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
|
|---|---|---|---|---|---|---|
integral_withDensity_eq_integral_smul
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) :
∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ
|
case pos.refine_2.hf
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x => ↑(f x))
u u' : X → E
a✝ : Disjoint (support u) (support u')
u_int : Integrable u (μ.withDensity fun x => ↑(f x))
u'_int : Integrable u' (μ.withDensity fun x => ↑(f x))
h : (∫ (x : X), u x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • u x ∂μ
h' : (∫ (x : X), u' x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • u' x ∂μ
⊢ Integrable (fun x => f x • u x) μ
|
exact (integrable_withDensity_iff_integrable_smul f_meas).1 u_int
|
no goals
|
cd239c0816bb7a0c
|
Field.primitive_element_inf_aux
|
Mathlib/FieldTheory/PrimitiveElement.lean
|
theorem primitive_element_inf_aux [Algebra.IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algebraMap E (Polynomial.map ιFE g).SplittingField
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
((Polynomial.map (algebraMap F ↥F⟮γ⟯) f).comp (C (AdjoinSimple.gen F γ) - C ⟨(algebraMap F E) c, ⋯⟩ * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd ((Polynomial.map ιFE f).comp (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : h.Separable
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : (Polynomial.map ιFE g).SplittingField
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (ιEE'.comp ιFE) c
⊢ x = ιEE' β
|
by_contra a
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algebraMap E (Polynomial.map ιFE g).SplittingField
c : F
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
EuclideanDomain.gcd
((Polynomial.map (algebraMap F ↥F⟮γ⟯) f).comp (C (AdjoinSimple.gen F γ) - C ⟨(algebraMap F E) c, ⋯⟩ * X))
(Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd ((Polynomial.map ιFE f).comp (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : h.Separable
h_root : eval β h = 0
h_splits : Splits ιEE' h
x : (Polynomial.map ιFE g).SplittingField
hx : eval₂ ιEE' x h = 0
hc : -(ιEE' γ - ιEE' (ιFE c) * x - ιEE' α) / (x - ιEE' β) ≠ (ιEE'.comp ιFE) c
a : ¬x = ιEE' β
⊢ False
|
c759c025da469f0d
|
List.append_cancel_left_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/BasicAux.lean
|
theorem append_cancel_left_eq (as bs cs : List α) : (as ++ bs = as ++ cs) = (bs = cs)
|
α : Type u_1
as bs cs : List α
⊢ (as ++ bs = as ++ cs) = (bs = cs)
|
apply propext
|
case a
α : Type u_1
as bs cs : List α
⊢ as ++ bs = as ++ cs ↔ bs = cs
|
e65cb60e56da7011
|
Finset.smul_stabilizer_of_no_doubling_aux
|
Mathlib/Combinatorics/Additive/VerySmallDoubling.lean
|
@[to_additive]
private lemma smul_stabilizer_of_no_doubling_aux (hA : #(A * A) ≤ #A) (ha : a ∈ A) :
a •> (stabilizer G A : Set G) = A ∧ (stabilizer G A : Set G) <• a = A
|
case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A : Finset G
a : G
hA : #(A * A) ≤ #A
ha : a ∈ A
smul_A : ∀ {a : G}, a ∈ A → a •> A = A * A
A_smul : ∀ {a : G}, a ∈ A → A <• a = A * A
smul_A_eq_A_smul : ∀ {a : G}, a ∈ A → a •> A = A <• a
mul_mem_A_comm : ∀ {x a : G}, a ∈ A → (x * a ∈ A ↔ a * x ∈ A)
H : Subgroup G := stabilizer G A
inv_smul_A : ∀ {a : G}, a ∈ A → a⁻¹ •> ↑A = ↑H
⊢ a⁻¹ •> (↑A <• a) = ↑A
|
norm_cast
|
case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A : Finset G
a : G
hA : #(A * A) ≤ #A
ha : a ∈ A
smul_A : ∀ {a : G}, a ∈ A → a •> A = A * A
A_smul : ∀ {a : G}, a ∈ A → A <• a = A * A
smul_A_eq_A_smul : ∀ {a : G}, a ∈ A → a •> A = A <• a
mul_mem_A_comm : ∀ {x a : G}, a ∈ A → (x * a ∈ A ↔ a * x ∈ A)
H : Subgroup G := stabilizer G A
inv_smul_A : ∀ {a : G}, a ∈ A → a⁻¹ •> ↑A = ↑H
⊢ a⁻¹ •> (A <• a) = A
|
edbd4fa4141174b5
|
Std.Sat.AIG.denote_mkIfCached
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/If.lean
|
theorem denote_mkIfCached {aig : AIG α} {input : TernaryInput aig} :
⟦aig.mkIfCached input, assign⟧
=
if ⟦aig, input.discr, assign⟧ then ⟦aig, input.lhs, assign⟧ else ⟦aig, input.rhs, assign⟧
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.TernaryInput
⊢ ⟦assign, aig.mkIfCached input⟧ =
(⟦assign, { aig := aig, ref := input.discr }⟧ && ⟦assign, { aig := aig, ref := input.lhs }⟧ ||
!⟦assign, { aig := aig, ref := input.discr }⟧ && ⟦assign, { aig := aig, ref := input.rhs }⟧)
|
unfold mkIfCached
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.TernaryInput
⊢ ⟦assign,
let res := aig.mkAndCached { lhs := input.discr, rhs := input.lhs };
let aig_1 := res.aig;
let lhsRef := res.ref;
let input_1 := input.cast ⋯;
let res_1 := aig_1.mkNotCached input_1.discr;
let aig_2 := res_1.aig;
let notDiscr := res_1.ref;
let input_2 := input_1.cast ⋯;
let res_2 := aig_2.mkAndCached { lhs := notDiscr, rhs := input_2.rhs };
let aig_3 := res_2.aig;
let rhsRef := res_2.ref;
let lhsRef := lhsRef.cast ⋯;
aig_3.mkOrCached { lhs := lhsRef, rhs := rhsRef }⟧ =
(⟦assign, { aig := aig, ref := input.discr }⟧ && ⟦assign, { aig := aig, ref := input.lhs }⟧ ||
!⟦assign, { aig := aig, ref := input.discr }⟧ && ⟦assign, { aig := aig, ref := input.rhs }⟧)
|
d6ac530231c33d37
|
Polynomial.natSepDegree_eq_of_splits
|
Mathlib/FieldTheory/SeparableDegree.lean
|
theorem natSepDegree_eq_of_splits [DecidableEq E] (h : f.Splits (algebraMap F E)) :
f.natSepDegree = (f.aroots E).toFinset.card
|
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
f : F[X]
inst✝ : DecidableEq E
h : Splits (algebraMap F E) f
⊢ f.natSepDegree = (f.aroots E).toFinset.card
|
rw [aroots, ← (SplittingField.lift f h).comp_algebraMap, ← map_map,
roots_map _ ((splits_id_iff_splits _).mpr <| SplittingField.splits f),
Multiset.toFinset_map, Finset.card_image_of_injective _ (RingHom.injective _), natSepDegree]
|
no goals
|
85af115cea509baa
|
div_le_egauge_closedBall
|
Mathlib/Analysis/Convex/EGauge.lean
|
lemma div_le_egauge_closedBall (r : ℝ≥0) (x : E) : ‖x‖ₑ / r ≤ egauge 𝕜 (closedBall 0 r) x
|
case intro.intro
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖(fun x => c • x) y‖ₑ / ↑r ≤ ‖c‖ₑ
|
rw [enorm_smul]
|
case intro.intro
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖c‖ₑ * ‖y‖ₑ / ↑r ≤ ‖c‖ₑ
|
62824bbc48d3b01b
|
Finset.Nat.antidiagonal_filter_fst_le_of_le
|
Mathlib/Data/Finset/NatAntidiagonal.lean
|
@[simp] lemma antidiagonal_filter_fst_le_of_le {n k : ℕ} (h : k ≤ n) :
(antidiagonal n).filter (fun a ↦ a.fst ≤ k) = (antidiagonal k).map
(Embedding.prodMap (Embedding.refl ℕ) ⟨_, add_left_injective (n - k)⟩)
|
n k : ℕ
h : k ≤ n
aux₁ : (fun a => a.1 ≤ k) = (fun a => a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm
aux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j
⊢ filter (fun a => a.1 ≤ k) (map (Equiv.prodComm ℕ ℕ).toEmbedding (antidiagonal n)) =
map ((Embedding.refl ℕ).prodMap { toFun := fun x => x + (n - k), inj' := ⋯ }) (antidiagonal k)
|
simp_rw [aux₁, ← map_filter, antidiagonal_filter_snd_le_of_le h, map_map]
|
n k : ℕ
h : k ≤ n
aux₁ : (fun a => a.1 ≤ k) = (fun a => a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm
aux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j
⊢ map (({ toFun := fun x => x + (n - k), inj' := ⋯ }.prodMap (Embedding.refl ℕ)).trans (Equiv.prodComm ℕ ℕ).toEmbedding)
(antidiagonal k) =
map ((Embedding.refl ℕ).prodMap { toFun := fun x => x + (n - k), inj' := ⋯ }) (antidiagonal k)
|
25436f958f45eaed
|
CategoryTheory.Functor.final_of_final_comp
|
Mathlib/CategoryTheory/Limits/Final.lean
|
theorem final_of_final_comp [hF : Final F] [hFG : Final (F ⋙ G)] : Final G
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
hFG : (F ⋙ G).Final
s₁ : C ≌ AsSmall C := AsSmall.equiv
s₂ : D ≌ AsSmall D := AsSmall.equiv
hF :
∀ (G : AsSmall D ⥤ Type (max (max (max (max (max u₁ u₂) u₃) v₁) v₂) v₃)),
IsIso (colimit.pre G (s₁.inverse ⋙ F ⋙ s₂.functor))
s₃ : E ≌ AsSmall E := AsSmall.equiv
_i : s₁.inverse ⋙ (F ⋙ G) ⋙ s₃.functor ≅ (s₁.inverse ⋙ F ⋙ s₂.functor) ⋙ s₂.inverse ⋙ G ⋙ s₃.functor :=
isoWhiskerLeft (s₁.inverse ⋙ F) (isoWhiskerRight s₂.unitIso (G ⋙ s₃.functor))
⊢ ∀ (G_1 : AsSmall E ⥤ Type (max (max (max (max (max u₁ u₂) u₃) v₁) v₂) v₃)),
IsIso (colimit.pre G_1 (s₂.inverse ⋙ G ⋙ s₃.functor))
|
rw [final_iff_comp_equivalence (F ⋙ G) s₃.functor, final_iff_equivalence_comp s₁.inverse,
final_natIso_iff _i, final_iff_isIso_colimit_pre] at hFG
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
s₁ : C ≌ AsSmall C := AsSmall.equiv
s₂ : D ≌ AsSmall D := AsSmall.equiv
hF :
∀ (G : AsSmall D ⥤ Type (max (max (max (max (max u₁ u₂) u₃) v₁) v₂) v₃)),
IsIso (colimit.pre G (s₁.inverse ⋙ F ⋙ s₂.functor))
s₃ : E ≌ AsSmall E := AsSmall.equiv
hFG :
∀ (G_1 : AsSmall E ⥤ Type (max (max (max (max (max u₁ u₂) u₃) v₁) v₂) v₃)),
