumber ≤ d
|
cases d with
| top => simp
| coe n =>
let cG : G.Coloring (Fin n) := (chromaticNumber_le_iff_colorable.mp hG).some
let cH : H.Coloring (Fin n) := (chromaticNumber_le_iff_colorable.mp hH).some
exact chromaticNumber_le_iff_colorable.mpr (Nonempty.intro (cG.sum cH))
|
no goals
|
14daea5911720f3b
|
RootPairing.range_polarization_domRestrict_le_span_coroot
|
Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean
|
theorem range_polarization_domRestrict_le_span_coroot :
LinearMap.range (P.Polarization.domRestrict P.rootSpan) ≤ P.corootSpan
|
case intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
y : N
x : ↥P.rootSpan
hx : (P.Polarization.domRestrict P.rootSpan) x = y
⊢ ∃ c, ∑ i : ι, c i • P.coroot i = ∑ i : ι, (P.coroot' i) ↑x • P.coroot i
|
use fun i => (P.toPerfectPairing x) (P.coroot i)
|
case h
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
y : N
x : ↥P.rootSpan
hx : (P.Polarization.domRestrict P.rootSpan) x = y
⊢ ∑ i : ι, (fun i => (P.toPerfectPairing ↑x) (P.coroot i)) i • P.coroot i = ∑ i : ι, (P.coroot' i) ↑x • P.coroot i
|
f62e36c9fced14b2
|
WittVector.coeff_add_of_disjoint
|
Mathlib/RingTheory/WittVector/InitTail.lean
|
theorem coeff_add_of_disjoint (x y : 𝕎 R) (h : ∀ n, x.coeff n = 0 ∨ y.coeff n = 0) :
(x + y).coeff n = x.coeff n + y.coeff n
|
case h
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n => y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n => if P n then x.coeff n else y.coeff n
n : ℕ
⊢ (select P z).coeff n = x.coeff n
|
rw [select, coeff_mk, coeff_mk]
|
case h
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n => y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n => if P n then x.coeff n else y.coeff n
n : ℕ
⊢ (if P n then if P n then x.coeff n else y.coeff n else 0) = x.coeff n
|
e3aa7fcb498bb341
|
precise_refinement_set
|
Mathlib/Topology/Compactness/Paracompact.lean
|
theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s) (u : ι → Set X)
(uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :
∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i
|
ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
inst✝ : ParacompactSpace X
s : Set X
hs : IsClosed s
u : ι → Set X
uo : ∀ (i : ι), IsOpen (u i)
us : s ⊆ ⋃ i, u i
⊢ ⋃ i, Option.elim' sᶜ u i = univ
|
apply Subset.antisymm (subset_univ _)
|
ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
inst✝ : ParacompactSpace X
s : Set X
hs : IsClosed s
u : ι → Set X
uo : ∀ (i : ι), IsOpen (u i)
us : s ⊆ ⋃ i, u i
⊢ univ ⊆ ⋃ i, Option.elim' sᶜ u i
|
ecc14a28630beea6
|
RingHom.FormallyUnramified.propertyIsLocal
|
Mathlib/RingTheory/RingHom/Unramified.lean
|
lemma propertyIsLocal :
PropertyIsLocal FormallyUnramified
|
case localizationAwayPreserves
R S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u_3
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
H✝ : f.FormallyUnramified
algInst✝² : Algebra R S := f.toAlgebra
algInst✝¹ : Algebra R S' := ((algebraMap S S').comp f).toAlgebra
algInst✝ : Algebra R' S' := (IsLocalization.Away.map R' S' f r).toAlgebra
scalarTowerInst✝ : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S'))
algebraizeInst✝ : Algebra.FormallyUnramified R S
this✝¹ : Algebra.FormallyUnramified S S'
this✝ : Algebra.FormallyUnramified R S'
H : Submonoid.powers r ≤ Submonoid.comap f (Submonoid.powers (f r))
this : IsScalarTower R R' S'
⊢ (IsLocalization.Away.map R' S' f r).FormallyUnramified
|
exact Algebra.FormallyUnramified.of_comp R R' S'
|
no goals
|
f955845a3f1095b7
|
Stream'.Seq.cons_injective2
|
Mathlib/Data/Seq/Seq.lean
|
theorem cons_injective2 : Function.Injective2 (cons : α → Seq α → Seq α) := fun x y s t h =>
⟨by rw [← Option.some_inj, ← get?_cons_zero, h, get?_cons_zero],
Seq.ext fun n => by simp_rw [← get?_cons_succ x s n, h, get?_cons_succ]⟩
|
α : Type u
x y : α
s t : Seq α
h : cons x s = cons y t
n : ℕ
⊢ s.get? n = t.get? n
|
simp_rw [← get?_cons_succ x s n, h, get?_cons_succ]
|
no goals
|
c62240eacb0d23c4
|
Polynomial.Monic.comp
|
Mathlib/Algebra/Polynomial/Monic.lean
|
lemma comp (hp : p.Monic) (hq : q.Monic) (h : q.natDegree ≠ 0) : (p.comp q).Monic
|
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p.Monic
hq : q.Monic
h : q.natDegree ≠ 0
a✝ : Nontrivial R
this : (p.comp q).natDegree = p.natDegree * q.natDegree
⊢ (p.comp q).Monic
|
rw [Monic.def, Polynomial.leadingCoeff, this, coeff_comp_degree_mul_degree h, hp.leadingCoeff,
hq.leadingCoeff, one_pow, mul_one]
|
no goals
|
6796f7c453197a8d
|
IsVisible.of_convexHull_of_pos
|
Mathlib/Analysis/Convex/Visible.lean
|
/-- If a point `x` sees a convex combination of points of a set `s` through `convexHull ℝ s ∌ x`,
then it sees all terms of that combination.
