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
|
|---|---|---|---|---|---|---|
Polynomial.coeff_divModByMonicAux_mem_span_pow_mul_span
|
Mathlib/Algebra/Polynomial/CoeffMem.lean
|
lemma coeff_divModByMonicAux_mem_span_pow_mul_span : ∀ (p q : S[X]) (hq : q.Monic) (i),
(p.divModByMonicAux hq).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧
(p.divModByMonicAux hq).2.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)
| p, q, hq, i => by
rw [divModByMonicAux]
have H₀ (i) : p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)
|
case neg
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
p q : S[X]
hq : q.Monic
i : ℕ
H₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)
hpq : ¬(q.degree ≤ p.degree ∧ p ≠ 0)
⊢ (0, p).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧ (0, p).2.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)
|
simpa using H₀ _
|
no goals
|
f188ae6d502b95a1
|
List.findM?_eq_findSomeM?
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Control.lean
|
theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
as.findM? p = as.findSomeM? fun a => return if (← p a) then some a else none
|
case cons.e_a.h
m : Type → Type u_1
α : Type
inst✝¹ : Monad m
inst✝ : LawfulMonad m
p : α → m Bool
a : α
as : List α
ih :
findM? p as =
findSomeM?
(fun a => do
let __do_lift ← p a
pure (if __do_lift = true then some a else none))
as
b : Bool
⊢ (match b with
| true => pure (some a)
| false => findSomeM? (fun a => (fun a_1 => if a_1 = true then some a else none) <$> p a) as) =
match if b = true then some a else none with
| some b => pure (some b)
| none => findSomeM? (fun a => (fun a_1 => if a_1 = true then some a else none) <$> p a) as
|
cases b <;> simp
|
no goals
|
ce5d2af180ea00d8
|
circleIntegrable_sub_zpow_iff
|
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
theorem circleIntegrable_sub_zpow_iff {c w : ℂ} {R : ℝ} {n : ℤ} :
CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ sphere c |R|
|
case mp.intro.intro
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ sphere c |R|
⊢ ¬IntervalIntegrable (fun θ => (circleMap 0 R θ * I) • (circleMap c R θ - w) ^ n) volume 0 (2 * π)
|
rw [← image_circleMap_Ioc] at hw
|
case mp.intro.intro
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ circleMap c R '' Ioc 0 (2 * π)
⊢ ¬IntervalIntegrable (fun θ => (circleMap 0 R θ * I) • (circleMap c R θ - w) ^ n) volume 0 (2 * π)
|
8d16963f19b22485
|
AlgebraicGeometry.AffineTargetMorphismProperty.respectsIso_mk
|
Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
|
theorem respectsIso_mk {P : AffineTargetMorphismProperty}
(h₁ : ∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) [IsAffine Z], P f → P (e.hom ≫ f))
(h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [IsAffine Y],
P f → @P _ _ (f ≫ e.hom) (isAffine_of_isIso e.inv)) :
P.toProperty.RespectsIso
|
case hpostcomp.intro
P : AffineTargetMorphismProperty
h₁ : ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z], P f → P (e.hom ≫ f)
h₂ : ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [inst : IsAffine Y], P f → P (f ≫ e.hom)
X Y Z : Scheme
e : Y ≅ Z
f : X ⟶ Y
a : IsAffine Y
h : P f
⊢ P.toProperty (f ≫ e.hom)
|
exact ⟨isAffine_of_isIso e.inv, h₂ e f h⟩
|
no goals
|
a0abeb70d6935d67
|
TopologicalSpace.Closeds.coe_sup
|
Mathlib/Topology/Sets/Closeds.lean
|
theorem coe_sup (s t : Closeds α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t
|
α : Type u_2
inst✝ : TopologicalSpace α
s t : Closeds α
⊢ ↑(s ⊔ t) = ↑s ∪ ↑t
|
rfl
|
no goals
|
d62cbb5ccdecb25d
|
Order.Ideal.isProper_of_not_mem
|
Mathlib/Order/Ideal.lean
|
theorem isProper_of_not_mem {I : Ideal P} {p : P} (nmem : p ∉ I) : IsProper I :=
⟨fun hp ↦ by
have := mem_univ p
rw [← hp] at this
exact nmem this⟩
|
P : Type u_1
inst✝ : LE P
I : Ideal P
p : P
nmem : p ∉ I
hp : ↑I = univ
this : p ∈ univ
⊢ False
|
rw [← hp] at this
|
P : Type u_1
inst✝ : LE P
I : Ideal P
p : P
nmem : p ∉ I
hp : ↑I = univ
this : p ∈ ↑I
⊢ False
|
eedecb497d07f70f
|
integral_sin_pow_aux
|
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
theorem integral_sin_pow_aux :
(∫ x in a..b, sin x ^ (n + 2)) =
(sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ x in a..b, sin x ^ n) -
(↑n + 1) * ∫ x in a..b, sin x ^ (n + 2)
|
a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x = C + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ 2 * sin x ^ n
|
simp [H, h, sq]
|
a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ -(sin b ^ (n + 1) * cos b) + sin a ^ (n + 1) * cos a = C
|
f362bf75d26c9d57
|
Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
|
Mathlib/Analysis/MeanInequalities.lean
|
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C
|
case intro.intro.intro.intro.intro.intro
ι : Type u
p q : ℝ
hpq : p.IsConjExponent q
f g : ι → ℝ≥0
A B : ℝ≥0
hf_sum : HasSum (fun a => f a ^ p) (A ^ p)
hg_sum : HasSum (fun a => g a ^ q) (B ^ q)
C : ℝ≥0
hC : C ≤ A * B
H : HasSum (fun i => f i * g i) C
⊢ HasSum (fun i => ↑(f i) * ↑(g i)) ↑C
|
norm_cast
|
no goals
|
164a3a6315ccf271
|
WeierstrassCurve.exists_variableChange_of_char_three_of_j_eq_zero
|
Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean
|
private lemma exists_variableChange_of_char_three_of_j_eq_zero
[E.IsShortNF] [E'.IsShortNF] :
∃ C : VariableChange F, E.variableChange C = E'
|
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
⊢ 2 ≤ 3
|
norm_num
|
no goals
|
c130cdcad82e2167
|
CategoryTheory.MorphismProperty.colimitsOfShape_le_of_final
|
Mathlib/CategoryTheory/MorphismProperty/Limits.lean
|
lemma colimitsOfShape_le_of_final {J' : Type*} [Category J'] (F : J ⥤ J') [F.Final] :
W.colimitsOfShape J' ≤ W.colimitsOfShape J
|
C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝² : Category.{u_4, u_1} J
J' : Type u_2
inst✝¹ : Category.{u_3, u_2} J'
F : J ⥤ J'
inst✝ : F.Final
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X₁ X₂ : J' ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J' f
⊢ W.colimitsOfShape J (h₁.desc { pt := c₂.pt, ι := f ≫ c₂.ι })
|
have h₁' : IsColimit (c₁.whisker F) := (Functor.Final.isColimitWhiskerEquiv F c₁).symm h₁
|
C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝² : Category.{u_4, u_1} J
J' : Type u_2
inst✝¹ : Category.{u_3, u_2} J'
F : J ⥤ J'
inst✝ : F.Final
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X₁ X₂ : J' ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J' f
h₁' : IsColimit (Cocone.whisker F c₁)
⊢ W.colimitsOfShape J (h₁.desc { pt := c₂.pt, ι := f ≫ c₂.ι })
|
3cb77bf936c9d4d9
|
Orientation.linearIsometryEquiv_comp_rightAngleRotation
|
Mathlib/Analysis/InnerProductSpace/TwoDim.lean
|
theorem linearIsometryEquiv_comp_rightAngleRotation (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x : E) : φ (J x) = J (φ x)
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ φ (o.rightAngleRotation x) = o.rightAngleRotation (φ x)
|
convert (o.rightAngleRotation_map φ (φ x)).symm
|
case h.e'_2.h.e'_6.h.e'_6
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ x = φ.symm (φ x)
case h.e'_3.h.e'_5.h.e'_5
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ o = (map (Fin 2) φ.toLinearEquiv) o
|
92cca6cb1f7f9fef
|
map_le_lineMap_iff_slope_le_slope
|
Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
|
theorem map_le_lineMap_iff_slope_le_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f c b
|
k : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField k
inst✝² : OrderedAddCommGroup E
inst✝¹ : Module k E
inst✝ : OrderedSMul k E
f : k → E
a b r : k
hab : a < b
h₀ : 0 < r
h₁ : r < 1
⊢ f ((lineMap a b) r) ≤ (lineMap (f a) (f b)) r ↔ slope f a ((lineMap a b) r) ≤ slope f ((lineMap a b) r) b
|
rw [map_le_lineMap_iff_slope_le_slope_left (mul_pos h₀ (sub_pos.2 hab)), ←
lineMap_slope_lineMap_slope_lineMap f a b r, right_le_lineMap_iff_le h₁]
|
no goals
|
22cc763cce3393ca
|
MeasureTheory.Measure.haar_singleton
|
Mathlib/MeasureTheory/Group/Measure.lean
|
theorem haar_singleton [IsTopologicalGroup G] [BorelSpace G] (g : G) : μ {g} = μ {(1 : G)}
|
case h.e'_2.h.e'_6
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsTopologicalGroup G
inst✝ : BorelSpace G
g : G
⊢ {g} = (fun h => g⁻¹ * h) ⁻¹' {1}
|
simp only [mul_one, preimage_mul_left_singleton, inv_inv]
|
no goals
|
f4a174aeac7f91b9
|
CoxeterSystem.isLeftInversion_simple_iff_isLeftDescent
|
Mathlib/GroupTheory/Coxeter/Inversion.lean
