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
|
|---|---|---|---|---|---|---|
FormalMultilinearSeries.comp_assoc
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem comp_assoc (r : FormalMultilinearSeries 𝕜 G H) (q : FormalMultilinearSeries 𝕜 F G)
(p : FormalMultilinearSeries 𝕜 E F) : (r.comp q).comp p = r.comp (q.comp p)
|
case h.H
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
⊢ ∑ c : (a : Composition n) × Composition a.length, f c =
∑ c : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)), g c
|
rw [← (sigmaEquivSigmaPi n).sum_comp]
|
case h.H
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
fun c => (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
fun c => (r c.fst.length) fun i => (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
⊢ ∑ c : (a : Composition n) × Composition a.length, f c =
∑ i : (a : Composition n) × Composition a.length, g ((sigmaEquivSigmaPi n) i)
|
e6d21021eddd0ef4
|
Asymptotics.isBigO_iff''
|
Mathlib/Analysis/Asymptotics/Defs.lean
|
theorem isBigO_iff'' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖
|
case mp
α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
h : f =O[l] g
⊢ ∃ c > 0, ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
case mpr
α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
h : ∃ c > 0, ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
⊢ f =O[l] g
|
case mp =>
rw [isBigO_iff'] at h
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
filter_upwards [hc] with x hx
rwa [inv_mul_le_iff₀ (by positivity)]
|
case mpr
α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
h : ∃ c > 0, ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
⊢ f =O[l] g
|
4acf1af7764c6d02
|
Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin
|
Mathlib/RingTheory/Adjoin/Basic.lean
|
theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
[IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
(hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B'
|
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ (l.sum fun a b => r ^ n • b • ↑a) ∈ B'
|
refine B'.toSubmodule.sum_mem ?_
|
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ ∀ c ∈ l.support, (fun a b => r ^ n • b • ↑a) c (l c) ∈ Subalgebra.toSubmodule B'
|
c0ccc6b7c63e5bab
|
RCLike.tendsto_add_mul_div_add_mul_atTop_nhds
|
Mathlib/Analysis/SpecificLimits/RCLike.lean
|
theorem RCLike.tendsto_add_mul_div_add_mul_atTop_nhds (a b c : 𝕜) {d : 𝕜} (hd : d ≠ 0) :
Tendsto (fun k : ℕ ↦ (a + c * k) / (b + d * k)) atTop (𝓝 (c / d))
|
case hl
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
hd : d ≠ 0
⊢ ?f₁ =ᶠ[atTop] fun k => (a + c * ↑k) / (b + d * ↑k)
case h
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
hd : d ≠ 0
⊢ Tendsto ?f₁ atTop (𝓝 (c / d))
case f₁
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
hd : d ≠ 0
⊢ ℕ → 𝕜
|
case f₁ => exact fun k ↦ (a * (↑k)⁻¹ + c) / (b * (↑k)⁻¹ + d)
|
case hl
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
hd : d ≠ 0
⊢ (fun k => (a * (↑k)⁻¹ + c) / (b * (↑k)⁻¹ + d)) =ᶠ[atTop] fun k => (a + c * ↑k) / (b + d * ↑k)
case h
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
hd : d ≠ 0
⊢ Tendsto (fun k => (a * (↑k)⁻¹ + c) / (b * (↑k)⁻¹ + d)) atTop (𝓝 (c / d))
|
6d8178cde5b196f7
|
Ordinal.enumOrd_isNormal_iff_isClosed
|
Mathlib/SetTheory/Ordinal/Topology.lean
|
theorem enumOrd_isNormal_iff_isClosed (hs : ¬ BddAbove s) :
IsNormal (enumOrd s) ↔ IsClosed s
|
s : Set Ordinal.{u}
hs : ¬BddAbove s
Hs : StrictMono (enumOrd s)
h : IsNormal (enumOrd s)
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso s hs).symm ⟨f i, ⋯⟩
this : enumOrd s (⨆ i, g i) = ⨆ i, f i