IsIso (colimit.pre G_1 ((s₁.inverse ⋙ F ⋙ s₂.functor) ⋙ s₂.inverse ⋙ G ⋙ s₃.functor))
_i : s₁.inverse ⋙ (F ⋙ G) ⋙ s₃.functor ≅ (s₁.inverse ⋙ F ⋙ s₂.functor) ⋙ s₂.inverse ⋙ G ⋙ s₃.functor :=
isoWhiskerLeft (s₁.inverse ⋙ F) (isoWhiskerRight s₂.unitIso (G ⋙ s₃.functor))
⊢ ∀ (G_1 : AsSmall E ⥤ Type (max (max (max (max (max u₁ u₂) u₃) v₁) v₂) v₃)),
IsIso (colimit.pre G_1 (s₂.inverse ⋙ G ⋙ s₃.functor))
|
dddeec27ae17bdc0
|
Pell.IsFundamental.exists_of_not_isSquare
|
Mathlib/NumberTheory/Pell.lean
|
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
∃ a : Solution₁ d, IsFundamental a
|
case intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
⊢ ∃ a, IsFundamental a
|
obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
|
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ x', 1 < x' ∧ ∃ y', 0 < y' ∧ ↑x' ^ 2 - d * y' ^ 2 = 1
x₁ : ℕ := Nat.find P
hx : 1 < Nat.find P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ : ↑(Nat.find P) ^ 2 - d * y₁ ^ 2 = 1
⊢ ∃ a, IsFundamental a
|
c0439655d6c2a144
|
Turing.TM2to1.tr_respects_aux₁
|
Mathlib/Computability/TuringMachine.lean
|
theorem tr_respects_aux₁ {k} (o q v) {S : List (Γ k)} {L : ListBlank (∀ k, Option (Γ k))}
(hL : L.map (proj k) = ListBlank.mk (S.map some).reverse) (n) (H : n ≤ S.length) :
Reaches₀ (TM1.step (tr M)) ⟨some (go k o q), v, Tape.mk' ∅ (addBottom L)⟩
⟨some (go k o q), v, (Tape.move Dir.right)^[n] (Tape.mk' ∅ (addBottom L))⟩
|
case succ
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
k : K
o : StAct K Γ σ k
q : TM2.Stmt Γ Λ σ
v : σ
S : List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ListBlank.map (proj k) L = ListBlank.mk (List.map some S).reverse
n : ℕ
IH :
n ≤ S.length →
Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom L) }
{ l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' ∅ (addBottom L)) }
H : n + 1 ≤ S.length
⊢ some
(bif (some S.reverse[n]).isNone then
TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' ∅ (addBottom L)))
else
{ l := some (go k o q), var := v,
Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' ∅ (addBottom L))) }) =
some
{ l := some (go k o q), var := v,
Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' ∅ (addBottom L))) }
|
rfl
|
no goals
|
89a4408472c01123
|
MeasureTheory.condExp_bot'
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
|
theorem condExp_bot' [hμ : NeZero μ] (f : α → E) :
μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ
|
case pos.intro
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hμ : NeZero μ
f : α → E
hμ_finite : IsFiniteMeasure μ
h_meas : StronglyMeasurable (μ[f|⊥])
c : E
h_eq : μ[f|⊥] = fun x => c
⊢ (fun x => c) = fun x => (μ Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂μ
|
have h_integral : ∫ x, (μ[f|⊥]) x ∂μ = ∫ x, f x ∂μ := integral_condExp bot_le
|
case pos.intro
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hμ : NeZero μ
f : α → E
hμ_finite : IsFiniteMeasure μ
h_meas : StronglyMeasurable (μ[f|⊥])
c : E
h_eq : μ[f|⊥] = fun x => c
h_integral : ∫ (x : α), (μ[f|⊥]) x ∂μ = ∫ (x : α), f x ∂μ
⊢ (fun x => c) = fun x => (μ Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂μ
|
46233d8f5b2c46f3
|
WithTop.isGLB_sInf'
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
theorem isGLB_sInf' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)}
(hs : BddBelow s) : IsGLB s (sInf s)
|
case neg.some.intro.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
hb : none ∈ lowerBounds s
⊢ False
|
apply h
|
case neg.some.intro.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
h : ¬s ⊆ {⊤}
a : β
ha : Option.some a ∈ s
hb : none ∈ lowerBounds s
⊢ s ⊆ {⊤}
|
ef7c47c8bf1f8b74
|
Array.unzip_zipIdx_eq_prod
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Range.lean
|
theorem unzip_zipIdx_eq_prod (l : Array α) {n : Nat} :
(l.zipIdx n).unzip = (l, range' n l.size)
|
α : Type u_1
l : Array α
n : Nat
⊢ (l.zipIdx n).unzip = (l, range' n l.size)
|
simp only [zipIdx_eq_zip_range', unzip_zip, size_range']
|
no goals
|
fc318d30133c04de
|
List.rtakeWhile_eq_nil_iff
|
Mathlib/Data/List/DropRight.lean
|
theorem rtakeWhile_eq_nil_iff : rtakeWhile p l = [] ↔ ∀ hl : l ≠ [], ¬p (l.getLast hl)
|
case append_singleton.refine_2
α : Type u_1
p : α → Bool
l✝ l : List α
a : α
a✝ : rtakeWhile p l = [] ↔ ∀ (hl : l ≠ []), ¬p (l.getLast hl) = true
h : ¬l ++ [a] = [] → ¬p a = true
⊢ (match p a with
| true => a :: takeWhile p l.reverse
| false => []).reverse =
[]
|
simp [h]
|
no goals
|
76d022d188e44678
|
ZMod.val_neg_of_ne_zero
|
Mathlib/Data/ZMod/Basic.lean
|
theorem val_neg_of_ne_zero {n : ℕ} [nz : NeZero n] (a : ZMod n) [na : NeZero a] :
(- a).val = n - a.val
|
n : ℕ
nz : NeZero n
a : ZMod n
na : NeZero a
⊢ (-a).val = n - a.val
|
simp_all [neg_val a, na.out]
|
no goals
|
81f31a342a7ea1f8
|
CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker_aux
|
Mathlib/RingTheory/Idempotents.lean
|
lemma CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker_aux
(h : ∀ x ∈ RingHom.ker f, IsNilpotent x)
{n} {e : Fin n → S} (he : CompleteOrthogonalIdempotents e) (he' : ∀ i, e i ∈ f.range) :
∃ e' : Fin n → R, CompleteOrthogonalIdempotents e' ∧ f ∘ e' = e
|
case inl
R : Type u_1
S : Type u_2
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
n : ℕ
e : Fin n → S
he : CompleteOrthogonalIdempotents e
he' : ∀ (i : Fin n), e i ∈ f.range
h✝ : Subsingleton R
⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
|
choose e' he' using he'
|
case inl
R : Type u_1
S : Type u_2
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
n : ℕ
e : Fin n → S
he : CompleteOrthogonalIdempotents e
h✝ : Subsingleton R
e' : Fin n → R
he' : ∀ (i : Fin n), f (e' i) = e i
⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
|
25d0a2b72a62cc0b
|
Array.getElem_swap_right
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem getElem_swap_right (a : Array α) {i j : Nat} {hi hj} :
(a.swap i j hi hj)[j]'(by simpa using hj) = a[i]
|
α : Type u_1
a : Array α
i j : Nat
hi : i < a.size
hj : j < a.size
⊢ (a.swap i j hi hj)[j] = a[i]
|
simp [swap_def, getElem_set]
|
no goals
|
08a360bc701e5699
|
List.dropInfix?_go_eq_some_iff
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
p = acc.reverse ++ p' ∧
-- `i` is an infix up to `==`
(∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
-- and there is no shorter prefix for which that is the case
(∀ p'' i'' s'', l = p'' ++ i'' ++ s'' → i'' == i → p''.length ≥ p'.length)
|
case h_2.h_1.mp.intro.intro.intro.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
p' i' : List α
h₂✝ : (i' == i) = true
w : ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i'' s'' : List α
h : (i'' ++ s'').dropPrefix? i = none
h₂ : (i'' == i) = true
h₁ : i'' ++ s'' = a :: (p' ++ (i' ++ s))
this : (i'' ++ s'').dropPrefix? i = some s''
⊢ p'.length + 1 ≤ [].length
|
simp_all
|
no goals
|
2a30116d5cc9db54
|
MeasureTheory.Content.outerMeasure_caratheodory
|
Mathlib/MeasureTheory/Measure/Content.lean
|
theorem outerMeasure_caratheodory (A : Set G) :
MeasurableSet[μ.outerMeasure.caratheodory] A ↔
∀ U : Opens G, μ.outerMeasure (U ∩ A) + μ.outerMeasure (U \ A) ≤ μ.outerMeasure U
|
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
⊢ MeasurableSet A ↔
∀ (U : Set G) (hU : IsOpen U),
μ.outerMeasure (↑{ carrier := U, is_open' := hU } ∩ A) + μ.outerMeasure (↑{ carrier := U, is_open' := hU } \ A) ≤
μ.outerMeasure ↑{ carrier := U, is_open' := hU }
|
apply inducedOuterMeasure_caratheodory
|
case msU
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
⊢ ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
μ.innerContent { carrier := ⋃ i, f i, is_open' := ⋯ } ≤ ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := ⋯ }
case m_mono
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
⊢ ∀ ⦃s₁ s₂ : Set G⦄ (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂),
s₁ ⊆ s₂ → μ.innerContent { carrier := s₁, is_open' := hs₁ } ≤ μ.innerContent { carrier := s₂, is_open' := hs₂ }
case PU
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
⊢ ∀ ⦃f : ℕ → Set G⦄, (∀ (i : ℕ), IsOpen (f i)) → IsOpen (⋃ i, f i)
|
6934270734572b0d
|
Matrix.det_mul_comm
|
Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
|
theorem det_mul_comm (M N : Matrix m m R) : det (M * N) = det (N * M)
|
m : Type u_1
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
M N : Matrix m m R
⊢ (M * N).det = (N * M).det
|
rw [det_mul, det_mul, mul_comm]
|
no goals
|
f1ac106dfe851275
|
CFC.nnrpow_nnrpow_inv
|
Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Basic.lean
|
lemma nnrpow_nnrpow_inv (a : A) {x : ℝ≥0} (hx : x ≠ 0) (ha : 0 ≤ a
|
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ≥0 fun a => 0 ≤ a
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