|
theorem isLeftInversion_simple_iff_isLeftDescent (w : W) (i : B) :
cs.IsLeftInversion w (s i) ↔ cs.IsLeftDescent w i
|
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
⊢ cs.IsLeftInversion w (cs.simple i) ↔ cs.IsLeftDescent w i
|
simp [IsLeftInversion, IsLeftDescent, cs.isReflection_simple i]
|
no goals
|
99132206d5581048
|
exists_seq_of_forall_finset_exists
|
Mathlib/Data/Fintype/Basic.lean
|
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
(h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n)
|
α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
this : Nonempty α
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)
|
choose! F hF using h
|
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)
|
84f3321d3efdbd70
|
Polynomial.coeff_hermite_explicit
|
Mathlib/RingTheory/Polynomial/Hermite/Basic.lean
|
theorem coeff_hermite_explicit :
∀ n k : ℕ, coeff (hermite (2 * n + k)) k = (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
| 0, _ => by simp
| n + 1, 0 => by
convert coeff_hermite_succ_zero (2 * n + 1) using 1
-- Porting note: ring_nf did not solve the goal on line 165
rw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] :
2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one, Nat.choose_zero_right,
Nat.choose_one_right, pow_succ]
push_cast
ring
| n + 1, k + 1 => by
let hermite_explicit : ℕ → ℕ → ℤ := fun n k =>
(-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
have hermite_explicit_recur :
∀ n k : ℕ,
hermite_explicit (n + 1) (k + 1) =
hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2)
|
case e_a.e_a
n✝ k✝ : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
n k : ℕ
⊢ (2 * (n + 1) - 1)‼ * (2 * (n + 1) + (k + 1)).choose (k + 1) =
(2 * (n + 1) - 1)‼ * (2 * (n + 1) + k).choose k + (2 * n - 1)‼ * ((2 * n + (k + 2)).choose (k + 2) * (k + 2))
|
rw [(by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),
Nat.doubleFactorial_add_one, mul_comm (2 * n + 1)]
|
case e_a.e_a
n✝ k✝ : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
n k : ℕ
⊢ (2 * n - 1)‼ * (2 * n + 1) * (2 * (n + 1) + (k + 1)).choose (k + 1) =
(2 * n - 1)‼ * (2 * n + 1) * (2 * (n + 1) + k).choose k +
(2 * n - 1)‼ * ((2 * n + (k + 2)).choose (k + 2) * (k + 2))
|
76b4bea6433d05b5
|
MeasureTheory.llr_smul_right
|
Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
|
lemma llr_smul_right [IsFiniteMeasure μ] [Measure.HaveLebesgueDecomposition μ ν]
(hμν : μ ≪ ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ∞) :
llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log c.toReal
|
case h.hx
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : μ.rnDeriv (c • ν) =ᶠ[ae ν] c⁻¹ • μ.rnDeriv ν
x : α
hx_eq : μ.rnDeriv (c • ν) x = (c⁻¹ • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤
⊢ c⁻¹ ≠ 0 ∧ c⁻¹ ≠ ⊤
|
simp [hc, hc_ne_top]
|
no goals
|
f2c098759e8140a2
|
LucasLehmer.X.ext
|
Mathlib/NumberTheory/LucasLehmer.lean
|
theorem ext {x y : X q} (h₁ : x.1 = y.1) (h₂ : x.2 = y.2) : x = y
|
case mk.mk
q : ℕ+
fst✝¹ snd✝¹ fst✝ snd✝ : ZMod ↑q
h₁ : (fst✝¹, snd✝¹).1 = (fst✝, snd✝).1
h₂ : (fst✝¹, snd✝¹).2 = (fst✝, snd✝).2
⊢ (fst✝¹, snd✝¹) = (fst✝, snd✝)
|
congr
|
no goals
|
b7ee4edacbd6f2eb
|
BoxIntegral.Box.subbox_induction_on
|
Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean
|
theorem subbox_induction_on {p : Box ι → Prop} (I : Box ι)
(H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J)
(H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ),
z ∈ Box.Icc J → Box.Icc J ⊆ U →
(∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
p I
|
ι : Type u_1
inst✝ : Fintype ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J
H_nhds :
∀ z ∈ Box.Icc I,
∃ U ∈ 𝓝[Box.Icc I] z,
∀ J ≤ I,
∀ (m : ℕ),
z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
⊢ p I
|
refine subbox_induction_on' I (fun J hle hs => H_ind J hle fun J' h' => ?_) H_nhds
|
ι : Type u_1
inst✝ : Fintype ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J
H_nhds :
∀ z ∈ Box.Icc I,
∃ U ∈ 𝓝[Box.Icc I] z,
∀ J ≤ I,
∀ (m : ℕ),
z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : Box ι
hle : J ≤ I
hs : ∀ (s : Set ι), p (J.splitCenterBox s)
J' : Box ι
h' : J' ∈ splitCenter J
⊢ p J'
|
7bb7c3e429c684e0
|
HahnModule.add_smul
|
Mathlib/RingTheory/HahnSeries/Multiplication.lean
|
theorem add_smul [AddCommMonoid R] [SMulWithZero R V] {x y : HahnSeries Γ R}
{z : HahnModule Γ' R V} (h : ∀ (r s : R) (u : V), (r + s) • u = r • u + s • u) :
(x + y) • z = x • z + y • z
|
case h.h
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : AddCommMonoid R
inst✝ : SMulWithZero R V
x y : HahnSeries Γ R
z : HahnModule Γ' R V
h✝ : ∀ (r s : R) (u : V), (r + s) • u = r • u + s • u
a : Γ'
b : Γ
h : x.coeff b = 0 ∧ y.coeff b = 0
⊢ x.coeff b + y.coeff b = 0
|
rw [h.1, h.2, add_zero]
|
no goals
|
d6350673eea7c23e
|
Subalgebra.LinearDisjoint.of_linearDisjoint_finite_left
|
Mathlib/RingTheory/LinearDisjoint.lean
|
theorem of_linearDisjoint_finite_left [Algebra.IsIntegral R A]
(H : ∀ A' : Subalgebra R S, A' ≤ A → [Module.Finite R A'] → A'.LinearDisjoint B) :
A.LinearDisjoint B
|
case intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
h : {x, y} ⊆ ↑(LinearMap.range (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)))
⊢ x = y
|
obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp)
|
case intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
h : {x, y} ⊆ ↑(LinearMap.range (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)))
x' : ↥(toSubmodule A') ⊗[R] ↥(toSubmodule B)
hx' : (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) x' = x
⊢ x = y
|
1d56e99b263dd6b5
|
Filter.inter_eventuallyEq_left
|
Mathlib/Order/Filter/Basic.lean
|
theorem inter_eventuallyEq_left {s t : Set α} {l : Filter α} :
(s ∩ t : Set α) =ᶠ[l] s ↔ ∀ᶠ x in l, x ∈ s → x ∈ t
|
α : Type u
s t : Set α
l : Filter α
⊢ s ∩ t =ᶠ[l] s ↔ ∀ᶠ (x : α) in l, x ∈ s → x ∈ t
|
simp only [eventuallyEq_set, mem_inter_iff, and_iff_left_iff_imp]
|
no goals
|
2d1c75287f901022
|
Ideal.absNorm_bot
|
Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean
|
theorem absNorm_bot : absNorm (⊥ : Ideal S) = 0
|
S : Type u_1
inst✝³ : CommRing S
inst✝² : Nontrivial S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
⊢ absNorm ⊥ = 0
|
rw [← Ideal.zero_eq_bot, _root_.map_zero]
|
no goals
|
73fd61bc09b59419
|
GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I
|
Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean
|
theorem integral_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
∫ x : ℝ, cexp (-b * (x + c * I) ^ 2) = (π / b) ^ (1 / 2 : ℂ)
|
b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (-↑T + ↑y * I) ^ 2)
T : ℝ
⊢ Differentiable ℂ fun z => z ^ 2
|
exact differentiable_pow 2
|
no goals
|
8f46f2d622ac75ef
|
Real.Angle.cos_eq_iff_coe_eq_or_eq_neg
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
cos θ = cos ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) = -ψ
|
case mp
θ ψ : ℝ
Hcos : (∃ n, ↑n * π = (θ + ψ) / 2) ∨ ∃ n, ↑n * π = (θ - ψ) / 2
⊢ ↑θ = ↑ψ ∨ ↑θ = -↑ψ
|
rcases Hcos with (⟨n, hn⟩ | ⟨n, hn⟩)
|
case mp.inl.intro
θ ψ : ℝ
n : ℤ
hn : ↑n * π = (θ + ψ) / 2
⊢ ↑θ = ↑ψ ∨ ↑θ = -↑ψ
case mp.inr.intro
θ ψ : ℝ
n : ℤ
hn : ↑n * π = (θ - ψ) / 2
⊢ ↑θ = ↑ψ ∨ ↑θ = -↑ψ
|
231d1921c276f456
|
ENNReal.rpow_le_one_of_one_le_of_neg
|
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
theorem rpow_le_one_of_one_le_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : z < 0) : x ^ z ≤ 1
|
case coe
z : ℝ
hz : z < 0
x✝ : ℝ≥0
hx : 1 ≤ ↑x✝
⊢ ↑x✝ ^ z ≤ 1
|
simp only [one_le_coe_iff, some_eq_coe] at hx
|
case coe
z : ℝ
hz : z < 0
x✝ : ℝ≥0
hx : 1 ≤ x✝
⊢ ↑x✝ ^ z ≤ 1
|
9fb379d2f486f39d
|
ENNReal.exists_pos_sum_of_countable
|
Mathlib/Analysis/SpecificLimits/Basic.lean
|
theorem exists_pos_sum_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι) [Countable ι] :
∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ (∑' i, (ε' i : ℝ≥0∞)) < ε
|
case intro.intro
ε : ℝ≥0∞
hε : ε ≠ 0
ι : Type u_4
inst✝ : Countable ι
r : ℝ≥0∞
h0r : 0 < r
hrε : r < ε
⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ↑(ε' i) < ε
|
rcases lt_iff_exists_coe.1 hrε with ⟨x, rfl, _⟩
|
case intro.intro.intro.intro
ε : ℝ≥0∞
hε : ε ≠ 0
ι : Type u_4
inst✝ : Countable ι
x : ℝ≥0
right✝ : ↑x < ε
h0r : 0 < ↑x
hrε : ↑x < ε
⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ↑(ε' i) < ε
|
6474cb59d23a91c6
|