Note that the converse does not hold. -/
lemma IsVisible.of_convexHull_of_pos {ι : Type*} {t : Finset ι} {a : ι → V} {w : ι → 𝕜}
(hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i ∈ t, w i = 1) (ha : ∀ i ∈ t, a i ∈ s)
(hx : x ∉ convexHull 𝕜 s) (hw : IsVisible 𝕜 (convexHull 𝕜 s) x (∑ i ∈ t, w i • a i)) {i : ι}
(hi : i ∈ t) (hwi : 0 < w i) : IsVisible 𝕜 (convexHull 𝕜 s) x (a i)
|
case h.e'_10
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝ : 0 < w i
hwi : w i = 1
⊢ a i = ∑ i ∈ t, w i • a i
|
rw [← one_smul 𝕜 (a i), ← hwi, eq_comm]
|
case h.e'_10
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝ : 0 < w i
hwi : w i = 1
⊢ ∑ i ∈ t, w i • a i = w i • a i
|
7c7847924d1006dd
|
MvPowerSeries.coeff_mul_of_add_lexOrder
|
Mathlib/RingTheory/MvPowerSeries/LexOrder.lean
|
theorem coeff_mul_of_add_lexOrder {φ ψ : MvPowerSeries σ R}
{p q : σ →₀ ℕ} (hp : lexOrder φ = toLex p) (hq : lexOrder ψ = toLex q) :
coeff R (p + q) (φ * ψ) = coeff R p φ * coeff R q ψ
|
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
⊢ ∑ p ∈ Finset.antidiagonal (p + q), (coeff R p.1) φ * (coeff R p.2) ψ = (coeff R p) φ * (coeff R q) ψ
|
apply Finset.sum_eq_single (⟨p, q⟩ : (σ →₀ ℕ) × (σ →₀ ℕ))
|
case h₀
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
⊢ ∀ b ∈ Finset.antidiagonal (p + q), b ≠ (p, q) → (coeff R b.1) φ * (coeff R b.2) ψ = 0
case h₁
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
⊢ (p, q) ∉ Finset.antidiagonal (p + q) → (coeff R (p, q).1) φ * (coeff R (p, q).2) ψ = 0
|
0c23704cc23b543a
|
isPrimePow_iff_factorization_eq_single
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
theorem isPrimePow_iff_factorization_eq_single {n : ℕ} :
IsPrimePow n ↔ ∃ p k : ℕ, 0 < k ∧ n.factorization = Finsupp.single p k
|
n : ℕ
⊢ IsPrimePow n ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k
|
rw [isPrimePow_nat_iff]
|
n : ℕ
⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k
|
f949a688297a79db
|
Algebra.FormallyUnramified.iff_exists_tensorProduct
|
Mathlib/RingTheory/Unramified/Finite.lean
|
theorem iff_exists_tensorProduct [EssFiniteType R S] :
FormallyUnramified R S ↔ ∃ t : S ⊗[R] S,
(∀ s, ((1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) * t = 0) ∧ TensorProduct.lmul' R t = 1
|
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
|
have : ∀ t : S ⊗[R] S, TensorProduct.lmul' R t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S := by
intro t
simp only [KaehlerDifferential.ideal, RingHom.mem_ker, map_sub, map_one,
sub_eq_zero, @eq_comm S 1]
|
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
this : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
⊢ (∃ e, IsIdempotentElem e ∧ KaehlerDifferential.ideal R S = Submodule.span (S ⊗[R] S) {e}) ↔
∃ t, (∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * t = 0) ∧ (TensorProduct.lmul' R) t = 1
|
242718edd9112c3c
|
Set.Countable.image
|
Mathlib/Data/Set/Countable.lean
|
theorem Countable.image {s : Set α} (hs : s.Countable) (f : α → β) : (f '' s).Countable
|
α : Type u
β : Type v
s : Set α
hs : s.Countable
f : α → β
this : Countable ↑s
⊢ (range fun x => f ↑x).Countable
|
apply countable_range
|
no goals
|
f0d955afdd57dc60
|
IsMulFreimanHom.prod
|
Mathlib/Combinatorics/Additive/FreimanHom.lean
|
@[to_additive]
lemma IsMulFreimanHom.prod (h₁ : IsMulFreimanHom n A₁ B₁ f₁) (h₂ : IsMulFreimanHom n A₂ B₂ f₂) :
IsMulFreimanHom n (A₁ ×ˢ A₂) (B₁ ×ˢ B₂) (Prod.map f₁ f₂) where
mapsTo := h₁.mapsTo.prodMap h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h
|
α₁ : Type u_5
α₂ : Type u_6
β₁ : Type u_7
β₂ : Type u_8
inst✝³ : CommMonoid α₁
inst✝² : CommMonoid α₂
inst✝¹ : CommMonoid β₁
inst✝ : CommMonoid β₂
A₁ : Set α₁