hx : x ≠ 0
ha : autoParam (0 ≤ a) _auto✝
⊢ (a ^ x) ^ x⁻¹ = a
|
simp [mul_inv_cancel₀ hx, nnrpow_one _ ha]
|
no goals
|
a935765bd05916a2
|
MeasureTheory.GridLines.T_insert_le_T_lmarginal_singleton
|
Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
|
theorem T_insert_le_T_lmarginal_singleton [∀ i, SigmaFinite (μ i)] (hp₀ : 0 ≤ p) (s : Finset ι)
(hp : (s.card : ℝ) * p ≤ 1)
(i : ι) (hi : i ∉ s) {f : (∀ i, A i) → ℝ≥0∞} (hf : Measurable f) :
T μ p f (insert i s) ≤ T μ p (∫⋯∫⁻_{i}, f ∂μ) s
|
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
this : ∀ (t : A i), (∫⋯∫⁻_{i}, f ∂μ) (X t) = (∫⋯∫⁻_{i}, f ∂μ) x
⊢ ∫⁻ (t : A i), (∫⋯∫⁻_{i}, f ∂μ) x ^ p * (f (X t) ^ (1 - k * p) * ∏ j ∈ s, (∫⋯∫⁻_{j}, f ∂μ) (X t) ^ p) ∂μ i =
(∫⋯∫⁻_{i}, f ∂μ) x ^ p * ∫⁻ (t : A i), f (X t) ^ (1 - k * p) * ∏ j ∈ s, (∫⋯∫⁻_{j}, f ∂μ) (X t) ^ p ∂μ i
|
rw [lintegral_const_mul]
|
case hf
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
this : ∀ (t : A i), (∫⋯∫⁻_{i}, f ∂μ) (X t) = (∫⋯∫⁻_{i}, f ∂μ) x
⊢ Measurable fun t => f (X t) ^ (1 - k * p) * ∏ j ∈ s, (∫⋯∫⁻_{j}, f ∂μ) (X t) ^ p
|
0fcf193773af0ce8
|
Std.DHashMap.Internal.Raw₀.toListModel_eraseₘ
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean
|
theorem toListModel_eraseₘ [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α] {m : Raw₀ α β}
{a : α} (h : Raw.WFImp m.1) :
Perm (toListModel (m.eraseₘ a).1.buckets) (eraseKey a (toListModel m.1.buckets))
|
case isTrue
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
a : α
h : Raw.WFImp m.val
h✝ : m.containsₘ a = true
⊢ toListModel (m.eraseₘaux a).val.buckets ~ eraseKey a (toListModel m.val.buckets)
|
exact toListModel_eraseₘaux m a h
|
no goals
|
579b712bf51eaa66
|
isLindelof_of_countable_subcover
|
Mathlib/Topology/Compactness/Lindelof.lean
|
theorem isLindelof_of_countable_subcover
(h : ∀ {ι : Type u} (U : ι → Set X), (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) →
∃ t : Set ι, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i) :
IsLindelof s := fun f hf hfs ↦ by
contrapose! h
simp only [ClusterPt, not_neBot, ← disjoint_iff, SetCoe.forall',
(nhds_basis_opens _).disjoint_iff_left] at h
choose fsub U hU hUf using h
refine ⟨s, U, fun x ↦ (hU x).2, fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, (hU _).1 ⟩, ?_⟩
intro t ht h
have uinf := f.sets_of_superset (le_principal_iff.1 fsub) h
have uninf : ⋂ i ∈ t, (U i)ᶜ ∈ f := (countable_bInter_mem ht).mpr (fun _ _ ↦ hUf _)
rw [← compl_iUnion₂] at uninf
have uninf := compl_not_mem uninf
simp only [compl_compl] at uninf
contradiction
|
X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : CountableInterFilter f
fsub : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
⊢ ∃ ι U, (∀ (i : ι), IsOpen (U i)) ∧ s ⊆ ⋃ i, U i ∧ ∀ (t : Set ι), t.Countable → ¬s ⊆ ⋃ i ∈ t, U i
|
refine ⟨s, U, fun x ↦ (hU x).2, fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, (hU _).1 ⟩, ?_⟩
|
X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : CountableInterFilter f
fsub : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
⊢ ∀ (t : Set ↑s), t.Countable → ¬s ⊆ ⋃ i ∈ t, U i
|
55d21432f93098bb
|
Cardinal.mk_emptyCollection_iff
|
Mathlib/SetTheory/Cardinal/Basic.lean
|
theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : #s = 0 ↔ s = ∅
|
case mpr
α : Type u
⊢ #↑∅ = 0
|
exact mk_emptyCollection _
|
no goals
|
4ba248ce53138d76
|
Complex.countable_preimage_exp
|
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
|
theorem countable_preimage_exp {s : Set ℂ} : (exp ⁻¹' s).Countable ↔ s.Countable
|
s : Set ℂ
⊢ (cexp ⁻¹' s).Countable ↔ s.Countable
|
refine ⟨fun hs => ?_, fun hs => ?_⟩
|
case refine_1
s : Set ℂ
hs : (cexp ⁻¹' s).Countable
⊢ s.Countable
case refine_2
s : Set ℂ
hs : s.Countable
⊢ (cexp ⁻¹' s).Countable
|
e3fd0c8739ed6e00
|
LinearPMap.image_iff
|
Mathlib/LinearAlgebra/LinearPMap.lean
|
theorem image_iff {f : E →ₗ.[R] F} {x : E} {y : F} (hx : x ∈ f.domain) :
y = f ⟨x, hx⟩ ↔ (x, y) ∈ f.graph
|
case h
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
x : E
y : F
hx : x ∈ f.domain
h : y = ↑f ⟨x, hx⟩
⊢ ↑⟨x, hx⟩ = (x, y).1 ∧ ↑f ⟨x, hx⟩ = (x, y).2
|
simp [h]
|
no goals
|
e4b78bc1d49361b4
|
RCLike.conj_neg_I
|
Mathlib/Analysis/RCLike/Basic.lean
|
theorem conj_neg_I : conj (-I) = (I : K)
|
K : Type u_1
inst✝ : RCLike K
⊢ (starRingEnd K) (-I) = I
|
rw [map_neg, conj_I, neg_neg]
|
no goals
|
6cda6b26eb47e5ed
|
Equiv.Perm.count_le_one_of_centralizer_le_alternating
|
Mathlib/GroupTheory/SpecificGroups/Alternating/Centralizer.lean
|
theorem count_le_one_of_centralizer_le_alternating
(h : Subgroup.centralizer {g} ≤ alternatingGroup α) :
∀ i, g.cycleType.count i ≤ 1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm { x // x ∈ g.cycleFactorsFinset } := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
⊢ a.toCentralizer (⟨τ, hτ⟩ ^ 2) = 1
|
convert MonoidHom.map_one _
|
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm { x // x ∈ g.cycleFactorsFinset } := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
⊢ ⟨τ, hτ⟩ ^ 2 = 1
|
1a7e68d832b229d5
|
AlgebraicGeometry.IsIntegralHom.iff_universallyClosed_and_isAffineHom
|
Mathlib/AlgebraicGeometry/Morphisms/Integral.lean
|
lemma iff_universallyClosed_and_isAffineHom {X Y : Scheme.{u}} {f : X ⟶ Y} :
IsIntegralHom f ↔ UniversallyClosed f ∧ IsAffineHom f
|
X Y : Scheme
f : X ⟶ Y
⊢ IsIntegralHom f ↔ UniversallyClosed f ∧ IsAffineHom f
|
refine ⟨fun _ ↦ ⟨inferInstance, inferInstance⟩, fun ⟨H₁, H₂⟩ ↦ ?_⟩
|
X Y : Scheme
f : X ⟶ Y
x✝ : UniversallyClosed f ∧ IsAffineHom f
H₁ : UniversallyClosed f
H₂ : IsAffineHom f
⊢ IsIntegralHom f
|
f7362dfec0323cb8
|
rieszContentAux_union
|
Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean
|
lemma rieszContentAux_union {K₁ K₂ : TopologicalSpace.Compacts X}
(disj : Disjoint (K₁ : Set X) K₂) :
rieszContentAux Λ (K₁ ⊔ K₂) = rieszContentAux Λ K₁ + rieszContentAux Λ K₂
|
case intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
f_eq_sum : f = g₁ * f + g₂ * f
aux₁ : ∀ x ∈ K₁, 1 ≤ (g₁ * f) x
⊢ rieszContentAux Λ K₁ + rieszContentAux Λ K₂ ≤ Λ (g₁ * f) + Λ (g₂ * f)
|
have aux₂ : ∀ x ∈ K₂, 1 ≤ (g₂ * f) x := by
intro x x_in_K₂
simp [hg₂ x_in_K₂, hf x (mem_union_right _ x_in_K₂)]
|
case intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
f_eq_sum : f = g₁ * f + g₂ * f
aux₁ : ∀ x ∈ K₁, 1 ≤ (g₁ * f) x
aux₂ : ∀ x ∈ K₂, 1 ≤ (g₂ * f) x
⊢ rieszContentAux Λ K₁ + rieszContentAux Λ K₂ ≤ Λ (g₁ * f) + Λ (g₂ * f)
|
764e0d74e1f8077b
|
HurwitzZeta.hasSum_nat_completedSinZeta
|
Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
|
/-- Formula for `completedSinZeta` as a Dirichlet series in the convergence range
(second version, with sum over `ℕ`). -/
lemma hasSum_nat_completedSinZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
HasSum (fun n : ℕ ↦ Gammaℝ (s + 1) * Real.sin (2 * π * a * n) / (n : ℂ) ^ s)
(completedSinZeta a s)
|
case inr
a : ℝ
s : ℂ
hs : 1 < s.re
this :
HasSum
(fun n =>
((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
(s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
↑n ^ s /
2)
(completedSinZeta (↑a) s)
n : ℕ
h : n ≠ 0
⊢ (s + 1).Gammaℝ * ((cexp (-↑(2 * π * a * ↑n) * I) - cexp (↑(2 * π * a * ↑n) * I)) * I / 2) / ↑n ^ s =
((s + 1).Gammaℝ * -I * 1 * cexp (2 * ↑π * I * ↑a * ↑n) + (s + 1).Gammaℝ * -I * -1 * cexp (2 * ↑π * I * ↑a * -↑n)) /
2 /
↑n ^ s
|
simp only [← mul_div_assoc, push_cast, mul_assoc (Gammaℝ _), ← mul_add]
|
case inr
a : ℝ
s : ℂ
hs : 1 < s.re
this :
HasSum
(fun n =>
((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +
(s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /
↑n ^ s /
2)
(completedSinZeta (↑a) s)
n : ℕ
h : n ≠ 0
⊢ (s + 1).Gammaℝ * ((cexp (-(2 * ↑π * ↑a * ↑n) * I) - cexp (2 * ↑π * ↑a * ↑n * I)) * I) / 2 / ↑n ^ s =
(s + 1).Gammaℝ * (-I * 1 * cexp (2 * ↑π * I * ↑a * ↑n) + -I * -1 * cexp (2 * ↑π * I * ↑a * -↑n)) / 2 / ↑n ^ s
|
8aa52d71fc22a1d9
|
Partrec.option_some_iff
|
Mathlib/Computability/Partrec.lean
|
theorem option_some_iff {f : α →. σ} : (Partrec fun a => (f a).map Option.some) ↔ Partrec f :=
⟨fun h => (Nat.Partrec.ppred.comp h).of_eq fun n => by simp [Part.bind_assoc, bind_some_eq_map],
fun hf => hf.map (option_some.comp snd).to₂⟩
|
α : Type u_1
σ : Type u_4
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α →. σ
h : Partrec fun a => Part.map Option.some (f a)
n : ℕ
⊢ (do
let n ← (↑(decode n)).bind fun a => Part.map encode ((fun a => Part.map Option.some (f a)) a)
↑n.ppred) =
(↑(decode n)).bind fun a => Part.map encode (f a)
|
simp [Part.bind_assoc, bind_some_eq_map]
|
no goals
|
3633b675f3ed41db
|
StrictConvexOn.map_sum_eq_iff'
|
Mathlib/Analysis/Convex/Jensen.lean
|
/-- Canonical form of the **equality case of Jensen's equality**.