SameRay.trans
|
Mathlib/LinearAlgebra/Ray.lean
|
theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
SameRay R x z
|
case inr
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x y z : M
hxy : SameRay R x y
hyz : SameRay R y z
hy : y = 0 → x = 0 ∨ z = 0
hx : x ≠ 0
⊢ SameRay R x z
|
rcases eq_or_ne z 0 with (rfl | hz)
|
case inr.inl
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x y : M
hxy : SameRay R x y
hx : x ≠ 0
hyz : SameRay R y 0
hy : y = 0 → x = 0 ∨ 0 = 0
⊢ SameRay R x 0
case inr.inr
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x y z : M
hxy : SameRay R x y
hyz : SameRay R y z
hy : y = 0 → x = 0 ∨ z = 0
hx : x ≠ 0
hz : z ≠ 0
⊢ SameRay R x z
|
58153a221638fd7f
|
IsLocalization.bot_lt_comap_prime
|
Mathlib/RingTheory/Localization/Ideal.lean
|
theorem bot_lt_comap_prime [IsDomain R] (hM : M ≤ R⁰) (p : Ideal S) [hpp : p.IsPrime]
(hp0 : p ≠ ⊥) : ⊥ < Ideal.comap (algebraMap R S) p
|
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
inst✝ : IsDomain R
hM : M ≤ R⁰
p : Ideal S
hpp : p.IsPrime
hp0 : p ≠ ⊥
this : IsDomain S
⊢ Ideal.comap (algebraMap R S) ⊥ < Ideal.comap (algebraMap R S) p
|
convert (orderIsoOfPrime M S).lt_iff_lt.mpr (show (⟨⊥, Ideal.bot_prime⟩ :
{ p : Ideal S // p.IsPrime }) < ⟨p, hpp⟩ from hp0.bot_lt)
|
no goals
|
070619afe4075fe6
|
Set.SMulAntidiagonal.finite_of_isPWO
|
Mathlib/Data/Set/SMulAntidiagonal.lean
|
theorem finite_of_isPWO (hs : s.IsPWO) (ht : t.IsPWO) (a) : (smulAntidiagonal s t a).Finite
|
case refl
G : Type u_1
P : Type u_2
s : Set G
t : Set P
inst✝³ : PartialOrder G
inst✝² : PartialOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
hs : s.IsPWO
ht : t.IsPWO
a : P
h : (s.smulAntidiagonal t a).Infinite
h1 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x1 x2 => x1 ≤ x2)
h2 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x1 x2 => x1 ≤ x2)
⊢ ∀ (a : G × P), (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) a a
|
simp_all only [Order.Preimage, le_refl, Prod.forall, implies_true]
|
no goals
|
7f1573a75cfc8a75
|
PowerSeries.X_dvd_iff
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
theorem X_dvd_iff {φ : R⟦X⟧} : (X : R⟦X⟧) ∣ φ ↔ constantCoeff R φ = 0
|
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ X ∣ φ ↔ (constantCoeff R) φ = 0
|
rw [← pow_one (X : R⟦X⟧), X_pow_dvd_iff, ← coeff_zero_eq_constantCoeff_apply]
|
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ (∀ m < 1, (coeff R m) φ = 0) ↔ (coeff R 0) φ = 0
|
cf60b9c3c4ca0721
|
SetTheory.Game.small_setOf_birthday_lt
|
Mathlib/SetTheory/Game/Birthday.lean
|
theorem small_setOf_birthday_lt (o : Ordinal) : Small.{u} {x : Game.{u} // birthday x < o}
|
y : PGame
IH : ∀ k < Order.succ (birthday ⟦y⟧), Small.{u, u + 1} { x // x.birthday < k }
S : Set Game := ⋃ a ∈ Set.Iio (Order.succ (birthday ⟦y⟧)), {x | x.birthday < a}
H : Small.{u, u + 1} ↑S := small_biUnion (Set.Iio (Order.succ (birthday ⟦y⟧))) fun a h => {x | x.birthday < a}
f : Set ↑S × Set ↑S → Game :=
fun g =>
⟦PGame.mk (Shrink.{u, u + 1} ↑g.1) (Shrink.{u, u + 1} ↑g.2) (fun x => Quotient.out ↑↑((equivShrink ↑g.1).symm x))
fun x => Quotient.out ↑↑((equivShrink ↑g.2).symm x)⟧
hy' : y.birthday = birthday ⟦y⟧
i : y.LeftMoves
this : ∃ b ≤ y.birthday, birthday ⟦y.moveLeft i⟧ < b
⊢ ⟦y.moveLeft i⟧ ∈ S
|
simpa [S, hy'] using this
|
no goals
|
942b1277300d8c20
|
Real.lt_tan
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
|
theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x
|
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
⊢ x < tan x
|
have cos_pos {y : ℝ} (hy : y ∈ U) : 0 < cos y := by
exact cos_pos_of_mem_Ioo (Ico_subset_Ioo_left (neg_lt_zero.mpr half_pi_pos) hy)
|
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
⊢ x < tan x
|
beb6377e2b7c7c87
|
Profinite.exists_locallyConstant
|
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
theorem exists_locallyConstant {α : Type*} (hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom
|
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
S : DiscreteQuotient ↑C.pt.toTop := f.discreteQuotient
ff : Quotient S.toSetoid → α := ⇑f.lift
h✝ : IsEmpty (Quotient S.toSetoid)
this✝ : ∃ j, IsEmpty ↑(F.obj j).toTop
j : J
hj : IsEmpty ↑(F.obj j).toTop
A : Set α
this : (fun a => hj.elim a) ⁻¹' A = ∅
⊢ IsOpen ((fun a => hj.elim a) ⁻¹' A)
|
rw [this]
|
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
S : DiscreteQuotient ↑C.pt.toTop := f.discreteQuotient
ff : Quotient S.toSetoid → α := ⇑f.lift
h✝ : IsEmpty (Quotient S.toSetoid)
this✝ : ∃ j, IsEmpty ↑(F.obj j).toTop
j : J
hj : IsEmpty ↑(F.obj j).toTop
A : Set α
this : (fun a => hj.elim a) ⁻¹' A = ∅
⊢ IsOpen ∅
|
72c7aeebc62ad404
|
ContDiffWithinAt.contDiffOn'
|
Mathlib/Analysis/Calculus/ContDiff/Defs.lean
|
theorem ContDiffWithinAt.contDiffOn' (hm : m ≤ n) (h' : m = ∞ → n = ω)
(h : ContDiffWithinAt 𝕜 n f s x) :
∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u)
|
case intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
m✝ n : WithTop ℕ∞
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ω
m : ℕ
hm : ↑m ≤ n
h' : ↑m = ∞ → n = ω
t : Set E
ht : t ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn (↑m) f p t
u : Set E
huo : IsOpen u
hxu : x ∈ u
hut : u ∩ insert x s ⊆ t
⊢ ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 (↑m) f (insert x s ∩ u)
|
rw [inter_comm] at hut
|
case intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
m✝ n : WithTop ℕ∞
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ω
m : ℕ
hm : ↑m ≤ n
h' : ↑m = ∞ → n = ω
t : Set E
ht : t ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn (↑m) f p t
u : Set E
huo : IsOpen u
hxu : x ∈ u
hut : insert x s ∩ u ⊆ t
⊢ ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 (↑m) f (insert x s ∩ u)
|
252a1c681d4d0950
|
CochainComplex.mappingCone.triangleMapOfHomotopy_comm₃
|
Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean
|
@[reassoc]
lemma triangleMapOfHomotopy_comm₃ :
mapOfHomotopy H ≫ (triangle φ₂).mor₃ = (triangle φ₁).mor₃ ≫ a⟦1⟧'
|
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K₁ L₁ K₂ L₂ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
⊢ mapOfHomotopy H ≫ (triangle φ₂).mor₃ = (triangle φ₁).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map a
|
ext p
|
case h
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K₁ L₁ K₂ L₂ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
p : ℤ
⊢ (mapOfHomotopy H ≫ (triangle φ₂).mor₃).f p =
((triangle φ₁).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map a).f p
|
5e3487226dd990fa
|
CategoryTheory.Limits.Sigma.eqToHom_comp_ι
|
Mathlib/CategoryTheory/Limits/Shapes/Products.lean
|
theorem Sigma.eqToHom_comp_ι {J : Type*} (f : J → C) [HasCoproduct f] {j j' : J} (w : j = j') :
eqToHom (by simp [w]) ≫ Sigma.ι f j' = Sigma.ι f j
|
case refl
C : Type u
inst✝¹ : Category.{v, u} C
J : Type u_1
f : J → C
inst✝ : HasCoproduct f
j : J
⊢ eqToHom ⋯ ≫ ι f j = ι f j
|
simp
|
no goals
|
50733f72a97eb131
|
hasDerivAt_of_tendstoUniformlyOnFilter
|
Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
|
theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasDerivAt g (g' x) x
|
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
⊢ HasDerivAt g (g' x) x
|
let G' z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (g' z)
|
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
⊢ HasDerivAt g (g' x) x
|
3364bb7f5ca1a8f4
|
PrimeSpectrum.toPiLocalization_surjective_of_discreteTopology
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
theorem toPiLocalization_surjective_of_discreteTopology :
Function.Surjective (toPiLocalization R) := fun x ↦ by
have (p : PrimeSpectrum R) : ∃ f, (basicOpen f : Set _) = {p} :=
have ⟨_, ⟨f, rfl⟩, hpf, hfp⟩ := isTopologicalBasis_basic_opens.isOpen_iff.mp
(isOpen_discrete {p}) p rfl
⟨f, hfp.antisymm <| Set.singleton_subset_iff.mpr hpf⟩
choose f hf using this
let e := Equiv.ofInjective f fun p q eq ↦ Set.singleton_injective (hf p ▸ eq ▸ hf q)
have loc a : IsLocalization.AtPrime (Localization.Away a.1) (e.symm a).1 :=
(isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton <| hf _).mp <| by
simp_rw [e, Equiv.apply_ofInjective_symm]; infer_instance
let algE a := IsLocalization.algEquiv (e.symm a).1.primeCompl
(Localization.AtPrime (e.symm a).1) (Localization.Away a.1)
have span_eq : Ideal.span (Set.range f) = ⊤ := iSup_basicOpen_eq_top_iff.mp <| top_unique