A₂ : Set α₂
B₁ : Set β₁
B₂ : Set β₂
f₁ : α₁ → β₁
f₂ : α₂ → β₂
n : ℕ
h₁ : IsMulFreimanHom n A₁ B₁ f₁
h₂ : IsMulFreimanHom n A₂ B₂ f₂
s t : Multiset (α₁ × α₂)
hs : s.card = n
ht : t.card = n
hsA : (∀ (a : α₁) (b : α₂), (a, b) ∈ s → a ∈ A₁) ∧ ∀ (a : α₁) (b : α₂), (a, b) ∈ s → b ∈ A₂
htA : (∀ (a : α₁) (b : α₂), (a, b) ∈ t → a ∈ A₁) ∧ ∀ (a : α₁) (b : α₂), (a, b) ∈ t → b ∈ A₂
h : (map Prod.fst s).prod = (map Prod.fst t).prod ∧ (map Prod.snd s).prod = (map Prod.snd t).prod
⊢ ∀ ⦃x : α₂⦄, x ∈ map Prod.snd t → x ∈ A₂
|
simpa [@forall_swap α₁] using htA.2
|
no goals
|
33dda5fafc4888d5
|
WeierstrassCurve.variableChange_id
|
Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean
|
lemma variableChange_id : W.variableChange VariableChange.id = W
|
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ { a₁ := 1 * (W.a₁ + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),
a₂ :=
1 ^ 2 *
(W.a₂ - { u := 1, r := 0, s := 0, t := 0 }.s * W.a₁ + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -
{ u := 1, r := 0, s := 0, t := 0 }.s ^ 2),
a₃ := 1 ^ 3 * (W.a₃ + { u := 1, r := 0, s := 0, t := 0 }.r * W.a₁ + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),
a₄ :=
1 ^ 4 *
(W.a₄ - { u := 1, r := 0, s := 0, t := 0 }.s * W.a₃ + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a₂ -
({ u := 1, r := 0, s := 0, t := 0 }.t +
{ u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *
W.a₁ +
3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -
2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),
a₆ :=
1 ^ 6 *
(W.a₆ + { u := 1, r := 0, s := 0, t := 0 }.r * W.a₄ + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a₂ +
{ u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -
{ u := 1, r := 0, s := 0, t := 0 }.t * W.a₃ -
{ u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -
{ u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a₁) } =
W
|
ext <;> (dsimp only; ring1)
|
no goals
|
be30df7f818a1ddd
|
AlgebraicTopology.DoldKan.HigherFacesVanish.induction
|
Mathlib/AlgebraicTopology/DoldKan/Faces.lean
|
theorem induction {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ) :
HigherFacesVanish (q + 1) (φ ≫ (𝟙 _ + Hσ q).f (n + 1))
|
case neg
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : ¬m + 1 < q
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ ≫ X.δ j.succ
|
rw [X.δ_comp_σ_of_gt', j.pred_succ]
|
case neg
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : ¬m + 1 < q
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.δ j ≫ X.σ (⟨a, ⋯⟩.castLT ⋯)
case neg
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : ¬m + 1 < q
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
⊢ ⟨a, ⋯⟩.succ < j.succ
case neg
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : ¬m + 1 < q
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
⊢ ⟨a, ⋯⟩.succ < j.succ
|
69ef2d77296be0d2
|
minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow
|
Mathlib/FieldTheory/SeparableDegree.lean
|
theorem natSepDegree_eq_one_iff_eq_X_sub_C_pow : (minpoly F x).natSepDegree = 1 ↔
∃ n : ℕ, (minpoly F x).map (algebraMap F E) = (X - C x) ^ q ^ n
|
case refine_1.intro.intro
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Ring E
inst✝¹ : IsDomain E
inst✝ : Algebra F E
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
|
have hx := congr_arg (Polynomial.aeval x) h.symm
|
case refine_1.intro.intro
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Ring E
inst✝¹ : IsDomain E
inst✝ : Algebra F E
q : ℕ
hF : ExpChar F q
x : E
this✝ : ExpChar E q
this : ExpChar E[X] q
h✝ : (minpoly F x).natSepDegree = 1
n : ℕ
y : F
h : minpoly F x = X ^ q ^ n - C y
hx : (Polynomial.aeval x) (X ^ q ^ n - C y) = (Polynomial.aeval x) (minpoly F x)
⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n
|
731ae2c59cbb171a
|
Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
|
Mathlib/LinearAlgebra/Dimension/Finite.lean
|
theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
{t : Finset M} (h : finrank R M + 1 < t.card) :
∃ f : M → R, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0
|
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun x => x - x₀, inj' := ⋯ }
t' : Finset M := Finset.map shift (t.erase x₀)
h' : finrank R M < #t'
g : M → R
gsum : ∑ e ∈ t', g e • e = 0
f : M → R := fun z => if z = x₀ then -∑ z ∈ t.erase x₀, g (z - x₀) else g (z - x₀)
x₁ : M
x₁_mem' : x₁ ∈ t.erase x₀
x₁_mem : shift x₁ ∈ t'
nz : g (shift x₁) ≠ 0
this : x₁ ≠ x₀ ∧ x₁ ∈ t
⊢ x₁ ∈ t ∧ f x₁ ≠ 0
|
simpa only [f, Embedding.coeFn_mk, sub_add_cancel, this.2, true_and, if_neg this.1]
|
no goals
|
9ccb45a24efab47f
|
Submonoid.LocalizationWithZeroMap.leftCancelMulZero_of_le_isLeftRegular
|
Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean
|
theorem leftCancelMulZero_of_le_isLeftRegular
(f : LocalizationWithZeroMap S N) [IsLeftCancelMulZero M]
(h : ∀ ⦃x⦄, x ∈ S → IsLeftRegular x) : IsLeftCancelMulZero N
|
M : Type u_1
inst✝² : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoidWithZero N
f : S.LocalizationWithZeroMap N
inst✝ : IsLeftCancelMulZero M
h : ∀ ⦃x : M⦄, x ∈ S → IsLeftRegular x
fl : S.LocalizationMap N := f.toLocalizationMap
g : M →* N := f.toMap
a z w : N
ha : a ≠ 0
hazw : a * z = a * w
b : M × ↥S
hb : a * fl.toMap ↑b.2 = fl.toMap b.1
x : M × ↥S
hx : z * fl.toMap ↑x.2 = fl.toMap x.1
y : M × ↥S
hy : w * fl.toMap ↑y.2 = fl.toMap y.1
b1ne0 : b.1 ≠ 0
⊢ a * g ↑b.2 * (g ↑x.2 * (w * g ↑y.2)) = a * w * g ↑b.2 * (g ↑x.2 * g ↑y.2)
|
rw [← mul_assoc, ← mul_assoc _ w, mul_comm _ w, mul_assoc w, mul_assoc,
← mul_assoc w, ← mul_assoc w, mul_comm w]
|
no goals
|
395e8d90995bfe86
|
Ring.DirectLimit.lift_unique
|
Mathlib/Algebra/Colimit/Ring.lean
|
theorem lift_unique (F : DirectLimit G f →+* P) (x) :
F x = lift G f P (fun i ↦ F.comp <| of G f i) (fun i j hij x ↦ by simp) x
|
ι : Type u_1
inst✝² : Preorder ι
G : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
P : Type u_3
inst✝ : CommRing P
F : DirectLimit G f →+* P
x : FreeCommRing ((i : ι) × G i)
⊢ ∀ (x y : FreeCommRing ((i : ι) × G i)),
F
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, f i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
(∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
(∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
x) =
(lift G f P (fun i => F.comp (of G f i)) ⋯)
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, f i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
(∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
(∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
x) →
F
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, f i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
(∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
(∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
y) =
(lift G f P (fun i => F.comp (of G f i)) ⋯)
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, f i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
(∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
(∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
y) →
F
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, f i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
(∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
(∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
(x * y)) =
(lift G f P (fun i => F.comp (of G f i)) ⋯)
((Ideal.Quotient.mk
(Ideal.span