For a strictly convex function `f` and nonnegative weights `w`, we have
`f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i)` if and only if the points `p` with nonzero
weight are all equal (and in fact all equal to their center of mass wrt `w`). -/
lemma StrictConvexOn.map_sum_eq_iff' (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔
∀ j ∈ t, w j ≠ 0 → p j = ∑ i ∈ t, w i • p i
|
case h₀
𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : OrderedAddCommGroup β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
hf : StrictConvexOn 𝕜 s f
h₀ : ∀ i ∈ t, 0 ≤ w i
h₁ : ∑ i ∈ t, w i = 1
hmem : ∀ i ∈ t, p i ∈ s
hw : ∀ i ∈ t, w i • p i ≠ 0 → w i ≠ 0
hw' : ∀ i ∈ t, w i • f (p i) ≠ 0 → w i ≠ 0
⊢ ∀ i ∈ filter (fun x => w x ≠ 0) t, 0 < w i
|
simp +contextual [(h₀ _ _).gt_iff_ne]
|
no goals
|
41fab4df447312ea
|
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite
|
Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
|
theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite (h𝒜 : IsSetAlgebra 𝒜)
(S : μ.FiniteSpanningSetsIn 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) :
μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms hμs ε ε_pos
|
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := Accumulate S.set
n : ℕ
⊢ T n ∈ 𝒜
|
simpa using h𝒜.biUnion_mem {k | k ≤ n}.toFinset (fun k _ ↦ S.set_mem k)
|
no goals
|
9296ab2335ceb925
|
Polynomial.isNilpotent_reflect_iff
|
Mathlib/RingTheory/Polynomial/Nilpotent.lean
|
@[simp] lemma isNilpotent_reflect_iff {P : R[X]} {N : ℕ} (hN : P.natDegree ≤ N) :
IsNilpotent (reflect N P) ↔ IsNilpotent P
|
case refine_2.inr
R : Type u_1
inst✝ : CommRing R
P : R[X]
N : ℕ
hN : P.natDegree ≤ N
h : ∀ (i : ℕ), IsNilpotent (P.coeff i)
i : ℕ
hi : N < i
⊢ IsNilpotent ((reflect N P).coeff i)
|
simpa [revAt_eq_self_of_lt hi] using h i
|
no goals
|
1304bc3605015049
|
integral_exp_Iic
|
Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
|
theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c
|
c : ℝ
⊢ ∫ (x : ℝ) in Iic c, rexp x = rexp c
|
refine
tendsto_nhds_unique
(intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_
|
c : ℝ
⊢ Tendsto (fun i => ∫ (x : ℝ) in id i..c, rexp x) atBot (𝓝 (rexp c))
|
7661757092c3e442
|
MeasureTheory.convolution_assoc'
|
Mathlib/Analysis/Convolution.lean
|
theorem convolution_assoc' (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G} (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
(hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt g k x L₄ μ)
(hi : Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)) :
((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] g ⋆[L₄, μ] k) x₀ :=
calc
((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = ∫ t, L₂ (∫ s, L (f s) (g (t - s)) ∂ν) (k (x₀ - t)) ∂μ := rfl
_ = ∫ t, ∫ s, L₂ (L (f s) (g (t - s))) (k (x₀ - t)) ∂ν ∂μ :=
(integral_congr_ae (hfg.mono fun t ht => ((L₂.flip (k (x₀ - t))).integral_comp_comm ht).symm))
_ = ∫ t, ∫ s, L₃ (f s) (L₄ (g (t - s)) (k (x₀ - t))) ∂ν ∂μ
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt g k x L₄ μ
hi : Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)
⊢ ∫ (t : G), ∫ (s : G), (L₂ ((L (f s)) (g (t - s)))) (k (x₀ - t)) ∂ν ∂μ =
∫ (t : G), ∫ (s : G), (L₃ (f s)) ((L₄ (g (t - s))) (k (x₀ - t))) ∂ν ∂μ
|
simp_rw [hL]
|
no goals
|
c9badc48511fd05a
|
Ordinal.card_opow_omega0
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem card_opow_omega0 {a : Ordinal} (h : 1 < a) : card (a ^ ω) = max ℵ₀ a.card
|
a : Ordinal.{u_1}
h : 1 < a
⊢ (a ^ ω).card = ℵ₀ ⊔ a.card
|
rw [card_opow_eq_of_omega0_le_right h le_rfl, card_omega0, max_comm]
|
no goals
|
1db76db8aaa2c304
|
RootPairing.Base.root_sub_root_mem_of_mem_of_mem
|
Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean
|
/-- This is Lemma 2.5 (a) from [Geck](Geck2017). -/
lemma root_sub_root_mem_of_mem_of_mem (hk : α k + α i - α j ∈ Φ)
(hkj : k ≠ j) (hk' : α k + α i ∈ Φ) :
α k - α j ∈ Φ
|
case a
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : Finite ι
inst✝⁴ : CharZero R
inst✝³ : P.IsCrystallographic
inst✝² : NoZeroDivisors R
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : NoZeroSMulDivisors R N
b : P.Base
i j k : ι
hij : i ≠ j
hi : i ∈ b.support
hj : j ∈ b.support
hkj : k ≠ j
hk' : P.root k + P.root i ∈ range ⇑P.root
hm : P.pairingIn ℤ j k ≤ 0
l : ι
hl : P.root l = P.root k + P.root i - P.root j
hli : l ≠ i
hkl : P.pairingIn ℤ l k ≤ 0
⊢ P.pairing l k = 2 + P.pairing i k - P.pairing j k
|
simpa using (P.coroot' k : M →ₗ[R] R).congr_arg hl
|
no goals
|
83118c1462129643
|
Subfield.rangeOfWType_eq_top
|
Mathlib/SetTheory/Cardinal/Subfield.lean
|
private lemma rangeOfWType_eq_top : rangeOfWType s = ⊤ := top_le_iff.mp fun a _ ↦ by
rw [← SetLike.mem_coe, ← Subtype.val_injective.mem_set_image]
change ↑a ∈ map (closure s).subtype _
refine closure_le.mpr (fun a ha ↦ ?_) a.prop
exact ⟨⟨a, subset_closure ha⟩, ⟨WType.mk (.inr ⟨a, ha⟩) Empty.rec, rfl⟩, rfl⟩
|
α : Type u
s : Set α
inst✝ : DivisionRing α
a : ↥(closure s)
x✝ : a ∈ ⊤
⊢ a ∈ Subfield.rangeOfWType s
|
rw [← SetLike.mem_coe, ← Subtype.val_injective.mem_set_image]
|
α : Type u
s : Set α
inst✝ : DivisionRing α
a : ↥(closure s)
x✝ : a ∈ ⊤
⊢ ↑a ∈ Subtype.val '' ↑(Subfield.rangeOfWType s)
|
5e9759fd58ebfc99
|
ite_zero_mul
|
Mathlib/Algebra/Ring/Defs.lean
|
lemma ite_zero_mul : ite P a 0 * b = ite P (a * b) 0
|
α : Type u
inst✝¹ : MulZeroClass α
P : Prop
inst✝ : Decidable P
a b : α
⊢ (if P then a else 0) * b = if P then a * b else 0
|
simp
|
no goals
|
36825256d2657813
|
Finset.Colex.IsInitSeg.shadow
|
Mathlib/Combinatorics/SetFamily/KruskalKatona.lean
|
/-- The shadow of an initial segment is also an initial segment. -/
protected lemma IsInitSeg.shadow [Finite α] (h₁ : IsInitSeg 𝒜 r) : IsInitSeg (∂ 𝒜) (r - 1)
|
case intro
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Finite α
h₁ : IsInitSeg 𝒜 r
val✝ : Fintype α
⊢ IsInitSeg (∂ 𝒜) (r - 1)
|
obtain rfl | hr := Nat.eq_zero_or_pos r
|
case intro.inl
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
inst✝ : Finite α
val✝ : Fintype α
h₁ : IsInitSeg 𝒜 0
⊢ IsInitSeg (∂ 𝒜) (0 - 1)
case intro.inr
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Finite α
h₁ : IsInitSeg 𝒜 r
val✝ : Fintype α
hr : r > 0
⊢ IsInitSeg (∂ 𝒜) (r - 1)
|
95646344ed44f65c
|
ZetaAsymptotics.term_tsum_of_lt
|
Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean
|
/-- For `1 < s`, the topological sum of `ZetaAsymptotics.term (n + 1) s` over all `n : ℕ` is
`1 / (s - 1) - ζ s / s`.
-/
lemma term_tsum_of_lt {s : ℝ} (hs : 1 < s) :
term_tsum s = (1 / (s - 1) - 1 / s * ∑' n : ℕ, 1 / (n + 1 : ℝ) ^ s)
|
case ha.hg
s : ℝ
hs : 1 < s
⊢ Tendsto (fun x => 1 / s * (∑ n ∈ Finset.range x, 1 / (↑n + 1) ^ s - ↑x / (↑x + 1) ^ s)) atTop
(𝓝 (1 / s * ∑' (n : ℕ), 1 / (↑n + 1) ^ s))
|
rw [← sub_zero (tsum _)]
|
case ha.hg
s : ℝ
hs : 1 < s
⊢ Tendsto (fun x => 1 / s * (∑ n ∈ Finset.range x, 1 / (↑n + 1) ^ s - ↑x / (↑x + 1) ^ s)) atTop
(𝓝 (1 / s * (∑' (n : ℕ), 1 / (↑n + 1) ^ s - 0)))
|
11b22989b7bc0459
|
WeierstrassCurve.Projective.equation_smul
|
Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
|
lemma equation_smul (P : Fin 3 → R) {u : R} (hu : IsUnit u) : W'.Equation (u • P) ↔ W'.Equation P :=
have hP (u : R) {P : Fin 3 → R} (hP : W'.Equation P) : W'.Equation <| u • P
|
R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
u : R
hu : IsUnit u
hP : ∀ (u : R) {P : Fin 3 → R}, W'.Equation P → W'.Equation (u • P)
h : W'.Equation (u • P)
⊢ W'.Equation P
|
convert hP ↑hu.unit⁻¹ h
|
case h.e'_4
R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
u : R
hu : IsUnit u
hP : ∀ (u : R) {P : Fin 3 → R}, W'.Equation P → W'.Equation (u • P)
h : W'.Equation (u • P)
⊢ P = ↑hu.unit⁻¹ • u • P
|
05e310694fabdfc8
|
nhds_of_nhdsWithin_of_nhds
|
Mathlib/Topology/ContinuousOn.lean
|
theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) :
t ∈ 𝓝 a
|
case intro.intro
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
a : α
h1 : s ∈ 𝓝 a
h2 : t ∈ 𝓝[s] a
w✝ : Set α
Hw : w✝ ∈ 𝓝 a
hw : w✝ ∩ s ⊆ t
⊢ t ∈ 𝓝 a
|
exact (𝓝 a).sets_of_superset ((𝓝 a).inter_sets Hw h1) hw
|
no goals
|
b6c7be9aae52b9b2
|
integral_gaussian_complex
|
Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
|
theorem integral_gaussian_complex {b : ℂ} (hb : 0 < re b) :
∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2) = (π / b) ^ (1 / 2 : ℂ)
|
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ↑(∫ (x : ℝ), rexp (-1 * x ^ 2)) = ↑(π / 1) ^ ↑(1 / 2)
|
rw [← ofReal_cpow, ofReal_inj]
|
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ∫ (x : ℝ), rexp (-1 * x ^ 2) = (π / 1) ^ (1 / 2)
case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π / 1
|
76ad93bf18e975c2
|
MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory
|
case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}
Ssep : ∀ (n : ℕ), Metric.AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), Metric.AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ⊤
r n : ℕ
⊢ ∑ a ∈ Finset.range n, μ (S (2 * a + 1 + r) \ S (2 * a + r)) ≤ μ (⋃ n, S n)
|
rw [← hm.finset_iUnion_of_pairwise_separated]
|
case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}
Ssep : ∀ (n : ℕ), Metric.AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), Metric.AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ⊤
r n : ℕ
⊢ μ (⋃ i ∈ Finset.range n, S (2 * i + 1 + r) \ S (2 * i + r)) ≤ μ (⋃ n, S n)
case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}
Ssep : ∀ (n : ℕ), Metric.AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), Metric.AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ⊤
r n : ℕ
⊢ ∀ i ∈ Finset.range n,
∀ j ∈ Finset.range n,
i ≠ j → Metric.AreSeparated (S (2 * i + 1 + r) \ S (2 * i + r)) (S (2 * j + 1 + r) \ S (2 * j + r))
|
603b48ab3190787f
|
Set.ncard_eq_of_bijective
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_of_bijective {n : ℕ} (f : ∀ i, i < n → α)
(hf : ∀ a ∈ s, ∃ i, ∃ h : i < n, f i h = a) (hf' : ∀ (i) (h : i < n), f i h ∈ s)
(f_inj : ∀ (i j) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j) : s.ncard = n
|
α : Type u_1
s : Set α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ i, ∃ (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
f' : Fin n → α := fun i => f ↑i ⋯
⊢ s = f' '' univ
|
ext x
|
case h
α : Type u_1
s : Set α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ i, ∃ (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
f' : Fin n → α := fun i => f ↑i ⋯
x : α
⊢ x ∈ s ↔ x ∈ f' '' univ
|
e5caade2630c707b
|
affineIndependent_equiv
|
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
theorem affineIndependent_equiv {ι' : Type*} (e : ι ≃ ι') {p : ι' → P} :
AffineIndependent k (p ∘ e) ↔ AffineIndependent k p
|
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
ι' : Type u_5
e : ι ≃ ι'
p : ι' → P
⊢ AffineIndependent k (p ∘ ⇑e) ↔ AffineIndependent k p
|