fun p _ ↦ TopologicalSpace.Opens.mem_iSup.mpr ⟨p, (hf p).ge rfl⟩
replace hf a : (basicOpen a.1 : Set _) = {e.symm a}
|
R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
a : ↑(Set.range f)
⊢ IsLocalization.Away (↑a) (Localization.Away ↑a)
|
infer_instance
|
no goals
|
680284c76fd78d8c
|
MeasureTheory.Martingale.condExp_stoppedValue_stopping_time_ae_eq_restrict_le
|
Mathlib/Probability/Martingale/OptionalSampling.lean
|
theorem condExp_stoppedValue_stopping_time_ae_eq_restrict_le (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) [SigmaFinite (μ.trim hσ.measurableSpace_le)]
(hτ_le : ∀ x, τ x ≤ i) :
μ[stoppedValue f τ|hσ.measurableSpace] =ᵐ[μ.restrict {x : Ω | τ x ≤ σ x}] stoppedValue f τ
|
case refine_1
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → ι
f : ι → Ω → E
i : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
hτ_le : ∀ (x : Ω), τ x ≤ i
h_int : Integrable ({ω | τ ω ≤ σ ω}.indicator (stoppedValue (fun n => f n) τ)) μ
t : Set Ω
ht : MeasurableSet (t ∩ {ω | τ ω ≤ σ ω})
⊢ MeasurableSet (t ∩ {ω | τ ω ≤ σ ω})
|
rw [hτ.measurableSet_inter_le_iff hσ, IsStoppingTime.measurableSet_min_iff hτ hσ] at ht
|
case refine_1
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → ι
f : ι → Ω → E
i : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
hτ_le : ∀ (x : Ω), τ x ≤ i
h_int : Integrable ({ω | τ ω ≤ σ ω}.indicator (stoppedValue (fun n => f n) τ)) μ
t : Set Ω
ht : MeasurableSet (t ∩ {ω | τ ω ≤ σ ω}) ∧ MeasurableSet (t ∩ {ω | τ ω ≤ σ ω})
⊢ MeasurableSet (t ∩ {ω | τ ω ≤ σ ω})
|
5957e7231ee5fbd1
|
contDiff_norm_rpow
|
Mathlib/Analysis/InnerProductSpace/NormPow.lean
|
theorem contDiff_norm_rpow {p : ℝ} (hp : 1 < p) : ContDiff ℝ 1 (fun x : E ↦ ‖x‖ ^ p)
|
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
p : ℝ
hp : 1 < p
x : E
hx : x = 0
⊢ Filter.Tendsto (fderiv ℝ fun x => ‖x‖ ^ p) (𝓝 0) (𝓝 0)
|
rw [tendsto_zero_iff_norm_tendsto_zero]
|
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
p : ℝ
hp : 1 < p
x : E
hx : x = 0
⊢ Filter.Tendsto (fun x => ‖fderiv ℝ (fun x => ‖x‖ ^ p) x‖) (𝓝 0) (𝓝 0)
|
fdf902c8d0df4633
|
NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_eq_iff
|
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean
|
theorem integerSetToAssociates_eq_iff (a b : integerSet K) :
integerSetToAssociates K a = integerSetToAssociates K b ↔
∃ ζ : torsion K, ζ • a = b
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a b : ↑(integerSet K)
x✝ : ∃ ζ, ↑ζ • ↑a = ↑b
u : (𝓞 K)ˣ
property✝ : u ∈ torsion K
h : ↑⟨u, property✝⟩ • ↑a = ↑b
⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑(unitsNonZeroDivisorsEquiv.symm u)) * ↑a = ↑b
|
simpa using h
|
no goals
|
33418b1081f2de4d
|
CategoryTheory.colimitYonedaHomEquiv_π_apply
|
Mathlib/CategoryTheory/Limits/Indization/LocallySmall.lean
|
theorem colimitYonedaHomEquiv_π_apply (η : colimit (F ⋙ yoneda) ⟶ G) (i : Iᵒᵖ) :
limit.π (F.op ⋙ G) i (colimitYonedaHomEquiv F G η) =
η.app (op (F.obj i.unop)) ((colimit.ι (F ⋙ yoneda) i.unop).app _ (𝟙 _))
|
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type u₁
inst✝³ : Category.{v₁, u₁} I
inst✝² : HasColimitsOfShape I (Type v)
inst✝¹ : HasLimitsOfShape Iᵒᵖ (Type v)
inst✝ : HasLimitsOfShape Iᵒᵖ (Type (max u v))
F : I ⥤ C
G : Cᵒᵖ ⥤ Type v
η : colimit (F ⋙ yoneda) ⟶ G
i : Iᵒᵖ
this :
∀ (a : limit ((F.op ⋙ G) ⋙ uliftFunctor.{u, v})),
limit.π (F.op ⋙ G) i ((preservesLimitIso uliftFunctor.{u, v} (F.op ⋙ G)).inv a).down =
(limit.π ((F.op ⋙ G) ⋙ uliftFunctor.{u, v}) i a).down
⊢ ((limit.π (F.op ⋙ G ⋙ uliftFunctor.{u, v}) i ≫ (F.op.associator G uliftFunctor.{u, v}).symm.hom.app i)
((colimitYonedaHomIsoLimitOp F G).hom η)).down =
η.app (op (F.obj (unop i))) ((colimit.ι (F ⋙ yoneda) (unop i)).app (op (F.obj (unop i))) (𝟙 (F.obj (unop i))))
|
simp
|
no goals
|
ab277699a6c63a6c
|
CompositionSeries.mem_eraseLast
|
Mathlib/Order/JordanHolder.lean
|
theorem mem_eraseLast {s : CompositionSeries X} {x : X} (h : 0 < s.length) :
x ∈ s.eraseLast ↔ x ≠ s.last ∧ x ∈ s
|
case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
⊢ (fun i => s.toFun ⟨↑i, ⋯⟩) i ≠ last s ∧ (fun i => s.toFun ⟨↑i, ⋯⟩) i ∈ range s.toFun
|
have hi : (i : ℕ) < s.length := by
conv_rhs => rw [← Nat.add_one_sub_one s.length, Nat.succ_sub h]
exact i.2
|
case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
hi : ↑i < s.length
⊢ (fun i => s.toFun ⟨↑i, ⋯⟩) i ≠ last s ∧ (fun i => s.toFun ⟨↑i, ⋯⟩) i ∈ range s.toFun
|
2ab0afadecbebacb
|
ProbabilityTheory.Kernel.ae_kernel_lt_top
|
Mathlib/Probability/Kernel/Composition/CompProd.lean
|
theorem ae_kernel_lt_top (a : α) (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
∀ᵐ b ∂κ a, η (a, b) (Prod.mk b ⁻¹' s) < ∞
|
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ⊤
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
this : ∀ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ≤ (η (a, b)) (Prod.mk b ⁻¹' t)
ht : MeasurableSet t
⊢ ∀ᵐ (b : β) ∂κ a, (η (a, b)) (Prod.mk b ⁻¹' s) < ⊤
|
have h2t : (κ ⊗ₖ η) a t ≠ ∞ := by rwa [measure_toMeasurable]
|
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ⊤
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
this : ∀ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ≤ (η (a, b)) (Prod.mk b ⁻¹' t)
ht : MeasurableSet t
h2t : ((κ ⊗ₖ η) a) t ≠ ⊤
⊢ ∀ᵐ (b : β) ∂κ a, (η (a, b)) (Prod.mk b ⁻¹' s) < ⊤
|
8ecbe6a27811f30f
|
Matroid.map_isBasis_iff'
|
Mathlib/Data/Matroid/Map.lean
|
lemma map_isBasis_iff' {I X : Set β} {hf} :
(M.map f hf).IsBasis I X ↔ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ I = f '' I₀ ∧ X = f '' X₀
|
case refine_1.intro.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
I : Set α
hI : I ⊆ M.E
X : Set α
hX : X ⊆ M.E
h : M.IsBasis I X
⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ f '' I = f '' I₀ ∧ f '' X = f '' X₀
|
exact ⟨I, X, h, rfl, rfl⟩
|
no goals
|
c192441893919a44
|
Array.find?_range'_eq_none
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Range.lean
|
theorem find?_range'_eq_none {s n : Nat} {p : Nat → Bool} :
(range' s n).find? p = none ↔ ∀ i, s ≤ i → i < s + n → !p i
|
s n : Nat
p : Nat → Bool
⊢ find? p (range' s n) = none ↔ ∀ (i : Nat), s ≤ i → i < s + n → (!p i) = true
|
rw [← List.toArray_range']
|
s n : Nat
p : Nat → Bool
⊢ find? p (List.range' s n).toArray = none ↔ ∀ (i : Nat), s ≤ i → i < s + n → (!p i) = true
|
1040f6ec7d7fe1a6
|
List.getLast_map
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem getLast_map (f : α → β) (l : List α) (h) :
getLast (map f l) h = f (getLast l (by simpa using h))
|
α : Type ?u.133064
β : Type ?u.133073
f : α → β
l : List α
h : map f l ≠ []
⊢ l ≠ []
|
simpa using h
|
no goals
|
9abc0df8c509c2cd
|
AddCircle.gcd_mul_addOrderOf_div_eq
|
Mathlib/Topology/Instances/AddCircle.lean
|
theorem gcd_mul_addOrderOf_div_eq {n : ℕ} (m : ℕ) (hn : 0 < n) :
m.gcd n * addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n
|
case h
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
p : 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ IsOfFinAddOrder ↑(p / ↑n)
|
rwa [← addOrderOf_pos_iff, addOrderOf_period_div hn]
|
no goals
|
dbb8668096a7f28d
|
CategoryTheory.Pretriangulated.Opposite.complete_distinguished_triangle_morphism
|
Mathlib/CategoryTheory/Triangulated/Opposite/Pretriangulated.lean
|
lemma complete_distinguished_triangle_morphism (T₁ T₂ : Triangle Cᵒᵖ)
(hT₁ : T₁ ∈ distinguishedTriangles C) (hT₂ : T₂ ∈ distinguishedTriangles C)
(a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧
T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃
|
case intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ T₂ : Triangle Cᵒᵖ
hT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.distinguishedTriangles
hT₂ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₂) ∈ Pretriangulated.distinguishedTriangles
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
c :
(Opposite.unop ((triangleOpEquivalence C).inverse.obj T₂)).obj₁ ⟶
(Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁)).obj₁
hc₁ :
(Opposite.unop ((triangleOpEquivalence C).inverse.obj T₂)).mor₁ ≫ b.unop =
c ≫ (Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁)).mor₁
hc₂ :
(Opposite.unop ((triangleOpEquivalence C).inverse.obj T₂)).mor₃ ≫ (shiftFunctor C 1).map c =
a.unop ≫ (Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁)).mor₃
⊢ ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map a = c ≫ T₂.mor₃