{a |
(∃ i j, ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, f i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
(∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
(∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
(x * y))
|
simp+contextual
|
no goals
|
051b292ba4e666cf
|
Mon_.mul_braiding
|
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
theorem mul_braiding {X Y : Mon_ C} :
(X ⊗ Y).mul ≫ (β_ X.X Y.X).hom = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (Y ⊗ X).mul
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : SymmetricCategory C
X Y : Mon_ C
⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫
((((((Y.X ⊗ X.X) ◁ (β_ X.X Y.X).hom ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫
Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫
Y.X ◁ (α_ Y.X X.X X.X).hom) ≫
(α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫
(Y.mul ⊗ X.mul) =
((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫
(α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫
Y.X ◁ (α_ X.X Y.X X.X).inv ≫
Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)
|
slice_lhs 1 2 =>
rw [← tensorHom_def]
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : SymmetricCategory C
X Y : Mon_ C
⊢ (((((((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫
Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫
Y.X ◁ (α_ Y.X X.X X.X).hom) ≫
(α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫
(Y.mul ⊗ X.mul) =
((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫
(α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫
Y.X ◁ (α_ X.X Y.X X.X).inv ≫
Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)
|
6be9ea682fb53a86
|
pairwise_coprime_iff_coprime_prod
|
Mathlib/RingTheory/Coprime/Lemmas.lean
|
theorem pairwise_coprime_iff_coprime_prod [DecidableEq I] :
Pairwise (IsCoprime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsCoprime (s i) (∏ j ∈ t \ {i}, s j)
|
case refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t : Finset I
inst✝ : DecidableEq I
hp : Pairwise (IsCoprime on fun i => s ↑i)
i : I
hi : i ∈ t
j : I
hj : j ∈ t ∧ ¬j = i
⊢ IsCoprime (s i) (s j)
|
obtain ⟨hj, ji⟩ := hj
|
case refine_1.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t : Finset I
inst✝ : DecidableEq I
hp : Pairwise (IsCoprime on fun i => s ↑i)
i : I
hi : i ∈ t
j : I
hj : j ∈ t
ji : ¬j = i
⊢ IsCoprime (s i) (s j)
|
d6dd92ae2640356e
|
WeierstrassCurve.Jacobian.Y_ne_negY_of_Y_ne
|
Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
|
lemma Y_ne_negY_of_Y_ne [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
(hQ : W'.Equation Q) (hx : P x * Q z ^ 2 = Q x * P z ^ 2) (hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3) :
P y ≠ W'.negY P
|
R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3
⊢ P y ≠ W'.negY P
|
have hy' : P y * Q z ^ 3 - W'.negY Q * P z ^ 3 = 0 :=
(mul_eq_zero.mp <| Y_sub_Y_mul_Y_sub_negY hP hQ hx).resolve_left <| sub_ne_zero_of_ne hy
|
R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3
hy' : P y * Q z ^ 3 - W'.negY Q * P z ^ 3 = 0
⊢ P y ≠ W'.negY P
|
1dce46f4b895ab5b
|
linearIndependent_iff_card_eq_finrank_span
|
Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
|
theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K
|
case mp
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
ι : Type u_2
inst✝ : Fintype ι
b : ι → V
h : LinearIndependent K b
⊢ Fintype.card ι = Set.finrank K (range b)
|
exact (finrank_span_eq_card h).symm
|
no goals
|
911125b85a6b1f1b
|
RCLike.nnnorm_natCast
|
Mathlib/Analysis/RCLike/Basic.lean
|
@[simp, rclike_simps, norm_cast] lemma nnnorm_natCast (n : ℕ) : ‖(n : K)‖₊ = n
|
K : Type u_1