⊢ ⨆ i, f i ∈ s
|
rw [← this]
|
s : Set Ordinal.{u}
hs : ¬BddAbove s
Hs : StrictMono (enumOrd s)
h : IsNormal (enumOrd s)
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso s hs).symm ⟨f i, ⋯⟩
this : enumOrd s (⨆ i, g i) = ⨆ i, f i
⊢ enumOrd s (⨆ i, g i) ∈ s
|
0e0eef18c9d95dc4
|
Int.natAbs_eq_natAbs_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem natAbs_eq_natAbs_iff {a b : Int} : a.natAbs = b.natAbs ↔ a = b ∨ a = -b
|
case mpr.inl
b : Int
⊢ b.natAbs = b.natAbs
|
rfl
|
no goals
|
e42c147e74a9944d
|
isPathConnected_sphere
|
Mathlib/Analysis/NormedSpace/Connected.lean
|
theorem isPathConnected_sphere (h : 1 < Module.rank ℝ E) (x : E) {r : ℝ} (hr : 0 ≤ r) :
IsPathConnected (sphere x r)
|
case h₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y => x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ
B : IsPathConnected {0}ᶜ
C : IsPathConnected (f '' {0}ᶜ)
⊢ f '' {0}ᶜ ⊆ sphere x r
|
rintro - ⟨y, hy, rfl⟩
|
case h₁.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : 1 < Module.rank ℝ E
x : E
r : ℝ
hr : 0 ≤ r
rpos : 0 < r
f : E → E := fun y => x + (r * ‖y‖⁻¹) • y
A : ContinuousOn f {0}ᶜ
B : IsPathConnected {0}ᶜ
C : IsPathConnected (f '' {0}ᶜ)
y : E
hy : y ∈ {0}ᶜ
⊢ f y ∈ sphere x r
|
8571eaf7ed341480
|
Pell.exists_of_not_isSquare
|
Mathlib/NumberTheory/Pell.lean
|
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0
|
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite
m : ℤ
hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Infinite
hm₀ : m ≠ 0
this : NeZero m.natAbs
f : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q => (↑q.num, ↑q.den)
q₁ : ℚ
h₁ : q₁.num ^ 2 - d * ↑q₁.den ^ 2 = m
q₂ : ℚ
h₂ : q₂.num ^ 2 - d * ↑q₂.den ^ 2 = m
hne : q₁ ≠ q₂
hqf : f q₁ = f q₂
hq1 : ↑q₁.num = ↑q₂.num
hq2 : ↑q₁.den = ↑q₂.den
⊢ ↑q₂.num ^ 2 - ↑d * ↑q₂.den ^ 2 = 0
|
norm_cast
|
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite
m : ℤ
hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Infinite
hm₀ : m ≠ 0
this : NeZero m.natAbs
f : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q => (↑q.num, ↑q.den)
q₁ : ℚ
h₁ : q₁.num ^ 2 - d * ↑q₁.den ^ 2 = m
q₂ : ℚ
h₂ : q₂.num ^ 2 - d * ↑q₂.den ^ 2 = m
hne : q₁ ≠ q₂
hqf : f q₁ = f q₂
hq1 : ↑q₁.num = ↑q₂.num
hq2 : ↑q₁.den = ↑q₂.den
⊢ ↑(q₂.num ^ 2 - d * ↑(q₂.den ^ 2)) = 0
|
e113d5ba2b34ead4
|
HomologicalComplex.HomologySequence.epi_homologyMap_τ₃
|
Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
|
lemma epi_homologyMap_τ₃ (i : ι)
(h₁ : Epi (homologyMap φ.τ₂ i))
(h₂ : ∀ j, c.Rel i j → Epi (homologyMap φ.τ₁ j))
(h₃ : ∀ j, c.Rel i j → Mono (homologyMap φ.τ₂ j)) :
Epi (homologyMap φ.τ₃ i)
|
case pos
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
hi : ∃ j, c.Rel i j
⊢ Epi (homologyMap φ.τ₃ i)
|
obtain ⟨j, hij⟩ := hi
|
case pos.intro
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
j : ι
hij : c.Rel i j
⊢ Epi (homologyMap φ.τ₃ i)
|
92d458a0d828d6da
|
cfcₙ_cases
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
|
lemma cfcₙ_cases (P : A → Prop) (a : A) (f : R → R) (h₀ : P 0)
(haf : ∀ (hf : ContinuousOn f (σₙ R a)) h0 ha, P (cfcₙHom ha ⟨⟨_, hf.restrict⟩, h0⟩)) :
P (cfcₙ f a)
|
case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
P ((cfcₙHom ha) { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)
|
rwa [cfcₙ_apply_of_not_predicate _ h]
|
no goals
|
3d5960a87dec8e02
|
contDiffOn_succ_iff_hasFDerivWithinAt
|
Mathlib/Analysis/Calculus/ContDiff/Defs.lean
|
theorem contDiffOn_succ_iff_hasFDerivWithinAt (hn : n ≠ ∞) :
ContDiffOn 𝕜 (n + 1) f s ↔
∀ x ∈ s, ∃ u ∈ 𝓝[insert x s] x, (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f' : E → E →L[𝕜] F,
(∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u
|
case mp.intro.intro.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
n : WithTop ℕ∞
hn : n ≠ ∞
h : ContDiffOn 𝕜 (n + 1) f s
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[s] x
f_an : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
hf' : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
vu : v ∈ 𝓝[insert x u] x
v'u : v ⊆ insert x u
hv : ContDiffOn 𝕜 n f' v
xu : x ∈ u
⊢ ∃ u ∈ 𝓝[s] x, (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u
|
rw [insert_eq_of_mem xu] at vu v'u
|
case mp.intro.intro.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
n : WithTop ℕ∞
hn : n ≠ ∞
h : ContDiffOn 𝕜 (n + 1) f s
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[s] x
f_an : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
hf' : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
vu : v ∈ 𝓝[u] x
v'u : v ⊆ u
hv : ContDiffOn 𝕜 n f' v
xu : x ∈ u
⊢ ∃ u ∈ 𝓝[s] x, (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u
|
e4422b8510aa07a9
|
Filter.tendsto_div_const_atBot_iff_neg
|
Mathlib/Order/Filter/AtTopBot/Field.lean
|
/-- If `f` tends to infinity along a nontrivial filter,
`fun x ↦ f x / r` tends to negative infinity if and only if `r < 0`. -/
lemma tendsto_div_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
Tendsto (fun x ↦ f x / r) l atBot ↔ r < 0
|
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrderedField α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atTop
⊢ Tendsto (fun x => f x / r) l atBot ↔ r < 0
|
simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff_neg h]
|
no goals
|
3c6c9b852805b0a3
|
ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot
|
Mathlib/Probability/Independence/ZeroOne.lean
|
theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
(h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1
|
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeInf ι
inst✝¹ : NoMinOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
t : Set Ω
ht_tail : MeasurableSet t
a : α
ha : IsProbabilityMeasure (κ a)
⊢ IsFiniteMeasure (κ a)
|
infer_instance
|
no goals
|
35288fc2d8b5a0f6
|
Finmap.not_mem_erase_self
|
Mathlib/Data/Finmap.lean
|
theorem not_mem_erase_self {a : α} {s : Finmap β} : ¬a ∈ erase a s
|
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
s : Finmap β
⊢ a ∉ erase a s
|
rw [mem_erase, not_and_or, not_not]
|
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
s : Finmap β
⊢ a = a ∨ a ∉ s
|
400c19d554d88abf
|
CategoryTheory.IsPushout.inl_isoPushout_inv
|
Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
|
theorem inl_isoPushout_inv (h : IsPushout f g inl inr) [HasPushout f g] :
pushout.inl _ _ ≫ h.isoPushout.inv = inl
|
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
h : IsPushout f g inl inr
inst✝ : HasPushout f g
⊢ pushout.inl f g ≫ (colimit.isoColimitCocone { cocone := PushoutCocone.mk inl inr ⋯, isColimit := h.isColimit }).hom =
inl
|
simp
|
no goals
|
bb6667f750039bbb
|
SimpleGraph.three_le_egirth
|
Mathlib/Combinatorics/SimpleGraph/Girth.lean
|
lemma three_le_egirth : 3 ≤ G.egirth
|
case pos
α : Type u_1
G : SimpleGraph α
h : G.egirth = ⊤
⊢ 3 ≤ ⊤
|
apply le_top
|
no goals
|
ea32850d40d27a7e
|
Filter.HasBasis.lebesgue_number_lemma
|
Mathlib/Topology/UniformSpace/Compact.lean
|
theorem Filter.HasBasis.lebesgue_number_lemma {ι' ι : Sort*} {p : ι' → Prop}
{V : ι' → Set (α × α)} {U : ι → Set α} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K)
(hopen : ∀ j, IsOpen (U j)) (hcover : K ⊆ ⋃ j, U j) :
∃ i, p i ∧ ∀ x ∈ K, ∃ j, ball x (V i) ⊆ U j
|
α : Type ua
inst✝ : UniformSpace α
K : Set α
ι' : Sort u_2
ι : Sort u_3
p : ι' → Prop