|
dsimp at c hc₁ hc₂
|
case intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ T₂ : Triangle Cᵒᵖ
hT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.distinguishedTriangles
hT₂ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₂) ∈ Pretriangulated.distinguishedTriangles
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
c : Opposite.unop T₂.obj₃ ⟶ Opposite.unop T₁.obj₃
hc₁ : T₂.mor₂.unop ≫ b.unop = c ≫ T₁.mor₂.unop
hc₂ :
(((opShiftFunctorEquivalence C 1).unitIso.inv.app T₂.obj₁).unop ≫ (shiftFunctor C 1).map T₂.mor₃.unop) ≫
(shiftFunctor C 1).map c =
a.unop ≫ ((opShiftFunctorEquivalence C 1).unitIso.inv.app T₁.obj₁).unop ≫ (shiftFunctor C 1).map T₁.mor₃.unop
⊢ ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map a = c ≫ T₂.mor₃
|
1f895fdca2611d63
|
catalan_eq_centralBinom_div
|
Mathlib/Combinatorics/Enumerative/Catalan.lean
|
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1)
|
case ind
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)
⊢ ↑(catalan d.succ) = ↑d.succ.centralBinom / (↑d.succ + 1)
|
simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
|
case ind
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)
⊢ ∑ x : Fin d.succ, ↑(catalan ↑x) * ↑(catalan (d - ↑x)) = ↑d.succ.centralBinom / (↑d.succ + 1)
|
6311ac92f4458437
|
analyticAt_clog
|
Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
|
theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z
|
z : ℂ
m : z ∈ slitPlane
⊢ AnalyticAt ℂ log z
|
rw [analyticAt_iff_eventually_differentiableAt]
|
z : ℂ
m : z ∈ slitPlane
⊢ ∀ᶠ (z : ℂ) in 𝓝 z, DifferentiableAt ℂ log z
|
057972827dcb7615
|
PowerSeries.rescale_neg_one_X
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
theorem rescale_neg_one_X : rescale (-1 : A) X = -X
|
A : Type u_2
inst✝ : CommRing A
⊢ (rescale (-1)) X = -X
|
rw [rescale_X, map_neg, map_one, neg_one_mul]
|
no goals
|
008008a1724770fe
|
Polynomial.Gal.mul_splits_in_splittingField_of_mul
|
Mathlib/FieldTheory/PolynomialGaloisGroup.lean
|
theorem mul_splits_in_splittingField_of_mul {p₁ q₁ p₂ q₂ : F[X]} (hq₁ : q₁ ≠ 0) (hq₂ : q₂ ≠ 0)
(h₁ : p₁.Splits (algebraMap F q₁.SplittingField))
(h₂ : p₂.Splits (algebraMap F q₂.SplittingField)) :
(p₁ * p₂).Splits (algebraMap F (q₁ * q₂).SplittingField)
|
case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : Splits (algebraMap F q₁.SplittingField) p₁
h₂ : Splits (algebraMap F q₂.SplittingField) p₂
⊢ Splits (algebraMap F (q₁ * q₂).SplittingField) p₁
|
rw [←
(SplittingField.lift q₁
(splits_of_splits_of_dvd (algebraMap F (q₁ * q₂).SplittingField) (mul_ne_zero hq₁ hq₂)
(SplittingField.splits _) (dvd_mul_right q₁ q₂))).comp_algebraMap]
|
case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : Splits (algebraMap F q₁.SplittingField) p₁
h₂ : Splits (algebraMap F q₂.SplittingField) p₂
⊢ Splits ((↑(SplittingField.lift q₁ ⋯)).comp (algebraMap F q₁.SplittingField)) p₁
|
12f0bbd6ecef4d4b
|
Nat.Prime.emultiplicity_choose'
|
Mathlib/Data/Nat/Multiplicity.lean
|
theorem emultiplicity_choose' {p n k b : ℕ} (hp : p.Prime) (hnb : log p (n + k) < b) :
emultiplicity p (choose (n + k) k) = #{i ∈ Ico 1 b | p ^ i ≤ k % p ^ i + n % p ^ i}
|
p n k b : ℕ
hp : Prime p
hnb : log p (n + k) < b
this : (n + k).choose k * k ! * n ! = (n + k)!
⊢ emultiplicity p ((n + k).choose k * k ! * n !) =
↑(#(filter (fun i => p ^ i ≤ k % p ^ i + n % p ^ i) (Ico 1 b))) + emultiplicity p (k ! * n !)
|
rw [this, hp.emultiplicity_factorial hnb, hp.emultiplicity_mul,
hp.emultiplicity_factorial ((log_mono_right (le_add_left k n)).trans_lt hnb),
hp.emultiplicity_factorial ((log_mono_right (le_add_left n k)).trans_lt
(add_comm n k ▸ hnb)), multiplicity_choose_aux hp (le_add_left k n)]
|
p n k b : ℕ
hp : Prime p
hnb : log p (n + k) < b
this : (n + k).choose k * k ! * n ! = (n + k)!
⊢ ↑(∑ i ∈ Ico 1 b, k / p ^ i + ∑ i ∈ Ico 1 b, (n + k - k) / p ^ i +
#(filter (fun i => p ^ i ≤ k % p ^ i + (n + k - k) % p ^ i) (Ico 1 b))) =
↑(#(filter (fun i => p ^ i ≤ k % p ^ i + n % p ^ i) (Ico 1 b))) +
(↑(∑ i ∈ Ico 1 b, k / p ^ i) + ↑(∑ i ∈ Ico 1 b, n / p ^ i))
|
ed202d3f67c3f305
|
tendsto_exp_div_rpow_atTop
|
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
theorem tendsto_exp_div_rpow_atTop (s : ℝ) : Tendsto (fun x : ℝ => exp x / x ^ s) atTop atTop
|
s : ℝ
⊢ Tendsto (fun x => rexp x / x ^ s) atTop atTop
|
obtain ⟨n, hn⟩ := archimedean_iff_nat_lt.1 Real.instArchimedean s
|
case intro
s : ℝ
n : ℕ
hn : s < ↑n
⊢ Tendsto (fun x => rexp x / x ^ s) atTop atTop
|
2529d965a6e29a2f
|
CompleteOrthogonalIdempotents.of_ker_isNilpotent_of_isMulCentral
|
Mathlib/RingTheory/Idempotents.lean
|
theorem CompleteOrthogonalIdempotents.of_ker_isNilpotent_of_isMulCentral
(h : ∀ x ∈ RingHom.ker f, IsNilpotent x)
(he : ∀ i, IsIdempotentElem (e i))
(he' : ∀ i, IsMulCentral (e i))
(he'' : CompleteOrthogonalIdempotents (f ∘ e)) :
CompleteOrthogonalIdempotents e
|
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
e : I → R
inst✝ : Fintype I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
he : ∀ (i : I), IsIdempotentElem (e i)
he' : ∀ (i : I), IsMulCentral (e i)
he'' : CompleteOrthogonalIdempotents (⇑f ∘ e)
e' : I → R
h₁ : CompleteOrthogonalIdempotents e'
h₂ : ⇑f ∘ e' = ⇑f ∘ e
⊢ e = e'
|
ext i
|
case h
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
e : I → R
inst✝ : Fintype I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
he : ∀ (i : I), IsIdempotentElem (e i)
he' : ∀ (i : I), IsMulCentral (e i)
he'' : CompleteOrthogonalIdempotents (⇑f ∘ e)
e' : I → R
h₁ : CompleteOrthogonalIdempotents e'
h₂ : ⇑f ∘ e' = ⇑f ∘ e
i : I
⊢ e i = e' i
|
d1848a670d0365bc
|
hasFDerivAt_norm_rpow
|
Mathlib/Analysis/InnerProductSpace/NormPow.lean
|
theorem hasFDerivAt_norm_rpow (x : E) {p : ℝ} (hp : 1 < p) :
HasFDerivAt (fun x : E ↦ ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • innerSL ℝ x) x
|
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
⊢ HasFDerivAt (fun x => ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) x
|
simp only [hx, norm_zero, map_zero, smul_zero]
|
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
⊢ HasFDerivAt (fun x => ‖x‖ ^ p) 0 0
|
df712761f8e3d685
|
PartialHomeomorph.nhds_eq_comap_inf_principal
|
Mathlib/Topology/PartialHomeomorph.lean
|
theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) :
𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source
|
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
x : X
hx : x ∈ e.source
⊢ 𝓝 x = comap (↑e) (𝓝 (↑e x)) ⊓ 𝓟 e.source
|
lift x to e.source using hx
|
case intro
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
x : { x // x ∈ e.source }
⊢ 𝓝 ↑x = comap (↑e) (𝓝 (↑e ↑x)) ⊓ 𝓟 e.source
|
ef44c68f626bbbb6
|
MeasureTheory.unifTight_of_subsingleton
|
Mathlib/MeasureTheory/Function/UnifTight.lean
|
theorem unifTight_of_subsingleton [Subsingleton ι] (hp_top : p ≠ ∞)
{f : ι → α → β} (hf : ∀ i, MemLp (f i) p μ) : UnifTight f p μ := fun ε hε ↦ by
by_cases hε_top : ε = ∞
· exact ⟨∅, by measurability, fun _ => hε_top.symm ▸ le_top⟩
by_cases hι : Nonempty ι
case neg => exact ⟨∅, (by measurability), fun i => False.elim <| hι <| Nonempty.intro i⟩
obtain ⟨i⟩ := hι
obtain ⟨s, _, hμs, hfε⟩ := (hf i).exists_eLpNorm_indicator_compl_lt hp_top (coe_ne_zero.2 hε.ne')
refine ⟨s, ne_of_lt hμs, fun j => ?_⟩
convert hfε.le
|
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ↑ε = ⊤
⊢ ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
|
exact ⟨∅, by measurability, fun _ => hε_top.symm ▸ le_top⟩
|
no goals
|
6576e3dc8dfe236e
|
MeasureTheory.Measure.haar.mul_left_index_le
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
theorem mul_left_index_le {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty)
(g : G) : index ((fun h => g * h) '' K) V ≤ index K V
|
case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
h2s : s.card = index K V
g₁ g₂ : G
hg₂ : g₂ ∈ s
hg₁ : g₂ * g₁ ∈ V
⊢ (fun h => g * h) g₁ ∈ ⋃ g_1 ∈ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s, (fun h => g_1 * h) ⁻¹' V
|
simp only [exists_prop, mem_iUnion, Finset.mem_map, Equiv.coe_mulRight,
exists_exists_and_eq_and, mem_preimage, Equiv.toEmbedding_apply]
|
case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
h2s : s.card = index K V
g₁ g₂ : G
hg₂ : g₂ ∈ s
hg₁ : g₂ * g₁ ∈ V
⊢ ∃ a ∈ s, a * g⁻¹ * (g * g₁) ∈ V
|
d7d19e4b8c244509
|
FreeGroup.reduce_invRev
|