inst✝ : RCLike K
n : ℕ
⊢ ‖↑n‖₊ = ↑n
|
simp [nnnorm]
|
no goals
|
20b378915ca882c2
|
PFun.mem_fix_iff
|
Mathlib/Data/PFun.lean
|
theorem mem_fix_iff {f : α →. β ⊕ α} {a : α} {b : β} :
b ∈ f.fix a ↔ Sum.inl b ∈ f a ∨ ∃ a', Sum.inr a' ∈ f a ∧ b ∈ f.fix a' :=
⟨fun h => by
let ⟨h₁, h₂⟩ := Part.mem_assert_iff.1 h
rw [WellFounded.fixFEq] at h₂
simp only [Part.mem_assert_iff] at h₂
obtain ⟨h₂, h₃⟩ := h₂
split at h₃
next e => simp only [Part.mem_some_iff] at h₃; subst b; exact Or.inl ⟨h₂, e⟩
next e => exact Or.inr ⟨_, ⟨_, e⟩, Part.mem_assert _ h₃⟩,
fun h => by
simp only [fix, Part.mem_assert_iff]
rcases h with (⟨h₁, h₂⟩ | ⟨a', h, h₃⟩)
· refine ⟨⟨_, fun y h' => ?_⟩, ?_⟩
· injection Part.mem_unique ⟨h₁, h₂⟩ h'
· rw [WellFounded.fixFEq]
-- Porting note: used to be simp [h₁, h₂]
apply Part.mem_assert h₁
split
next e =>
injection h₂.symm.trans e with h; simp [h]
next e =>
injection h₂.symm.trans e
· simp only [fix, Part.mem_assert_iff] at h₃
obtain ⟨h₃, h₄⟩ := h₃
refine ⟨⟨_, fun y h' => ?_⟩, ?_⟩
· injection Part.mem_unique h h' with e
exact e ▸ h₃
· obtain ⟨h₁, h₂⟩ := h
rw [WellFounded.fixFEq]
-- Porting note: used to be simp [h₁, h₂, h₄]
apply Part.mem_assert h₁
split
next e =>
injection h₂.symm.trans e
next e =>
injection h₂.symm.trans e; subst a'; exact h₄⟩
|
case inl.intro.refine_2
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a : α
b : β
h₁ : (f a).Dom
h₂ : (f a).get h₁ = Sum.inl b
⊢ b ∈
match e : (f a).get h₁ with
| Sum.inl b => Part.some b
| Sum.inr a' =>
(fun y p =>
WellFounded.fixF
(fun a IH =>
Part.assert (f a).Dom fun hf =>
match e : (f a).get hf with
| Sum.inl b => Part.some b
| Sum.inr a' => IH a' ⋯)
y ⋯)
a' ⋯
|
split
|
case inl.intro.refine_2.h_1
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a : α
b : β
h₁ : (f a).Dom
h₂ : (f a).get h₁ = Sum.inl b
b✝ : β
heq✝ : (f a).get h₁ = Sum.inl b✝
⊢ b ∈ Part.some b✝
case inl.intro.refine_2.h_2
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a : α
b : β
h₁ : (f a).Dom
h₂ : (f a).get h₁ = Sum.inl b
a'✝ : α
heq✝ : (f a).get h₁ = Sum.inr a'✝
⊢ b ∈
(fun y p =>
WellFounded.fixF
(fun a IH =>
Part.assert (f a).Dom fun hf =>
match e : (f a).get hf with
| Sum.inl b => Part.some b
| Sum.inr a' => IH a' ⋯)
y ⋯)
a'✝ ⋯
|
b0fcdf86c1a0e2fe
|
List.mem_permutationsAux_of_perm
|
Mathlib/Data/List/Permutation.lean
|
theorem mem_permutationsAux_of_perm :
∀ {ts is l : List α},
l ~ is ++ ts → (∃ (is' : _) (_ : is' ~ is), l = is' ++ ts) ∨ l ∈ permutationsAux ts is
|
case inr
α : Type u_1
t : α
ts is : List α
IH1 :
∀ (l : List α), l ~ t :: is ++ ts → (∃ is', ∃ (_ : is' ~ t :: is), l = is' ++ ts) ∨ l ∈ ts.permutationsAux (t :: is)
IH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is', ∃ (_ : is' ~ []), l = is' ++ is) ∨ l ∈ is.permutationsAux []
l : List α
p : l ~ is ++ t :: ts
m : l ∈ ts.permutationsAux (t :: is)
⊢ (∃ is', ∃ (_ : is' ~ is), l = is' ++ t :: ts) ∨
l ∈ ts.permutationsAux (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ is.permutations ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts
|
exact Or.inr (Or.inl m)
|
no goals
|
c51a06076c0dbbc1
|
Pell.xz_sub
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
theorem xz_sub {m n} (h : n ≤ m) :
xz a1 (m - n) = xz a1 m * xz a1 n - d a1 * yz a1 m * yz a1 n
|
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ xz a1 (m - n) = xz a1 m * xz a1 n - ↑(Pell.d a1) * yz a1 m * yz a1 n
|
rw [sub_eq_add_neg, ← mul_neg]
|
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ xz a1 (m - n) = xz a1 m * xz a1 n + ↑(Pell.d a1) * yz a1 m * -yz a1 n
|
8c1df321155b9108
|
OmegaCompletePartialOrder.ωSup_eq_of_isLUB
|
Mathlib/Order/OmegaCompletePartialOrder.lean
|