refine ⟨?_, AffineIndependent.comp_embedding e.toEmbedding⟩
|
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
ι' : Type u_5
e : ι ≃ ι'
p : ι' → P
⊢ AffineIndependent k (p ∘ ⇑e) → AffineIndependent k p
|
1095c6848a2c441d
|
List.take_eq_self_iff
|
Mathlib/Data/List/TakeDrop.lean
|
@[simp] lemma take_eq_self_iff (x : List α) {n : ℕ} : x.take n = x ↔ x.length ≤ n :=
⟨fun h ↦ by rw [← h]; simp; omega, take_of_length_le⟩
|
α : Type u
x : List α
n : ℕ
h : take n x = x
⊢ x.length ≤ n
|
rw [← h]
|
α : Type u
x : List α
n : ℕ
h : take n x = x
⊢ (take n x).length ≤ n
|
bb60c7701b0127c7
|
ValuationSubring.valuation_unit
|
Mathlib/RingTheory/Valuation/ValuationSubring.lean
|
theorem valuation_unit (a : Aˣ) : A.valuation a = 1
|
K : Type u
inst✝ : Field K
A : ValuationSubring K
a : (↥A)ˣ
⊢ ∃ a_1, ↑↑a_1 * 1 = ↑↑a
|
use a
|
case h
K : Type u
inst✝ : Field K
A : ValuationSubring K
a : (↥A)ˣ
⊢ ↑↑a * 1 = ↑↑a
|
bfe56c9cfacbd2e1
|
CategoryTheory.toNerve₂.mk_naturality_δ1i
|
Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean
|
lemma toNerve₂.mk_naturality_δ1i (i : Fin 3) : toNerve₂.mk.naturalityProperty F (δ₂ i)
|
case h.«1»
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
x : X.obj (op { obj := [1 + 1], property := ⋯ })
⊢ ComposableArrows.mk₁ (F.map { edge := X.map (δ 1).op x, src_eq := ⋯, tgt_eq := ⋯ }) =
(nerve C).map (δ 1).op (ComposableArrows.mk₂ (F.map (ev01₂ x)) (F.map (ev12₂ x)))
|
show _ = (nerve C).δ 1 _
|
case h.«1»
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
x : X.obj (op { obj := [1 + 1], property := ⋯ })
⊢ ComposableArrows.mk₁ (F.map { edge := X.map (δ 1).op x, src_eq := ⋯, tgt_eq := ⋯ }) =
SimplicialObject.δ (nerve C) 1 (ComposableArrows.mk₂ (F.map (ev01₂ x)) (F.map (ev12₂ x)))
|
4128d2c817244193
|
CategoryTheory.Presieve.isSheafFor_of_factorsThru
|
Mathlib/CategoryTheory/Sites/Coverage.lean
|
lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type*)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f) :
T.IsSheafFor P
|
case refine_2
C : Type u_3
inst✝ : Category.{u_2, u_3} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type u_1
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong S f
hS : IsSeparatedFor P S ∧ ∀ (x : FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : x.Compatible
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) ⋯)
hy : y.Compatible
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t
⊢ ∃ t, x.IsAmalgamation t
|
obtain ⟨z, hz⟩ := h2' y hy
|
case refine_2.intro
C : Type u_3
inst✝ : Category.{u_2, u_3} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type u_1
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong S f
hS : IsSeparatedFor P S ∧ ∀ (x : FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t
W : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C
i : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → Z ⟶ W a
e : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → (a : S g) → W a ⟶ X
h1 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), T (e a)
h2 : ∀ ⦃Z : C⦄ ⦃g : Z ⟶ X⦄ (a : S g), i a ≫ e a = g
x : FamilyOfElements P T
hx : x.Compatible
y : FamilyOfElements P S := fun Y g hg => P.map (i hg).op (x (e hg) ⋯)
hy : y.Compatible
left✝ : IsSeparatedFor P S
h2' : ∀ (x : FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t
z : P.obj (Opposite.op X)
hz : y.IsAmalgamation z
⊢ ∃ t, x.IsAmalgamation t
|
d4cb88a64ea278af
|
Real.tendsto_rightDeriv_mul_log_atTop
|
Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
|
lemma tendsto_rightDeriv_mul_log_atTop :
Tendsto (fun x ↦ derivWithin (fun x ↦ x * log x) (Set.Ioi x) x) atTop atTop
|
⊢ Tendsto (fun x => derivWithin (fun x => x * log x) (Set.Ioi x) x) atTop atTop
|
refine (tendsto_congr' ?_).mpr (tendsto_log_atTop.atTop_add (tendsto_const_nhds (x := 1)))
|
⊢ (fun x => derivWithin (fun x => x * log x) (Set.Ioi x) x) =ᶠ[atTop] fun x => log x + 1
|
963a998f76dc66b7
|
exists_idempotent_of_compact_t2_of_continuous_mul_left
|
Mathlib/Topology/Algebra/Semigroup.lean
|
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m
|
case hts
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x => x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun x => x * m) '' N ⊆ N
|
rintro _ ⟨m', hm', rfl⟩
|
case hts.intro.intro
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x => x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
m' : M
hm' : m' ∈ N
⊢ (fun x => x * m) m' ∈ N
|
fc87b9dbd8289f3d
|
IsSeparable.of_algebra_isSeparable_of_isSeparable
|
Mathlib/FieldTheory/SeparableDegree.lean
|
theorem IsSeparable.of_algebra_isSeparable_of_isSeparable [Algebra E K] [IsScalarTower F E K]
[Algebra.IsSeparable F E] {x : K} (hsep : IsSeparable E x) : IsSeparable F x
|
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E
x : K
f : E[X] := minpoly E x
hf : f = minpoly E x
E' : IntermediateField F E := adjoin F ↑f.coeffs
this✝⁵ : FiniteDimensional F ↥E'
g : (↥E')[X]
h : Polynomial.map (algebraMap (↥E') E) g = f
hx : x ∈ restrictScalars F (↥E')⟮x⟯
hzero : (aeval x) g = 0
halg : IsIntegral (↥E') x
hsep : (minpoly (↥E') x).Separable
this✝⁴ : Algebra.IsSeparable F ↥E'
this✝³ : Algebra.IsSeparable ↥E' ↥(↥E')⟮x⟯
this✝² : FiniteDimensional ↥E' ↥(↥E')⟮x⟯
this✝¹ : FiniteDimensional F ↥(↥E')⟮x⟯
this✝ : Algebra.IsAlgebraic ↥E' ↥(↥E')⟮x⟯
this : Algebra.IsSeparable F ↥(↥E')⟮x⟯
⊢ IsSeparable F x
|
change Algebra.IsSeparable F (restrictScalars F E'⟮x⟯) at this
|
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E
x : K
f : E[X] := minpoly E x
hf : f = minpoly E x
E' : IntermediateField F E := adjoin F ↑f.coeffs
this✝⁵ : FiniteDimensional F ↥E'
g : (↥E')[X]
h : Polynomial.map (algebraMap (↥E') E) g = f
hx : x ∈ restrictScalars F (↥E')⟮x⟯
hzero : (aeval x) g = 0
halg : IsIntegral (↥E') x
hsep : (minpoly (↥E') x).Separable
this✝⁴ : Algebra.IsSeparable F ↥E'
this✝³ : Algebra.IsSeparable ↥E' ↥(↥E')⟮x⟯
this✝² : FiniteDimensional ↥E' ↥(↥E')⟮x⟯
this✝¹ : FiniteDimensional F ↥(↥E')⟮x⟯
this✝ : Algebra.IsAlgebraic ↥E' ↥(↥E')⟮x⟯
this : Algebra.IsSeparable F ↥(restrictScalars F (↥E')⟮x⟯)
⊢ IsSeparable F x
|
6901ddfd7b743413
|
NonUnitalSubsemiring.closure_addSubmonoid_closure
|
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
theorem closure_addSubmonoid_closure {s : Set R} :
closure ↑(AddSubmonoid.closure s) = closure s
|
case h
R : Type u
inst✝ : NonUnitalNonAssocSemiring R
s : Set R
x : R
⊢ x ∈ closure ↑(AddSubmonoid.closure s) ↔ x ∈ closure s
|
refine ⟨fun hx => ?_, fun hx => closure_mono AddSubmonoid.subset_closure hx⟩
|
case h
R : Type u
inst✝ : NonUnitalNonAssocSemiring R
s : Set R
x : R
hx : x ∈ closure ↑(AddSubmonoid.closure s)
⊢ x ∈ closure s
|
904943feed2d9199
|
Complex.countable_preimage_exp
|
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
|
theorem countable_preimage_exp {s : Set ℂ} : (exp ⁻¹' s).Countable ↔ s.Countable
|
case refine_1
s : Set ℂ
hs : (cexp ⁻¹' s).Countable
⊢ s ⊆ s ∪ {0}
|
exact Set.subset_union_left
|
no goals
|
e3fd0c8739ed6e00
|
TensorProduct.uncurry_apply
|
Mathlib/LinearAlgebra/TensorProduct/Basic.lean
|
theorem uncurry_apply (f : M →ₗ[R] N →ₗ[R] P) (m : M) (n : N) :
uncurry R M N P f (m ⊗ₜ n) = f m n
|
R : Type u_1
inst✝⁶ : CommSemiring R
M : Type u_5
N : Type u_6
P : Type u_7
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
m : M
n : N
⊢ ((uncurry R M N P) f) (m ⊗ₜ[R] n) = (f m) n
|
rw [uncurry, LinearMap.flip_apply, lift.tmul]
|
R : Type u_1
inst✝⁶ : CommSemiring R
M : Type u_5
N : Type u_6
P : Type u_7
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
m : M
n : N
⊢ (((LinearMap.lflip ∘ₗ LinearMap.id.flip) m) n) f = (f m) n
|
5731556e40a15182
|
Algebra.IsSeparable.insepDegree_eq
|
Mathlib/FieldTheory/SeparableClosure.lean
|
theorem Algebra.IsSeparable.insepDegree_eq [Algebra.IsSeparable F E] : insepDegree F E = 1
|
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
⊢ insepDegree F E = 1
|
rw [insepDegree, (separableClosure.eq_top_iff F E).2 ‹_›, IntermediateField.rank_top]
|
no goals
|
f3a0133939d0621b
|
CategoryTheory.tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
theorem tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight {Y Y' Z : C} [ExactPairing Y Y']
(f : Y ⟶ Z) : (tensorRightHomEquiv _ Y _ _).symm (η_ Y Y' ≫ f ▷ Y') = (λ_ _).hom ≫ f :=
calc
_ = η_ Y Y' ▷ Y ⊗≫ (f ▷ (Y' ⊗ Y) ≫ Z ◁ ε_ Y Y') ⊗≫ 𝟙 _
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
Y Y' Z : C
inst✝ : ExactPairing Y Y'
f : Y ⟶ Z
⊢ η_ Y Y' ▷ Y ⊗≫ (Y ◁ ε_ Y Y' ≫ f ▷ 𝟙_ C) ⊗≫ 𝟙 Z = (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ⊗≫ f
|
monoidal
|
no goals
|
0f6e5fba767e74b3
|
Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div
|
Mathlib/Analysis/Complex/Schwarz.lean
|
theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div [CompleteSpace E] [StrictConvexSpace ℝ E]
(hd : DifferentiableOn ℂ f (ball c R₁)) (h_maps : Set.MapsTo f (ball c R₁) (ball (f c) R₂))
(h_z₀ : z₀ ∈ ball c R₁) (h_eq : ‖dslope f c z₀‖ = R₂ / R₁) :
Set.EqOn f (fun z => f c + (z - c) • dslope f c z₀) (ball c R₁)
|
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z₀ : ℂ
inst✝¹ : CompleteSpace E
inst✝ : StrictConvexSpace ℝ E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)
h_z₀ : z₀ ∈ ball c R₁
h_eq : ‖dslope f c z₀‖ = R₂ / R₁
⊢ EqOn f (fun z => f c + (z - c) • dslope f c z₀) (ball c R₁)
|
set g := dslope f c
|
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z₀ : ℂ
inst✝¹ : CompleteSpace E
inst✝ : StrictConvexSpace ℝ E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)
h_z₀ : z₀ ∈ ball c R₁
g : ℂ → E := dslope f c
h_eq : ‖g z₀‖ = R₂ / R₁
⊢ EqOn f (fun z => f c + (z - c) • g z₀) (ball c R₁)
|
b6a020ab208bf1a6
|
Seminorm.closedBall_smul_ball
|
Mathlib/Analysis/Seminorm.lean
|
theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
Metric.closedBall (0 : 𝕜) r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂)
|
𝕜 : Type u_3
E : Type u_7
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r₁ : ℝ
hr₁ : r₁ ≠ 0
r₂ : ℝ
⊢ ∀ (a : 𝕜), ‖a‖ ≤ r₁ → ∀ (b : E), p b < r₂ → ‖a‖ * p b < r₁ * r₂
|
refine fun a ha b hb ↦ mul_lt_mul' ha hb (apply_nonneg _ _) ?_
|
𝕜 : Type u_3
E : Type u_7
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r₁ : ℝ
hr₁ : r₁ ≠ 0
r₂ : ℝ
a : 𝕜
ha : ‖a‖ ≤ r₁
b : E
hb : p b < r₂
⊢ 0 < r₁
|
938a7d7f28bc0b1a
|
MonomialOrder.degree_sub_of_lt
|
Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean
|
theorem degree_sub_of_lt {f g : MvPolynomial σ R} (h : m.degree g ≺[m] m.degree f) :
m.degree (f - g) = m.degree f
|
case h
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
f g : MvPolynomial σ R
h : m.toSyn (m.degree g) < m.toSyn (m.degree f)
⊢ m.toSyn (m.degree (-g)) < m.toSyn (m.degree f)
|
simp only [degree_neg, h]
|
no goals
|
9b98f46c96318252
|
PrimeSpectrum.mem_image_comap_zeroLocus_sdiff
|
Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean
|
/-- Let `A` be an `R`-algebra.