Mathlib/GroupTheory/FreeGroup/Reduce.lean
|
theorem reduce_invRev {w : List (α × Bool)} : reduce (invRev w) = invRev (reduce w)
|
case H
α : Type u_1
inst✝ : DecidableEq α
w : List (α × Bool)
⊢ Red (reduce (invRev w)) (invRev (reduce w))
|
rw [← red_invRev_iff, invRev_invRev]
|
case H
α : Type u_1
inst✝ : DecidableEq α
w : List (α × Bool)
⊢ Red (invRev (reduce (invRev w))) (reduce w)
|
931dc9c3210eb8cc
|
IsConj.eq_of_left_mem_center
|
Mathlib/GroupTheory/Subgroup/Center.lean
|
theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h
|
M : Type u_2
inst✝ : Monoid M
g h : M
H : IsConj g h
Hg : g ∈ Set.center M
⊢ g = h
|
rcases H with ⟨u, hu⟩
|
case intro
M : Type u_2
inst✝ : Monoid M
g h : M
Hg : g ∈ Set.center M
u : Mˣ
hu : SemiconjBy (↑u) g h
⊢ g = h
|
7f39f38db800ef78
|
round_neg_two_inv
|
Mathlib/Algebra/Order/Round.lean
|
theorem round_neg_two_inv : round (-2⁻¹ : α) = 0
|
α : Type u_2
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
⊢ round (-2⁻¹) = 0
|
simp only [round_eq, ← one_div, neg_add_cancel, floor_zero]
|
no goals
|
421e31398e309b65
|
Nat.exists_most_significant_bit
|
Mathlib/Data/Nat/Bitwise.lean
|
theorem exists_most_significant_bit {n : ℕ} (h : n ≠ 0) :
∃ i, testBit n i = true ∧ ∀ j, i < j → testBit n j = false
|
case pos
b : Bool
n : ℕ
hn : n ≠ 0 → ∃ i, n.testBit i = true ∧ ∀ (j : ℕ), i < j → n.testBit j = false
h : bit b n ≠ 0
h' : n = 0
⊢ ∃ i, (bit b n).testBit i = true ∧ ∀ (j : ℕ), i < j → (bit b n).testBit j = false
|
subst h'
|
case pos
b : Bool
hn : 0 ≠ 0 → ∃ i, testBit 0 i = true ∧ ∀ (j : ℕ), i < j → testBit 0 j = false
h : bit b 0 ≠ 0
⊢ ∃ i, (bit b 0).testBit i = true ∧ ∀ (j : ℕ), i < j → (bit b 0).testBit j = false
|
164bb1ef025f0b32
|
FormalMultilinearSeries.comp_coeff_one
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem comp_coeff_one (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(v : Fin 1 → E) : (q.comp p) 1 v = q 1 fun _i => p 1 v
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
this : {Composition.ones 1} = Finset.univ
⊢ (q (Composition.ones 1).length) (p.applyComposition (Composition.ones 1) v) = (q 1) fun _i => (p 1) v
|
refine q.congr (by simp) fun i hi1 hi2 => ?_
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
this : {Composition.ones 1} = Finset.univ
i : ℕ
hi1 : i < (Composition.ones 1).length
hi2 : i < 1
⊢ p.applyComposition (Composition.ones 1) v ⟨i, hi1⟩ = (p 1) v
|
2ac1709b5f4213f3
|
FirstOrder.Language.HomClass.realize_term
|
Mathlib/ModelTheory/Semantics.lean
|
theorem HomClass.realize_term {F : Type*} [FunLike F M N] [HomClass L F M N]
(g : F) {t : L.Term α} {v : α → M} :
t.realize (g ∘ v) = g (t.realize v)
|
L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure M
inst✝² : L.Structure N
α : Type u'
F : Type u_4
inst✝¹ : FunLike F M N
inst✝ : L.HomClass F M N
g : F
t : L.Term α
v : α → M
⊢ Term.realize (⇑g ∘ v) t = g (Term.realize v t)
|
induction t
|
case var
L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure M
inst✝² : L.Structure N
α : Type u'
F : Type u_4
inst✝¹ : FunLike F M N
inst✝ : L.HomClass F M N
g : F
v : α → M
a✝ : α
⊢ Term.realize (⇑g ∘ v) (var a✝) = g (Term.realize v (var a✝))
case func
L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure M
inst✝² : L.Structure N
α : Type u'
F : Type u_4
inst✝¹ : FunLike F M N
inst✝ : L.HomClass F M N
g : F
v : α → M
l✝ : ℕ
_f✝ : L.Functions l✝
_ts✝ : Fin l✝ → L.Term α
_ts_ih✝ : ∀ (a : Fin l✝), Term.realize (⇑g ∘ v) (_ts✝ a) = g (Term.realize v (_ts✝ a))
⊢ Term.realize (⇑g ∘ v) (func _f✝ _ts✝) = g (Term.realize v (func _f✝ _ts✝))
|
df81aa8676069e92
|
ContinuousMap.exists_extension_forall_mem_of_isClosedEmbedding
|
Mathlib/Topology/TietzeExtension.lean
|
theorem exists_extension_forall_mem_of_isClosedEmbedding (f : C(X, ℝ)) {t : Set ℝ} {e : X → Y}
[hs : OrdConnected t] (hf : ∀ x, f x ∈ t) (hne : t.Nonempty) (he : IsClosedEmbedding e) :
∃ g : C(Y, ℝ), (∀ y, g y ∈ t) ∧ g ∘ e = f
|
case intro.intro.refine_2.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : C(X, ℝ)
t : Set ℝ
e : X → Y
hs : t.OrdConnected
hf : ∀ (x : X), f x ∈ t
hne : t.Nonempty
he : IsClosedEmbedding e
h : ℝ ≃o ↑(Ioo (-1) 1)
F : X →ᵇ ℝ := { toFun := Subtype.val ∘ ⇑h ∘ ⇑f, continuous_toFun := ⋯, map_bounded' := ⋯ }
t' : Set ℝ := Subtype.val ∘ ⇑h '' t
ht_sub : t' ⊆ Ioo (-1) 1
this : t'.OrdConnected
hFt : ∀ (x : X), F x ∈ t'
G : Y →ᵇ ℝ
hG : ∀ (y : Y), G y ∈ t'
hGF : ⇑G ∘ e = ⇑F
g : C(Y, ℝ) := { toFun := ⇑h.symm ∘ Set.codRestrict (⇑G) (Ioo (-1) 1) ⋯, continuous_toFun := ⋯ }
hgG : ∀ {y : Y} {a : ℝ}, g y = a ↔ G y = ↑(h a)
x : X
⊢ (⇑g ∘ e) x = f x
|
exact hgG.2 (congr_fun hGF _)
|
no goals
|
b5717c30218ed2f5
|
BoxIntegral.IntegrationParams.tendsto_embedBox_toFilteriUnion_top
|
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
|
theorem tendsto_embedBox_toFilteriUnion_top (l : IntegrationParams) (h : I ≤ J) :
Tendsto (TaggedPrepartition.embedBox I J h) (l.toFilteriUnion I ⊤)
(l.toFilteriUnion J (Prepartition.single J I h))
|
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
⊢ Tendsto (⇑(embedBox I J h)) (toFilteriUnion I ⊤) (toFilteriUnion J (Prepartition.single J I h))
|
simp only [toFilteriUnion, tendsto_iSup]
|
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
⊢ ∀ (i : ℝ≥0),
Tendsto (⇑(embedBox I J h)) (l.toFilterDistortioniUnion I i ⊤)
(⨆ c, l.toFilterDistortioniUnion J c (Prepartition.single J I h))
|
75f666dbd7c5bb0f
|
Asymptotics.isLittleO_principal
|
Mathlib/Analysis/Asymptotics/Lemmas.lean
|
theorem isLittleO_principal {s : Set α} : f'' =o[𝓟 s] g' ↔ ∀ x ∈ s, f'' x = 0
|
α : Type u_1
F' : Type u_7
E'' : Type u_9
inst✝¹ : SeminormedAddCommGroup F'
inst✝ : NormedAddCommGroup E''
g' : α → F'
f'' : α → E''
s : Set α
h : ∀ x ∈ s, f'' x = 0
⊢ (fun _x => 0) =ᶠ[𝓟 s] f''
|
exact fun x hx ↦ (h x hx).symm
|
no goals
|
9fb493bf550d3541
|
Multiset.count_map_eq_count'
|
Mathlib/Data/Multiset/Filter.lean
|
theorem count_map_eq_count' [DecidableEq β] (f : α → β) (s : Multiset α) (hf : Function.Injective f)
(x : α) : (s.map f).count (f x) = s.count x
|
case neg
α : Type u_1
β : Type v
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
s : Multiset α
hf : Injective f
x : α
H : x ∉ s
⊢ count (f x) (map f s) = count x s
|
rw [count_eq_zero_of_not_mem H, count_eq_zero, mem_map]
|
case neg
α : Type u_1
β : Type v
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
s : Multiset α
hf : Injective f
x : α
H : x ∉ s
⊢ ¬∃ a ∈ s, f a = f x
|
617a7c8c055ce103
|
surjOn_Icc_of_monotone_surjective
|
Mathlib/Order/Interval/Set/SurjOn.lean
|
theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
{a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b))
|
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a b : α
hab : a ≤ b
p : β
hp : p ∈ Icc (f a) (f b)
⊢ p ∈ f '' Icc a b
|
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')
|
case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a b : α
hab : a ≤ b
hp : f a ∈ Icc (f a) (f b)
⊢ f a ∈ f '' Icc a b
case inr.inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a b : α
hab : a ≤ b
hp : f b ∈ Icc (f a) (f b)
⊢ f b ∈ f '' Icc a b
case inr.inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a b : α
hab : a ≤ b
p : β
hp : p ∈ Icc (f a) (f b)
hp' : p ∈ Ioo (f a) (f b)
⊢ p ∈ f '' Icc a b
|
c609773ecb1a1f37
|
Nat.bit_mod_two_eq_zero_iff
|
Mathlib/Data/Nat/Bitwise.lean
|
lemma bit_mod_two_eq_zero_iff (a x) :
bit a x % 2 = 0 ↔ !a
|
a : Bool
x : ℕ
⊢ bit a x % 2 = 0 ↔ (!a) = true
|
simp
|
no goals
|
c6c9ec748d9d84d0
|
convexOn_iff_pairwise_pos
|
Mathlib/Analysis/Convex/Function.lean
|
theorem convexOn_iff_pairwise_pos {s : Set E} {f : E → β} :
ConvexOn 𝕜 s f ↔
Convex 𝕜 s ∧
s.Pairwise fun x y =>
∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y
|
𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : OrderedAddCommMonoid β
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 β
s : Set E
f : E → β
⊢ (Convex 𝕜 s ∧
∀ ⦃x : E⦄,
x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y) ↔
Convex 𝕜 s ∧ s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y
|
refine
and_congr_right'
⟨fun h x hx y hy _ a b ha hb hab => h hx hy ha hb hab, fun h x hx y hy a b ha hb hab => ?_⟩
|
𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : OrderedAddCommMonoid β
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 β
s : Set E
f : E → β
h : s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ a • f x + b • f y