lemma ωSup_eq_of_isLUB {c : Chain α} {a : α} (h : IsLUB (Set.range c) a) : a = ωSup c
|
case right
α : Type u_2
inst✝ : OmegaCompletePartialOrder α
c : Chain α
a : α
h : (∀ (a_1 : ℕ), c a_1 ≤ a) ∧ ∀ ⦃a_1 : α⦄, (∀ (a : ℕ), c a ≤ a_1) → a ≤ a_1
⊢ ∀ (i : ℕ), c i ≤ a
|
apply h.1
|
no goals
|
05205937dddfa3ff
|
Finset.offDiag_card
|
Mathlib/Data/Finset/Prod.lean
|
theorem offDiag_card : (offDiag s).card = s.card * s.card - s.card :=
suffices (diag s).card + (offDiag s).card = s.card * s.card by rw [s.diag_card] at this; omega
by rw [← card_product, diag, offDiag]
conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun a => a.1 = a.2)]
|
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
⊢ #(filter (fun a => a.1 = a.2) (s ×ˢ s)) + #(filter (fun a => a.1 ≠ a.2) (s ×ˢ s)) = #(s ×ˢ s)
|
conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun a => a.1 = a.2)]
|
no goals
|
953ae524eef68703
|
WithCStarModule.prod_norm_le_norm_add
|
Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
|
lemma prod_norm_le_norm_add (x : C⋆ᵐᵒᵈ (E × F)) : ‖x‖ ≤ ‖x.1‖ + ‖x.2‖
|
A : Type u_1
inst✝⁹ : NonUnitalCStarAlgebra A
inst✝⁸ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : Module ℂ E
inst✝⁵ : SMul Aᵐᵒᵖ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : Module ℂ F
inst✝² : SMul Aᵐᵒᵖ F
inst✝¹ : CStarModule A E
inst✝ : CStarModule A F
x : C⋆ᵐᵒᵈ (E × F)
⊢ ‖⟪x.1, x.1⟫_A‖ + ‖⟪x.2, x.2⟫_A‖ = ‖x.1‖ ^ 2 + 0 + ‖x.2‖ ^ 2
|
simp [norm_sq_eq]
|
no goals
|
530d4c0602d88bbf
|
Real.rpow_sub_intCast
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
lemma rpow_sub_intCast {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n
|
x : ℝ
hx : x ≠ 0
y : ℝ
n : ℕ
⊢ x ^ (y - ↑n) = x ^ y / x ^ n
|
simpa using rpow_add_intCast hx y (-n)
|
no goals
|
d0fad410096f2fd8
|
Equiv.symm_trans_swap_trans
|
Mathlib/Logic/Equiv/Basic.lean
|
theorem symm_trans_swap_trans [DecidableEq β] (a b : α) (e : α ≃ β) :
(e.symm.trans (swap a b)).trans e = swap (e a) (e b) :=
Equiv.ext fun x => by
have : ∀ a, e.symm x = a ↔ x = e a := fun a => by
rw [@eq_comm _ (e.symm x)]
constructor <;> intros <;> simp_all
simp only [trans_apply, swap_apply_def, this]
split_ifs <;> simp
|
α : Sort u_1
β : Sort u_4
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
a b : α
e : α ≃ β
x : β
⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x
|
have : ∀ a, e.symm x = a ↔ x = e a := fun a => by
rw [@eq_comm _ (e.symm x)]
constructor <;> intros <;> simp_all
|
α : Sort u_1
β : Sort u_4
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
a b : α
e : α ≃ β
x : β
this : ∀ (a : α), e.symm x = a ↔ x = e a
⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x
|
48f02afc5d1f6bd8
|
AlternatingMap.domCoprod.summand_eq_zero_of_smul_invariant
|
Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
|
theorem domCoprod.summand_eq_zero_of_smul_invariant (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁)
(b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : ιa ⊕ ιb → Mᵢ}
{i j : ιa ⊕ ιb} (hv : v i = v j) (hij : i ≠ j) :
swap i j • σ = σ → domCoprod.summand a b σ v = 0
|
case h.e'_2.h.e'_10
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
i' j' : ιb
hv : v (σ (Sum.inr i')) = v (σ (Sum.inr j'))
hij : σ (Sum.inr i') ≠ σ (Sum.inr j')
hσ✝ : swap (σ (Sum.inr i')) (σ (Sum.inr j')) • Quotient.mk'' σ = Quotient.mk'' σ
⊢ (↑b fun i => v (σ (Sum.inr i))) = 0
|
exact AlternatingMap.map_eq_zero_of_eq _ _ hv fun hij' => hij (hij' ▸ rfl)
|
no goals
|
3328e6875d9d0122
|
Metric.subsingleton_closedBall
|
Mathlib/Topology/MetricSpace/Defs.lean
|
theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton
|
case inr
γ : Type w
inst✝ : MetricSpace γ
x : γ
hr : 0 ≤ 0