`𝔭 : Spec R` is in the image of `Z(I) ∩ D(f) ⊆ Spec S`
if and only if `f` is not nilpotent on `κ(𝔭) ⊗ A ⧸ I`. -/
lemma mem_image_comap_zeroLocus_sdiff (f : A) (s : Set A) (x) :
x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f}) ↔
¬ IsNilpotent (algebraMap A ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField) f)
|
case mp.intro.intro.intro
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
q : PrimeSpectrum A
H : IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] ((comap (algebraMap R A)) q).asIdeal.ResidueField)) f)
hqg : s ⊆ ↑q.asIdeal
hqf : f ∉ q.asIdeal
hs : Ideal.span s ≤ RingHom.ker (algebraMap A q.asIdeal.ResidueField)
⊢ False
|
let F : (A ⧸ Ideal.span s) ⊗[R] (q.asIdeal.comap (algebraMap R A)).ResidueField →ₐ[A]
q.asIdeal.ResidueField :=
Algebra.TensorProduct.lift
(Ideal.Quotient.liftₐ (Ideal.span s) (Algebra.ofId A _) hs)
(Ideal.ResidueField.mapₐ _ _ rfl)
fun _ _ ↦ .all _ _
|
case mp.intro.intro.intro
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
q : PrimeSpectrum A
H : IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] ((comap (algebraMap R A)) q).asIdeal.ResidueField)) f)
hqg : s ⊆ ↑q.asIdeal
hqf : f ∉ q.asIdeal
hs : Ideal.span s ≤ RingHom.ker (algebraMap A q.asIdeal.ResidueField)
F : (A ⧸ Ideal.span s) ⊗[R] (Ideal.comap (algebraMap R A) q.asIdeal).ResidueField →ₐ[A] q.asIdeal.ResidueField :=
Algebra.TensorProduct.lift (Ideal.Quotient.liftₐ (Ideal.span s) (Algebra.ofId A q.asIdeal.ResidueField) hs)
(Ideal.ResidueField.mapₐ (Ideal.comap (algebraMap R A) q.asIdeal) q.asIdeal ⋯) ⋯
⊢ False
|
b5370dee55806ec3
|
IsPrimitiveRoot.pow_isRoot_minpoly
|
Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
|
theorem pow_isRoot_minpoly {m : ℕ} (hcop : Nat.Coprime m n) :
IsRoot (map (Int.castRingHom K) (minpoly ℤ μ)) (μ ^ m)
|
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
m : ℕ
hcop : m.Coprime n
⊢ (map (Int.castRingHom K) (minpoly ℤ μ)).IsRoot (μ ^ m)
|
simp only [minpoly_eq_pow_coprime h hcop, IsRoot.def, eval_map]
|
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
m : ℕ
hcop : m.Coprime n
⊢ eval₂ (Int.castRingHom K) (μ ^ m) (minpoly ℤ (μ ^ m)) = 0
|
21b43582b0c980ed
|
MeasureTheory.tendsto_Lp_finite_of_tendstoInMeasure
|
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
theorem tendsto_Lp_finite_of_tendstoInMeasure [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : MemLp g p μ) (hui : UnifIntegrable f p μ)
(hfg : TendstoInMeasure μ f atTop g) : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
⊢ Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
|
refine tendsto_of_subseq_tendsto fun ns hns => ?_
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ℕ → ℕ
hns : Tendsto ns atTop atTop
⊢ ∃ ms, Tendsto (fun n => eLpNorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)
|
6a9a7bb3da79aee9
|
PerfectPairing.exists_basis_basis_of_span_eq_top_of_mem_algebraMap
|
Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean
|
/-- If a perfect pairing over a field `L` takes values in a subfield `K` along two `K`-subspaces
whose `L` span is full, then these subspaces induce a `K`-structure in the sense of
[*Algebra I*, Bourbaki : Chapter II, §8.1 Definition 1][bourbaki1989]. -/
lemma exists_basis_basis_of_span_eq_top_of_mem_algebraMap
(M' : Submodule K M) (N' : Submodule K N)
(hM : span L (M' : Set M) = ⊤)
(hN : span L (N' : Set N) = ⊤)
(hp : ∀ᵉ (x ∈ M') (y ∈ N'), p x y ∈ (algebraMap K L).range) :
∃ (n : ℕ) (b : Basis (Fin n) L M) (b' : Basis (Fin n) K M'), ∀ i, b i = b' i
|
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairing L M N
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range
this✝¹ : IsReflexive L M
this✝ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃✝ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃✝ ⋯
this : Fintype ↑v
v' : ↑v → ↥M' := fun i => ⟨↑i, ⋯⟩
hv₃ : LinearIndependent K Subtype.val
⊢ LinearIndependent (ι := ↑v) K v'
|
rw [show ((↑) : v → M) = M'.subtype ∘ v' by ext; simp [v']] at hv₃
|
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairing L M N
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range
this✝¹ : IsReflexive L M
this✝ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃✝ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃✝ ⋯
this : Fintype ↑v
v' : ↑v → ↥M' := fun i => ⟨↑i, ⋯⟩
hv₃ : LinearIndependent K (⇑M'.subtype ∘ v')
⊢ LinearIndependent (ι := ↑v) K v'
|
28e714660261030d
|
IsPrimitiveRoot.prod_one_sub_pow_eq_order
|
Mathlib/RingTheory/RootsOfUnity/Lemmas.lean
|
/-- If `μ` is a primitive `n`th root of unity in `R`, then `∏(1≤k<n) (1-μ^k) = n`.