|
24d415d43f469181
|
MeasureTheory.L1.edist_def
|
Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean
|
theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ edist f g = ∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ
|
simp only [Lp.edist_def, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_enorm, Pi.sub_apply,
one_toReal, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ (if 1 = ⊤ then eLpNormEssSup (↑↑f - ↑↑g) μ else ∫⁻ (a : α), ‖↑↑f a - ↑↑g a‖ₑ ∂μ) =
∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ
|
fd4f8e4e94f651b9
|
Polynomial.leadingCoeff_cubic
|
Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean
|
theorem leadingCoeff_cubic (ha : a ≠ 0) :
leadingCoeff (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d) = a
|
R : Type u
a b c d : R
inst✝ : Semiring R
ha : a ≠ 0
⊢ (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).leadingCoeff = a
|
rw [add_assoc, add_assoc, ← add_assoc (C b * X ^ 2), add_comm,
leadingCoeff_add_of_degree_lt <| degree_quadratic_lt_degree_C_mul_X_cb ha,
leadingCoeff_C_mul_X_pow]
|
no goals
|
e4540a9c25cdab0b
|
UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto
|
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
theorem UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto
(hF : UniformCauchySeqOnFilter F p p')
(hF' : ∀ᶠ x : α in p', Tendsto (fun n => F n x) p (𝓝 (f x))) :
TendstoUniformlyOnFilter F f p p'
|
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
hF : UniformCauchySeqOnFilter F p p'
hF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))
⊢ TendstoUniformlyOnFilter F f p p'
|
rcases p.eq_or_neBot with rfl | _
|
case inl
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p' : Filter α
hF : UniformCauchySeqOnFilter F ⊥ p'
hF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) ⊥ (𝓝 (f x))
⊢ TendstoUniformlyOnFilter F f ⊥ p'
case inr
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
hF : UniformCauchySeqOnFilter F p p'
hF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))
h✝ : p.NeBot
⊢ TendstoUniformlyOnFilter F f p p'
|
be7f1c945998c45a
|
MeasureTheory.measurableSet_generateFrom_singleton_iff
|
Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
|
theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ
|
α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
p : Prop
hp : p ∈ x
hpneg : False ↔ p
⊢ False ∈ x
|
convert hp
|
no goals
|
39127bdfcef76ebf
|
Real.invariant
|
Mathlib/NumberTheory/DiophantineApproximation/Basic.lean
|
theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v)
|
case refine_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : ξ - ↑u / ↑v < (↑v * (2 * ↑v - 1))⁻¹
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v
|
rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h'
|
case refine_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : fract ξ < 2 / (2 * ↑v - 1)
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v
|
8eb0aea68cc1bde9
|
Doset.doset_union_rightCoset
|
Mathlib/GroupTheory/DoubleCoset.lean
|
theorem doset_union_rightCoset (H K : Subgroup G) (a : G) :
⋃ k : K, op (a * k) • ↑H = doset a H K
|
case h.mpr.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a x✝ x : G
hx : x ∈ H
y : G
hy : y ∈ K
hxy : x✝ = x * a * y
⊢ ∃ i, x✝ * ((↑i)⁻¹ * a⁻¹) ∈ H
|
refine ⟨⟨y, hy⟩, ?_⟩
|
case h.mpr.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a x✝ x : G
hx : x ∈ H
y : G
hy : y ∈ K
hxy : x✝ = x * a * y
⊢ x✝ * ((↑⟨y, hy⟩)⁻¹ * a⁻¹) ∈ H
|
1ad4289c4ad1f9f3
|
GenContFract.IntFractPair.coe_stream_nth_rat_eq
|
Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean
|
theorem coe_stream_nth_rat_eq (v_eq_q : v = (↑q : K)) (n : ℕ) :
((IntFractPair.stream q n).map (mapFr (↑)) : Option <| IntFractPair K) =
IntFractPair.stream v n
|
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
b : ℤ
fr : ℚ
stream_q_nth_eq : IntFractPair.stream q n = some { b := b, fr := fr }
fr_ne_zero : ¬fr = 0
IH : some { b := b, fr := ↑fr } = IntFractPair.stream (↑q) n
⊢ (↑fr)⁻¹ = ↑fr⁻¹
|
norm_cast
|
no goals
|
3c20096a5980fe60
|
SimpleGraph.connected_iff_exists_forall_reachable
|
Mathlib/Combinatorics/SimpleGraph/Path.lean
|
lemma connected_iff_exists_forall_reachable : G.Connected ↔ ∃ v, ∀ w, G.Reachable v w
|
V : Type u
G : SimpleGraph V
⊢ G.Preconnected ∧ Nonempty V ↔ ∃ v, ∀ (w : V), G.Reachable v w
|
constructor
|
case mp
V : Type u
G : SimpleGraph V
⊢ G.Preconnected ∧ Nonempty V → ∃ v, ∀ (w : V), G.Reachable v w
case mpr
V : Type u
G : SimpleGraph V
⊢ (∃ v, ∀ (w : V), G.Reachable v w) → G.Preconnected ∧ Nonempty V
|
1c445adc9e510c30
|
Real.Gamma_one_half_eq
|
Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
|
theorem Real.Gamma_one_half_eq : Real.Gamma (1 / 2) = √π
|
case h.e'_2
⊢ ∫ (x : ℝ) in Ioi 0, (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =
∫ (a : ℝ) in Ioi 0, 2 * rexp (-1 * a ^ 2)
|
refine setIntegral_congr_fun measurableSet_Ioi fun x hx => ?_
|
case h.e'_2
x : ℝ
hx : x ∈ Ioi 0
⊢ (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)
|
a42e522f111124af
|
QuasispectrumRestricts.cfc
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
|
theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalContinuousFunctionalCalculus R p where
predicate_zero := h0
compactSpace_quasispectrum a
|
case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
⊢ (nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯)
{ toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } =
a
case hom_map_spectrum
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(σₙ R a), R)₀), σₙ R ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) f_1) = range ⇑f_1
case predicate_hom
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(σₙ R a), R)₀), p ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) f_1)
|
case hom_id => exact ((h a).mp ha).2.nonUnitalStarAlgHom_id <| cfcₙHom_id ((h a).mp ha).1
|
case hom_map_spectrum
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(σₙ R a), R)₀), σₙ R ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) f_1) = range ⇑f_1
case predicate_hom
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(σₙ R a), R)₀), p ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) f_1)
|
3af5d542f22c6b37
|
GenContFract.coe_of_s_rat_eq
|
Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean
|
theorem coe_of_s_rat_eq (v_eq_q : v = (↑q : K)) :
((of q).s.map (Pair.map ((↑))) : Stream'.Seq <| Pair K) = (of v).s
|
case h.a
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
a✝ : Pair K
⊢ a✝ ∈ (Stream'.Seq.map (Pair.map Rat.cast) (of q).s).get? n ↔ a✝ ∈ Option.map (Pair.map Rat.cast) ((of q).s.get? n)
|
rfl
|
no goals
|
68950bb07f32bd54
|
MeasureTheory.IsStoppingTime.add_const_nat
|
Mathlib/Probability/Process/Stopping.lean
|
theorem add_const_nat {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) {i : ℕ} :
IsStoppingTime f fun ω => τ ω + i
|
case pos
Ω : Type u_1
m : MeasurableSpace Ω
f : Filtration ℕ m
τ : Ω → ℕ
hτ : IsStoppingTime f τ
i j : ℕ
hij : i ≤ j
⊢ MeasurableSet {ω | τ ω + i = j}
|
simp_rw [eq_comm, ← Nat.sub_eq_iff_eq_add hij, eq_comm]
|
case pos
Ω : Type u_1
m : MeasurableSpace Ω
f : Filtration ℕ m
τ : Ω → ℕ
hτ : IsStoppingTime f τ
i j : ℕ
hij : i ≤ j
⊢ MeasurableSet {ω | τ ω = j - i}
|
089b9068779bbc69
|
Array.forIn_map
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Monadic.lean
|
theorem forIn_map [Monad m] [LawfulMonad m]
(l : Array α) (g : α → β) (f : β → γ → m (ForInStep γ)) :
forIn (l.map g) init f = forIn l init fun a y => f (g a) y
|
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_4
γ : Type u_1
init : γ
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : Array α
g : α → β
f : β → γ → m (ForInStep γ)
⊢ forIn (map g l) init f = forIn l init fun a y => f (g a) y
|
cases l
|
case mk
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_4
γ : Type u_1
init : γ
inst✝¹ : Monad m
inst✝ : LawfulMonad m
g : α → β
f : β → γ → m (ForInStep γ)
toList✝ : List α
⊢ forIn (map g { toList := toList✝ }) init f = forIn { toList := toList✝ } init fun a y => f (g a) y
|
992467d3c065c582
|
Module.freeLocus_localization
|
Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean
|
lemma freeLocus_localization (S : Submonoid R) :
freeLocus (Localization S) (LocalizedModule S M) =
comap (algebraMap R _) ⁻¹' freeLocus R M