⊢ {x}.Subsingleton
|
exact subsingleton_singleton
|
no goals
|
c66d4373aa3b40df
|
Vector.mem_set
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean
|
theorem mem_set (v : Vector α n) (i : Nat) (hi : i < n) (a : α) :
a ∈ v.set i a hi
|
α : Type u_1
n : Nat
v : Vector α n
i : Nat
hi : i < n
a : α
⊢ ∃ i_1 h, (v.set i a hi)[i_1] = a
|
exact ⟨i, (by simpa using hi), by simp⟩
|
no goals
|
78ca1d27ff802219
|
FractionalIdeal.mem_zero_iff
|
Mathlib/RingTheory/FractionalIdeal/Basic.lean
|
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 :=
⟨fun ⟨x', x'_mem_zero, x'_eq_x⟩ => by
have x'_eq_zero : x' = 0 := x'_mem_zero
simp [x'_eq_x.symm, x'_eq_zero], fun hx => ⟨0, rfl, by simp [hx]⟩⟩
|
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
x : P
x✝ : x ∈ 0
x' : R
x'_mem_zero : x' ∈ ↑0
x'_eq_x : (Algebra.linearMap R P) x' = x
x'_eq_zero : x' = 0
⊢ x = 0
|
simp [x'_eq_x.symm, x'_eq_zero]
|
no goals
|
0682cefbc56573ad
|
List.dropInfix?_go_eq_some_iff
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
p = acc.reverse ++ p' ∧
-- `i` is an infix up to `==`
(∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
-- and there is no shorter prefix for which that is the case
(∀ p'' i'' s'', l = p'' ++ i'' ++ s'' → i'' == i → p''.length ≥ p'.length)
|
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
a' : List α
h : (a :: (a' ++ s)).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), a :: (a' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ [].length
h₂ : (a :: a' == i) = true
⊢ ∃ p',
acc.reverse ++ [] = (a :: acc).reverse ++ p' ∧
(∃ i', a' ++ s = p' ++ i' ++ s ∧ (i' == i) = true) ∧
∀ (p'' i'' s'' : List α), a' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
|
rw [← cons_append] at h
|
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
a' : List α
h : (a :: a' ++ s).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), a :: (a' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ [].length
h₂ : (a :: a' == i) = true
⊢ ∃ p',
acc.reverse ++ [] = (a :: acc).reverse ++ p' ∧
(∃ i', a' ++ s = p' ++ i' ++ s ∧ (i' == i) = true) ∧
∀ (p'' i'' s'' : List α), a' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
|
2a30116d5cc9db54
|
OrthogonalFamily.range_linearIsometry
|
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
theorem range_linearIsometry [∀ i, CompleteSpace (G i)] :
LinearMap.range hV.linearIsometry.toLinearMap =
(⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure
|
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ LinearMap.range hV.linearIsometry.toLinearMap = (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure
|
refine le_antisymm ?_ ?_
|
case refine_1
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ LinearMap.range hV.linearIsometry.toLinearMap ≤ (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure
case refine_2
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure ≤ LinearMap.range hV.linearIsometry.toLinearMap
|
e67b0f768537d5e1
|
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite
|
Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
|
theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite [IsFiniteMeasure μ]
(h𝒜 : IsSetAlgebra 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) : μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms
|
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ ∑ n ∈ Finset.range (N + 1), ε / (2 * (↑N + 1)) ≤ ε / 2
|
simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul,
Nat.cast_add, Nat.cast_one]
|
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (↑N + 1) * (ε / (2 * (↑N + 1))) ≤ ε / 2
|
92d6fe7992cc346b
|
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