(Stated with `n+1` in place of `n` to avoid the condition `n ≠ 0`.) -/
lemma prod_one_sub_pow_eq_order {n : ℕ} {μ : R} (hμ : IsPrimitiveRoot μ (n + 1)) :
∏ k ∈ range n, (1 - μ ^ (k + 1)) = n + 1
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
μ : R
hμ : IsPrimitiveRoot μ (n + 1)
this : eval 1 (∏ k ∈ range n, (X - C (μ ^ (k + 1) * 1))) = eval 1 (∑ i ∈ range (n + 1), X ^ i)
⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = ↑n + 1
|
simpa only [mul_one, map_pow, eval_prod, eval_sub, eval_X, eval_pow, eval_C, eval_geom_sum,
one_pow, sum_const, card_range, nsmul_eq_mul, Nat.cast_add, Nat.cast_one] using this
|
no goals
|
3f906da38fed75a1
|
mellin_hasDerivAt_of_isBigO_rpow
|
Mathlib/Analysis/MellinTransform.lean
|
theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
{f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
(hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
MellinConvergent (fun t => log t • f t) s ∧
HasDerivAt (mellin f) (mellin (fun t => log t • f t) s) s
|
case hf.refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t => (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t => (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
⊢ ContinuousOn log (Ioi 0)
|
exact continuousOn_log.mono (subset_compl_singleton_iff.mpr not_mem_Ioi_self)
|
no goals
|
3ad5d9a87042b6f9
|
LinearMap.trace_prodMap
|
Mathlib/LinearAlgebra/Trace.lean
|
theorem trace_prodMap :
trace R (M × N) ∘ₗ prodMapLinear R M N M N R =
(coprod id id : R × R →ₗ[R] R) ∘ₗ prodMap (trace R M) (trace R N)
|
R : Type u_1
inst✝⁸ : CommRing R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Free R M
inst✝² : Module.Finite R M
inst✝¹ : Free R N
inst✝ : Module.Finite R N
e : (Dual R M ⊗[R] M × Dual R N ⊗[R] N) ≃ₗ[R] (M →ₗ[R] M) × (N →ₗ[R] N) :=
(dualTensorHomEquiv R M M).prod (dualTensorHomEquiv R N N)
h : Function.Surjective ⇑↑e
⊢ (trace R (M × N) ∘ₗ prodMapLinear R M N M N R) ∘ₗ ↑e = (id.coprod id ∘ₗ (trace R M).prodMap (trace R N)) ∘ₗ ↑e
|
ext
|
case hl.a.h.h
R : Type u_1
inst✝⁸ : CommRing R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Free R M
inst✝² : Module.Finite R M
inst✝¹ : Free R N
inst✝ : Module.Finite R N
e : (Dual R M ⊗[R] M × Dual R N ⊗[R] N) ≃ₗ[R] (M →ₗ[R] M) × (N →ₗ[R] N) :=
(dualTensorHomEquiv R M M).prod (dualTensorHomEquiv R N N)
h : Function.Surjective ⇑↑e
x✝¹ : Dual R M
x✝ : M
⊢ ((AlgebraTensorModule.curry
(((trace R (M × N) ∘ₗ prodMapLinear R M N M N R) ∘ₗ ↑e) ∘ₗ inl R (Dual R M ⊗[R] M) (Dual R N ⊗[R] N)))
x✝¹)
x✝ =
((AlgebraTensorModule.curry
(((id.coprod id ∘ₗ (trace R M).prodMap (trace R N)) ∘ₗ ↑e) ∘ₗ inl R (Dual R M ⊗[R] M) (Dual R N ⊗[R] N)))
x✝¹)
x✝
case hr.a.h.h
R : Type u_1
inst✝⁸ : CommRing R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Free R M
inst✝² : Module.Finite R M
inst✝¹ : Free R N
inst✝ : Module.Finite R N
e : (Dual R M ⊗[R] M × Dual R N ⊗[R] N) ≃ₗ[R] (M →ₗ[R] M) × (N →ₗ[R] N) :=
(dualTensorHomEquiv R M M).prod (dualTensorHomEquiv R N N)
h : Function.Surjective ⇑↑e
x✝¹ : Dual R N
x✝ : N
⊢ ((AlgebraTensorModule.curry
(((trace R (M × N) ∘ₗ prodMapLinear R M N M N R) ∘ₗ ↑e) ∘ₗ inr R (Dual R M ⊗[R] M) (Dual R N ⊗[R] N)))
x✝¹)
x✝ =
((AlgebraTensorModule.curry
(((id.coprod id ∘ₗ (trace R M).prodMap (trace R N)) ∘ₗ ↑e) ∘ₗ inr R (Dual R M ⊗[R] M) (Dual R N ⊗[R] N)))
x✝¹)
x✝
|
9f1ec8a0dddb51eb
|
Algebra.discr_isIntegral
|
Mathlib/RingTheory/Discriminant.lean
|
theorem discr_isIntegral {b : ι → L} (h : ∀ i, IsIntegral R (b i)) : IsIntegral R (discr K b)
|
ι : Type w
inst✝⁹ : DecidableEq ι
inst✝⁸ : Fintype ι
K : Type u
L : Type v
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
inst✝⁴ : Module.Finite K L
R : Type z
inst✝³ : CommRing R
inst✝² : Algebra R K
inst✝¹ : Algebra R L
inst✝ : IsScalarTower R K L
b : ι → L
h : ∀ (i : ι), IsIntegral R (b i)
⊢ IsIntegral R (traceMatrix K b).det
|
exact IsIntegral.det fun i j ↦ isIntegral_trace ((h i).mul (h j))
|
no goals
|
63e7b8a9e0f0411b
|
summable_iff_cauchySeq_finset_and_tsum_mem
|
Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean
|
theorem summable_iff_cauchySeq_finset_and_tsum_mem (f : β → α) :
Summable f ↔ CauchySeq (fun s : Finset β ↦ ∑ b ∈ s, f b) ∧
∑' i, toCompl (f i) ∈ Set.range toCompl
|
case mpr.intro
α : Type u_1
β : Type u_2
inst✝² : AddCommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
f : β → α
h_cauchy : CauchySeq fun s => ∑ b ∈ s, f b
h_tsum : ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
⊢ Summable (⇑toCompl ∘ f) ∧ ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
|
constructor
|
case mpr.intro.left
α : Type u_1
β : Type u_2
inst✝² : AddCommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
f : β → α
h_cauchy : CauchySeq fun s => ∑ b ∈ s, f b
h_tsum : ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
⊢ Summable (⇑toCompl ∘ f)
case mpr.intro.right
α : Type u_1
β : Type u_2
inst✝² : AddCommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformAddGroup α
f : β → α
h_cauchy : CauchySeq fun s => ∑ b ∈ s, f b
h_tsum : ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
⊢ ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
|
8ed3f0cc17cd0b17
|
ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite
|
Mathlib/Probability/Kernel/MeasurableLIntegral.lean
|
theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t)
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t₁ : Set α
ht₁ : MeasurableSet t₁
t₂ : Set β
ht₂ : MeasurableSet t₂
⊢ (fun a => (κ a) (if a ∈ t₁ then t₂ else ∅)) = fun a => if a ∈ t₁ then (κ a) t₂ else 0
|
ext1 a
|
case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t₁ : Set α
ht₁ : MeasurableSet t₁
t₂ : Set β
ht₂ : MeasurableSet t₂
a : α
⊢ (κ a) (if a ∈ t₁ then t₂ else ∅) = if a ∈ t₁ then (κ a) t₂ else 0
|
4c14e8ca6e5d1721
|
Topology.IsInducing.le_functorObj_iff
|
Mathlib/Topology/Category/TopCat/Opens.lean
|
lemma le_functorObj_iff {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) {U : Opens X}
{V : Opens Y} : V ≤ hf.functorObj U ↔ (Opens.map f).obj V ≤ U
|
X Y : TopCat
f : X ⟶ Y
hf : IsInducing ⇑(ConcreteCategory.hom f)
U : Opens ↑X
V : Opens ↑Y
⊢ V ≤ hf.functorObj U ↔ (Opens.map f).obj V ≤ U
|
obtain ⟨U, hU⟩ := U
|
case mk
X Y : TopCat
f : X ⟶ Y
hf : IsInducing ⇑(ConcreteCategory.hom f)
V : Opens ↑Y
U : Set ↑X
hU : IsOpen U
⊢ V ≤ hf.functorObj { carrier := U, is_open' := hU } ↔ (Opens.map f).obj V ≤ { carrier := U, is_open' := hU }
|
f5d72770c98cc6d6
|
LSeries.positive_of_differentiable_of_eqOn
|
Mathlib/NumberTheory/LSeries/Positivity.lean
|
/-- If all values of `a : ℕ → ℂ` are nonnegative reals and `a 1`
is positive, and the L-series of `a` agrees with an entire function `f` on some open
right half-plane where it converges, then `f` is real and positive on `ℝ`. -/
lemma positive_of_differentiable_of_eqOn {a : ℕ → ℂ} (ha₀ : 0 ≤ a) (ha₁ : 0 < a 1) {f : ℂ → ℂ}
(hf : Differentiable ℂ f) {x : ℝ} (hx : abscissaOfAbsConv a ≤ x)
(hf' : {s | x < s.re}.EqOn f (LSeries a)) (y : ℝ) :
0 < f y
|
a : ℕ → ℂ
ha₀ : 0 ≤ a
ha₁ : 0 < a 1
f : ℂ → ℂ
hf : Differentiable ℂ f
x : ℝ
hx : abscissaOfAbsConv a ≤ ↑x
hf' : Set.EqOn f (LSeries a) {s | x < s.re}
y : ℝ
hxy : x < x ⊔ y + 1
hxy' : abscissaOfAbsConv a < ↑(x ⊔ y) + 1
hys : ↑(x ⊔ y) + 1 ∈ {s | x < s.re}
⊢ 0 < f (↑(x ⊔ y) + 1)
|
simpa only [hf' hys, ofReal_add, ofReal_one] using positive ha₀ ha₁ hxy'
|
no goals
|
69608794dd5901ab
|
Equiv.Perm.filter_parts_partition_eq_cycleType
|
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
theorem filter_parts_partition_eq_cycleType {σ : Perm α} :
((partition σ).parts.filter fun n => 2 ≤ n) = σ.cycleType
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
a : ℕ
h : a ∈ Multiset.replicate (Fintype.card α - σ.support.card) 1
⊢ ¬2 ≤ 1
|
decide
|
no goals
|
889c6c104ebfc117
|
Batteries.RBNode.min?_mem
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean
|
theorem min?_mem {t : RBNode α} (h : t.min? = some a) : a ∈ t
|
case cons.refl
α : Type u_1
a : α
t : RBNode α
tail✝ : List α
⊢ a ∈ a :: tail✝
|
constructor
|
no goals
|
589a118f7141b07d
|
IsSelfAdjoint.sq_nonneg
|
Mathlib/Algebra/Order/Star/Basic.lean
|
theorem IsSelfAdjoint.sq_nonneg {a : R} (ha : IsSelfAdjoint a) : 0 ≤ a ^ 2
|
R : Type u
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
a : R
ha : IsSelfAdjoint a
⊢ 0 ≤ a ^ 2
|
simp [sq, ha.mul_self_nonneg]
|
no goals
|
6fb21375459f007d
|
GaussianFourier.tendsto_verticalIntegral
|
Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean
|
theorem tendsto_verticalIntegral (hb : 0 < b.re) (c : ℝ) :
Tendsto (verticalIntegral b c) atTop (𝓝 0)
|
case hf
b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun x => b.re * x ^ 2 - 2 * |b.im| * |c| * x) atTop atTop
|
simp_rw [sq, ← mul_assoc, ← sub_mul]
|
case hf
b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun x => (b.re * x - 2 * |b.im| * |c|) * x) atTop atTop
|
d42274eb447fca3c
|
IsSeparatedMap.comp_right
|
Mathlib/Topology/SeparatedMap.lean
|
theorem IsSeparatedMap.comp_right {f : X → Y} (sep : IsSeparatedMap f) {g : A → X}
(cont : Continuous g) (inj : g.Injective) : IsSeparatedMap (f ∘ g)
|
X : Type u_1
Y : Sort u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
sep : IsClosed (Function.pullbackDiagonal f)
g : A → X
cont : Continuous g
inj : Function.Injective g
⊢ IsClosed (Function.mapPullback g id g ⋯ ⋯ ⁻¹' Function.pullbackDiagonal f)
|
exact sep.preimage (cont.mapPullback cont)
|
no goals
|
b4436a2198ba72e6
|
Array.le_iff_lt_or_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lex/Lemmas.lean
|
theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
[Std.Irrefl (· < · : α → α → Prop)]
[Std.Antisymm (¬ · < · : α → α → Prop)]
[Std.Total (¬ · < · : α → α → Prop)]
{l₁ l₂ : Array α} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂
|
α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : LT α
inst✝³ : DecidableLT α
inst✝² : Std.Irrefl fun x1 x2 => x1 < x2
inst✝¹ : Std.Antisymm fun x1 x2 => ¬x1 < x2
inst✝ : Std.Total fun x1 x2 => ¬x1 < x2
l₁ l₂ : Array α
⊢ l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂
|
simpa using List.le_iff_lt_or_eq (l₁ := l₁.toList) (l₂ := l₂.toList)
|
no goals
|
382d4a7da3893ff7
|
Equiv.Perm.CycleType.count_def
|
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
theorem CycleType.count_def {σ : Perm α} (n : ℕ) :
σ.cycleType.count n =
Fintype.card {c : σ.cycleFactorsFinset // (c : Perm α).support.card = n }
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
n : ℕ
⊢ Multiset.count n σ.cycleType = Fintype.card { c // (↑c).support.card = n }