|
case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
⊢ p ∈ freeLocus (Localization S) (LocalizedModule S M) ↔ (comap (algebraMap R (Localization S))) p ∈ freeLocus R M
|
let Mₚ := LocalizedModule p'.primeCompl M
|
case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
⊢ p ∈ freeLocus (Localization S) (LocalizedModule S M) ↔ (comap (algebraMap R (Localization S))) p ∈ freeLocus R M
|
ac4a766398c7061e
|
Std.DHashMap.Internal.Raw₀.getKeyD_modify_self
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem getKeyD_modify_self (h : m.1.WF) [Inhabited α] {k fallback : α} {f : β k → β k} :
(m.modify k f).getKeyD k fallback = if m.contains k then k else fallback
|
α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : LawfulBEq α
h : m.val.WF
inst✝ : Inhabited α
k fallback : α
f : β k → β k
⊢ (m.modify k f).getKeyD k fallback = if m.contains k = true then k else fallback
|
simp_to_model [modify] using List.getKeyD_modifyKey_self
|
no goals
|
37c14462b0d2c3a7
|
CategoryTheory.Presheaf.isSheaf_iff_multiequalizer
|
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
theorem isSheaf_iff_multiequalizer [∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)] :
IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P)
|
case refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
S : J.Cover X
⊢ IsIso (S.toMultiequalizer P) → Nonempty (IsLimit (S.multifork P))
|
intro h
|
case refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
S : J.Cover X
h : IsIso (S.toMultiequalizer P)
⊢ Nonempty (IsLimit (S.multifork P))
|
32f315e61f5eaf0d
|
isZero_Ext_succ_of_projective
|
Mathlib/CategoryTheory/Abelian/Ext.lean
|
/-- If `X : C` is projective and `n : ℕ`, then `Ext^(n + 1) X Y ≅ 0` for any `Y`. -/
lemma isZero_Ext_succ_of_projective (X Y : C) [Projective X] (n : ℕ) :
IsZero (((Ext R C (n + 1)).obj (Opposite.op X)).obj Y)
|
R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{u_3, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
⊢ 𝟙 (((linearYoneda R C).obj Y).obj (Opposite.op (((ChainComplex.single₀ C).obj X).X (n + 1)))) = 0
|
ext (x : _ ⟶ _)
|
case hf.h
R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{u_3, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
x : Opposite.unop (Opposite.op (((ChainComplex.single₀ C).obj X).X (n + 1))) ⟶ Y
⊢ (ModuleCat.Hom.hom (𝟙 (((linearYoneda R C).obj Y).obj (Opposite.op (((ChainComplex.single₀ C).obj X).X (n + 1))))))
x =
(ModuleCat.Hom.hom 0) x
|
40abdca1318b58b3
|
ProbabilityTheory.Kernel.IsCondKernel.isProbabilityMeasure_ae
|
Mathlib/Probability/Kernel/Disintegration/Basic.lean
|
/-- A conditional kernel is almost everywhere a probability measure. -/
lemma IsCondKernel.isProbabilityMeasure_ae [IsFiniteKernel κ.fst] [κ.IsCondKernel κCond] (a : α) :
∀ᵐ b ∂(κ.fst a), IsProbabilityMeasure (κCond (a, b))
|
case neg
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : 0 = κ
h_sfin : ¬IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond (a, b))
|
simp [h.symm]
|
no goals
|
d615b2a173689899
|
Ordinal.card_opow_le_of_omega0_le_left
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem card_opow_le_of_omega0_le_left {a : Ordinal} (ha : ω ≤ a) (b : Ordinal) :
(a ^ b).card ≤ max a.card b.card
|
case h
a : Ordinal.{u_1}
ha : ω ≤ a
b✝ b : Ordinal.{u_1}
hb : b.IsLimit
IH : ∀ o' < b, (a ^ o').card ≤ a.card ⊔ o'.card
⊢ ∀ (i : ↑(Iio b)), (a ^ ↑i).card ≤ a.card ⊔ b.card
|
intro c
|
case h
a : Ordinal.{u_1}
ha : ω ≤ a
b✝ b : Ordinal.{u_1}
hb : b.IsLimit
IH : ∀ o' < b, (a ^ o').card ≤ a.card ⊔ o'.card
c : ↑(Iio b)
⊢ (a ^ ↑c).card ≤ a.card ⊔ b.card
|
ca96380056897695
|
MeasurableSet.exists_isOpen_symmDiff_lt
|
Mathlib/MeasureTheory/Measure/Regular.lean
|
theorem _root_.MeasurableSet.exists_isOpen_symmDiff_lt [InnerRegularCompactLTTop μ]
[IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
{s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, IsOpen U ∧ μ U < ∞ ∧ μ (U ∆ s) < ε
|
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
this : ε / 2 ≠ 0
K : Set α
hKs : K ⊆ s
hKco : IsCompact K
hKcl : IsClosed K
hμK : μ (s \ K) < ε / 2
U : Set α
hKU : K ⊆ U
hUo : IsOpen U
hμU : μ U < μ K + ε / 2
⊢ μ K ≠ ⊤
|
exact ne_top_of_le_ne_top hμs (by gcongr)
|
no goals
|
498e42d44efdd46a
|
Real.tendsto_logb_atTop_of_base_lt_one
|
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot
|
b : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e : ℝ
⊢ ∃ i, ∀ (a : ℝ), i ≤ a → logb b a ≤ e
|
use 1 ⊔ b ^ e
|
case h
b : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e : ℝ
⊢ ∀ (a : ℝ), 1 ⊔ b ^ e ≤ a → logb b a ≤ e
|
294c4d8092b3c5ec
|
EReal.right_distrib_of_nonneg
|
Mathlib/Data/Real/EReal.lean
|
lemma right_distrib_of_nonneg {a b c : EReal} (ha : 0 ≤ a) (hb : 0 ≤ b) :
(a + b) * c = a * c + b * c
|
case inr.inr.inr.inr.h_real.h_bot
b : EReal
hb : 0 ≤ b
b_pos : 0 < b
a✝ : ℝ
c_pos : 0 < ↑a✝
ha : 0 ≤ ⊥
a_pos : 0 < ⊥
⊢ False
|
exact not_lt_bot a_pos
|
no goals
|
4e8a6473d4cbc325
|
isSimpleRing_iff_isField
|
Mathlib/RingTheory/SimpleRing/Field.lean
|
lemma isSimpleRing_iff_isField (A : Type*) [CommRing A] : IsSimpleRing A ↔ IsField A :=
⟨fun _ ↦ Subring.topEquiv.symm.toMulEquiv.isField _ <| by
rw [← Subring.center_eq_top A]; exact IsSimpleRing.isField_center A,
fun h ↦ letI := h.toField; inferInstance⟩
|
A : Type u_1
inst✝ : CommRing A
x✝ : IsSimpleRing A
⊢ IsField ↥(Subring.center A)
|
exact IsSimpleRing.isField_center A
|
no goals
|
5feac9988e67d3e2
|
MeasureTheory.lintegral_sub_le'
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) :
∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ
|
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ⊤
|
exact le_top
|
no goals
|
3c78cc5e3caf47ca
|
Profinite.exists_locallyConstant_finite_aux
|
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom
|
case intro.intro.e_f.h
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
a : α
⊢ (LocallyConstant.map ι f).flip a = (LocallyConstant.comap (TopCat.Hom.hom (C.π.app j0)) ggg).flip a
|
change ff a = _
|
case intro.intro.e_f.h
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
a : α
⊢ ff a = (LocallyConstant.comap (TopCat.Hom.hom (C.π.app j0)) ggg).flip a
|
ad55d9835f0fad2e
|
CompHausLike.sigmaComparison_eq_comp_isos
|
Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean
|
theorem sigmaComparison_eq_comp_isos : sigmaComparison X σ =
(X.mapIso (opCoproductIsoProduct'
(finiteCoproduct.isColimit.{u, u} (fun a ↦ of P (σ a)))
(productIsProduct fun x ↦ Opposite.op (of P (σ x))))).hom ≫
(PreservesProduct.iso X fun a ↦ ⟨of P (σ a)⟩).hom ≫
(Types.productIso.{u, max u w} fun a ↦ X.obj ⟨of P (σ a)⟩).hom
|
case h.h
P : TopCat → Prop
inst✝⁶ : HasExplicitFiniteCoproducts P
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝⁵ : PreservesFiniteProducts X
α : Type u
inst✝⁴ : Finite α
σ : α → Type u
inst✝³ : (a : α) → TopologicalSpace (σ a)
inst✝² : ∀ (a : α), CompactSpace (σ a)
inst✝¹ : ∀ (a : α), T2Space (σ a)
inst✝ : ∀ (a : α), HasProp P (σ a)
x : X.obj (Opposite.op (of P ((a : α) × σ a)))
a : α
⊢ sigmaComparison X σ x a =
Pi.π (fun a => X.obj (Opposite.op (of P (σ a)))) a
(piComparison X (fun a => Opposite.op (of P (σ a)))
(X.map
(opCoproductIsoProduct' (finiteCoproduct.isColimit fun a => of P (σ a))
(productIsProduct fun x => Opposite.op (of P (σ x)))).hom
x))
|
have := congrFun (piComparison_comp_π X (fun a ↦ ⟨of P (σ a)⟩) a)
|
case h.h
P : TopCat → Prop
inst✝⁶ : HasExplicitFiniteCoproducts P
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝⁵ : PreservesFiniteProducts X
α : Type u
inst✝⁴ : Finite α
σ : α → Type u
inst✝³ : (a : α) → TopologicalSpace (σ a)
inst✝² : ∀ (a : α), CompactSpace (σ a)
inst✝¹ : ∀ (a : α), T2Space (σ a)
inst✝ : ∀ (a : α), HasProp P (σ a)
x : X.obj (Opposite.op (of P ((a : α) × σ a)))
a : α
this :
∀ (a_1 : X.obj (∏ᶜ fun a => Opposite.op (of P (σ a)))),
((piComparison X fun a => Opposite.op (of P (σ a))) ≫ Pi.π (fun b => X.obj (Opposite.op (of P (σ b)))) a) a_1 =
X.map (Pi.π (fun a => Opposite.op (of P (σ a))) a) a_1
⊢ sigmaComparison X σ x a =
Pi.π (fun a => X.obj (Opposite.op (of P (σ a)))) a
(piComparison X (fun a => Opposite.op (of P (σ a)))
(X.map
(opCoproductIsoProduct' (finiteCoproduct.isColimit fun a => of P (σ a))
(productIsProduct fun x => Opposite.op (of P (σ x)))).hom
x))
|
6a810ee204114508
|
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