|
rw [cycleType, Multiset.count_eq_card_filter_eq]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
n : ℕ
⊢ (Multiset.filter (fun x => n = x) (Multiset.map (Finset.card ∘ support) σ.cycleFactorsFinset.val)).card =
Fintype.card { c // (↑c).support.card = n }
|
d7a2a31666eddd8f
|
MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE
|
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
theorem iSup_mem_measurableLE (f : ℕ → α → ℝ≥0∞) (hf : ∀ n, f n ∈ measurableLE μ ν) (n : ℕ) :
(fun x ↦ ⨆ (k) (_ : k ≤ n), f k x) ∈ measurableLE μ ν
|
case zero.right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
⊢ ∀ (A : Set α), MeasurableSet A → ∫⁻ (x : α) in A, (fun x => ⨆ k, ⨆ (_ : k ≤ 0), f k x) x ∂μ ≤ ν A
|
intro A hA
|
case zero.right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (x : α) in A, (fun x => ⨆ k, ⨆ (_ : k ≤ 0), f k x) x ∂μ ≤ ν A
|
252de3e134c10542
|
groupCohomology.resolution.diagonalSucc_hom_single
|
Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
|
theorem diagonalSucc_hom_single (f : Gⁿ⁺¹) (a : k) :
(diagonalSucc k G n).hom.hom (single f a) =
single (f 0) 1 ⊗ₜ single (fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) a
|
k G : Type u
inst✝¹ : CommRing k
n : ℕ
inst✝ : Group G
f : Fin (n + 1) → G
a : k
⊢ (ModuleCat.Hom.hom
(𝟙 ((linearization k G).obj (Action.leftRegular G)).V ⊗ (linearizationTrivialIso k G (Fin n → G)).hom.hom))
(single ((Equiv.refl (G × (Fin n → G))) (f 0, fun i => (f i.castSucc)⁻¹ * f i.succ)).1 1 ⊗ₜ[k]
single ((Equiv.refl (G × (Fin n → G))) (f 0, fun i => (f i.castSucc)⁻¹ * f i.succ)).2 a) =
single (f 0) 1 ⊗ₜ[k] single (fun i => (f i.castSucc)⁻¹ * f i.succ) a
|
rfl
|
no goals
|
33dec4a2165d0b2f
|
Mathlib.Meta.NormNum.minFacHelper_1
|
Mathlib/Tactic/NormNum/Prime.lean
|
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
(np : minFac n ≠ k) : MinFacHelper n k'
|
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : n.minFac ≠ k
h2✝ : k < n.minFac
h2 : k.succ = n.minFac
h3 : 2 ∣ n.minFac
⊢ 2 = n.minFac
|
rw [dvd_prime <| minFac_prime h.one_lt.ne'] at h3
|
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : n.minFac ≠ k
h2✝ : k < n.minFac
h2 : k.succ = n.minFac
h3 : 2 = 1 ∨ 2 = n.minFac
⊢ 2 = n.minFac
|
0bac89197e80246f
|
Complex.GammaIntegral_ofReal
|
Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
|
theorem GammaIntegral_ofReal (s : ℝ) :
GammaIntegral ↑s = ↑(∫ x : ℝ in Ioi 0, Real.exp (-x) * x ^ (s - 1))
|
s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
⊢ ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ↑(∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1))
|
conv_rhs => rw [this, ← _root_.integral_ofReal]
|
s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
⊢ ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x) * x ^ (s - 1))
|
52e5eb39fcb5c44d
|
EuclideanGeometry.Cospherical.affineIndependent
|
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
theorem Cospherical.affineIndependent {s : Set P} (hs : Cospherical s) {p : Fin 3 → P}
(hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p
|
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
⊢ ∀ (i : Fin 3), i ≠ 0 → f i = -2 * inner v (p 0 -ᵥ c) / inner v v
|
intro i hi
|
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
i : Fin 3
hi : i ≠ 0
⊢ f i = -2 * inner v (p 0 -ᵥ c) / inner v v
|
41419d3ed5b040ce
|
HasDerivAt.lhopital_zero_atTop
|
Mathlib/Analysis/Calculus/LHopital.lean
|
theorem lhopital_zero_atTop (hff' : ∀ᶠ x in atTop, HasDerivAt f (f' x) x)
(hgg' : ∀ᶠ x in atTop, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atTop, g' x ≠ 0)
(hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l
|
case intro.intro.intro.intro.intro.intro.intro
l✝ : Filter ℝ
f f' g g' : ℝ → ℝ
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l✝
s₁ : Set ℝ
hs₁ : s₁ ∈ atTop
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ atTop
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ atTop
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
l : ℝ
hl : ∀ b ≥ l, b ∈ s
hl' : Ioi l ⊆ s
⊢ Tendsto (fun x => f x / g x) atTop l✝
|
refine lhopital_zero_atTop_on_Ioi ?_ ?_ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>
intro x hx <;> apply_assumption <;> first | exact (hl' hx).1.1| exact (hl' hx).1.2
|
no goals
|
8b13f0337ff756ee
|
Set.ncard_eq_three
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_three : s.ncard = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
|
case refine_2
α : Type u_1
s : Set α
h : s.encard = 3
⊢ ↑s.encard.toNat = 3
|
simp [h]
|
no goals
|
81e51a925750236b
|
Finset.ofDual_max'
|
Mathlib/Data/Finset/Max.lean
|
theorem ofDual_max' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
ofDual (max' s hs) = min' (s.image ofDual) (hs.image _)
|
α : Type u_2
inst✝ : LinearOrder α
s : Finset αᵒᵈ
hs : s.Nonempty
⊢ ↑(ofDual (s.max' hs)) = ↑((image (⇑ofDual) s).min' ⋯)
|
simp only [max'_eq_sup', id_eq, ofDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',
inf_image]
|
α : Type u_2
inst✝ : LinearOrder α
s : Finset αᵒᵈ
hs : s.Nonempty
⊢ s.inf (WithTop.some ∘ fun x => ofDual x) = s.inf ((WithTop.some ∘ fun x => x) ∘ ⇑ofDual)
|
7560dd1c58b579c0
|
Bimod.TensorBimod.middle_assoc'
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
theorem middle_assoc' :
(actLeft P Q ▷ T.X) ≫ actRight P Q =
(α_ R.X _ T.X).hom ≫ (R.X ◁ actRight P Q) ≫ actLeft P Q
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫
(((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
(α_ R.X (P.X ⊗ Q.X) T.X).hom ≫
R.X ◁ (α_ P.X Q.X T.X).hom ≫
(α_ R.X P.X (Q.X ⊗ T.X)).inv ≫
(P.actLeft ▷ (Q.X ⊗ T.X) ≫ P.X ◁ Q.actRight) ≫
coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
|
simp
|
no goals
|
81405f27cb7d242f
|
Module.Finite.exists_smul_of_comp_eq_of_isLocalizedModule
|
Mathlib/Algebra/Module/FinitePresentation.lean
|
lemma Module.Finite.exists_smul_of_comp_eq_of_isLocalizedModule
[hM : Module.Finite R M] (g₁ g₂ : M →ₗ[R] N) (h : f.comp g₁ = f.comp g₂) :
∃ (s : S), s • g₁ = s • g₂
|
case h
R : Type u_1
M : Type u_2
N : Type u_3
N' : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup N'
inst✝¹ : Module R N'
S : Submonoid R
f : N →ₗ[R] N'
inst✝ : IsLocalizedModule S f
g₁ g₂ : M →ₗ[R] N
h : f ∘ₗ g₁ = f ∘ₗ g₂
s : M → ↥S
hs : ∀ (x : M), s x • g₁ x = s x • g₂ x
σ : Finset M
hσ : Submodule.span R ↑σ = ⊤
⊢ σ.prod s • g₁ = σ.prod s • g₂
|
rw [← sub_eq_zero, ← LinearMap.ker_eq_top, ← top_le_iff, ← hσ, Submodule.span_le]
|
case h
R : Type u_1
M : Type u_2
N : Type u_3
N' : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
inst✝² : AddCommGroup N'
inst✝¹ : Module R N'
S : Submonoid R
f : N →ₗ[R] N'
inst✝ : IsLocalizedModule S f
g₁ g₂ : M →ₗ[R] N
h : f ∘ₗ g₁ = f ∘ₗ g₂
s : M → ↥S
hs : ∀ (x : M), s x • g₁ x = s x • g₂ x
σ : Finset M
hσ : Submodule.span R ↑σ = ⊤
⊢ ↑σ ⊆ ↑(LinearMap.ker (σ.prod s • g₁ - σ.prod s • g₂))
|
014a59324e17703b
|
ContinuousLinearMap.opNorm_prod
|
Mathlib/Analysis/NormedSpace/OperatorNorm/Prod.lean
|
theorem opNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖ = ‖(f, g)‖ :=
le_antisymm
(opNorm_le_bound _ (norm_nonneg _) fun x => by
simpa only [prod_apply, Prod.norm_def, max_mul_of_nonneg, norm_nonneg] using
max_le_max (le_opNorm f x) (le_opNorm g x)) <|
max_le
(opNorm_le_bound _ (norm_nonneg _) fun x =>
(le_max_left _ _).trans ((f.prod g).le_opNorm x))
(opNorm_le_bound _ (norm_nonneg _) fun x =>
(le_max_right _ _).trans ((f.prod g).le_opNorm x))
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : NormedSpace 𝕜 G
f : E →L[𝕜] F
g : E →L[𝕜] G
x : E
⊢ ‖(f.prod g) x‖ ≤ ‖(f, g)‖ * ‖x‖
|
simpa only [prod_apply, Prod.norm_def, max_mul_of_nonneg, norm_nonneg] using
max_le_max (le_opNorm f x) (le_opNorm g x)
|
no goals
|
998a3d540cbc0d33
|
Polynomial.add_modByMonic
|
Mathlib/Algebra/Polynomial/Div.lean
|
lemma add_modByMonic (p₁ p₂ : R[X]) : (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q
|
case pos
R : Type u
inst✝ : CommRing R
q p₁ p₂ : R[X]
hq : q.Monic
⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q
|
rcases subsingleton_or_nontrivial R with hR | hR
|
case pos.inl
R : Type u
inst✝ : CommRing R
q p₁ p₂ : R[X]
hq : q.Monic
hR : Subsingleton R
⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q
case pos.inr
R : Type u
inst✝ : CommRing R
q p₁ p₂ : R[X]
hq : q.Monic
hR : Nontrivial R
⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q
|
f3e37a67abb7d804
|
CHSH_inequality_of_comm
|
Mathlib/Algebra/Star/CHSH.lean
|
theorem CHSH_inequality_of_comm [OrderedCommRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
[OrderedSMul ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2
|
R : Type u
inst✝⁴ : OrderedCommRing R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : Algebra ℝ R
inst✝ : OrderedSMul ℝ R
A₀ A₁ B₀ B₁ : R
T : IsCHSHTuple A₀ A₁ B₀ B₁
P : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁
idem : P * P = 4 * P
⊢ P = (1 / 4) • (P * P)
|
have h : 4 * P = (4 : ℝ) • P := by simp [map_ofNat, Algebra.smul_def]
|
R : Type u
inst✝⁴ : OrderedCommRing R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : Algebra ℝ R
inst✝ : OrderedSMul ℝ R
A₀ A₁ B₀ B₁ : R
T : IsCHSHTuple A₀ A₁ B₀ B₁
P : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁
idem : P * P = 4 * P
h : 4 * P = 4 • P
⊢ P = (1 / 4) • (P * P)
|
69323f291cd142bc
|
round_natCast
|
Mathlib/Algebra/Order/Round.lean
|
theorem round_natCast (n : ℕ) : round (n : α) = n
|
α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
n : ℕ
⊢ round ↑n = ↑n
|
simp [round]
|
no goals
|
e277d16a891225e1
|
TensorProduct.equivFinsuppOfBasisRight_symm
|
Mathlib/LinearAlgebra/TensorProduct/Basis.lean
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lemma TensorProduct.equivFinsuppOfBasisRight_symm :
(TensorProduct.equivFinsuppOfBasisRight 𝒞).symm.toLinearMap =
Finsupp.lsum R fun i ↦ (TensorProduct.mk R M N).flip (𝒞 i)
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case h.h
R : Type u_1
M : Type u_3
N : Type u_4
κ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : DecidableEq κ
𝒞 : Basis κ R N
a✝ : κ
x✝ : M
⊢ (↑(equivFinsuppOfBasisRight 𝒞).symm ∘ₗ Finsupp.lsingle a✝) x✝ =
(((Finsupp.lsum R) fun i => (mk R M N).flip (𝒞 i)) ∘ₗ Finsupp.lsingle a✝) x✝
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simp [equivFinsuppOfBasisRight]
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no goals
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0009cfbe297fadaa
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