V : ι' → Set (α × α)
U : ι → Set α
hbasis : (𝓤 α).HasBasis p V
hK : IsCompact K
hopen : ∀ (j : ι), IsOpen (U j)
hcover : K ⊆ ⋃ j, U j
⊢ ∀ ⦃s t : Set (α × α)⦄, s ⊆ t → (∀ x ∈ K, ∃ i, ball x t ⊆ U i) → ∀ x ∈ K, ∃ i, ball x s ⊆ U i
|
exact fun s t hst ht x hx ↦ (ht x hx).imp fun i hi ↦ Subset.trans (ball_mono hst _) hi
|
no goals
|
acfb123237652f8a
|
ascPochhammer_map
|
Mathlib/RingTheory/Polynomial/Pochhammer.lean
|
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n
|
case succ
S : Type u
inst✝¹ : Semiring S
T : Type v
inst✝ : Semiring T
f : S →+* T
n : ℕ
ih : map f (ascPochhammer S n) = ascPochhammer T n
⊢ map f (ascPochhammer S (n + 1)) = ascPochhammer T (n + 1)
|
simp [ih, ascPochhammer_succ_left, map_comp]
|
no goals
|
da57e8e75083539b
|
Fin.init_snoc
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem init_snoc : init (snoc p x) = p
|
case h
n : ℕ
α : Fin (n + 1) → Sort u_1
x : α (last n)
p : (i : Fin n) → α i.castSucc
i : Fin n
⊢ p (i.castSucc.castLT ⋯) = p i
|
convert cast_eq rfl (p i)
|
no goals
|
80c09a1e40ce8e6e
|
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastShiftRight
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean
|
theorem denote_blastShiftRight (aig : AIG α) (target : ArbitraryShiftTarget aig w0)
(lhs : BitVec w0) (rhs : BitVec target.n) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w0), ⟦aig, target.target.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < target.n), ⟦aig, target.distance.get idx hidx, assign⟧ = rhs.getLsbD idx) :
∀ (idx : Nat) (hidx : idx < w0),
⟦
(blastShiftRight aig target).aig,
(blastShiftRight aig target).vec.get idx hidx,
assign
⟧
=
(lhs >>> rhs).getLsbD idx
|
case mk
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w0 : Nat
aig : AIG α
lhs : BitVec w0
assign : α → Bool
idx : Nat
hidx : idx < w0
res : RefVecEntry α w0
n : Nat
target : aig.RefVec w0
distance : aig.RefVec n
rhs : BitVec { n := n, target := target, distance := distance }.n
hleft :
∀ (idx : Nat) (hidx : idx < w0),
⟦assign, { aig := aig, ref := { n := n, target := target, distance := distance }.target.get idx hidx }⟧ =
lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < { n := n, target := target, distance := distance }.n),
⟦assign, { aig := aig, ref := { n := n, target := target, distance := distance }.distance.get idx hidx }⟧ =
rhs.getLsbD idx
hres :
(if n = 0 then { aig := aig, vec := target }
else
blastShiftRight.go (blastShiftRight.twoPowShift aig { n := n, lhs := target, rhs := distance, pow := 0 }).aig
(distance.cast ⋯) 0
(blastShiftRight.twoPowShift aig { n := n, lhs := target, rhs := distance, pow := 0 }).vec) =
res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx }⟧ =
(lhs.ushiftRightRec rhs ({ n := n, target := target, distance := distance }.n - 1)).getLsbD idx
|
split at hres
|
case mk.isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w0 : Nat
aig : AIG α
lhs : BitVec w0
assign : α → Bool
idx : Nat
hidx : idx < w0
res : RefVecEntry α w0
n : Nat
target : aig.RefVec w0
distance : aig.RefVec n
rhs : BitVec { n := n, target := target, distance := distance }.n
hleft :
∀ (idx : Nat) (hidx : idx < w0),
⟦assign, { aig := aig, ref := { n := n, target := target, distance := distance }.target.get idx hidx }⟧ =
lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < { n := n, target := target, distance := distance }.n),
⟦assign, { aig := aig, ref := { n := n, target := target, distance := distance }.distance.get idx hidx }⟧ =
rhs.getLsbD idx
h✝ : n = 0
hres : { aig := aig, vec := target } = res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx }⟧ =
(lhs.ushiftRightRec rhs ({ n := n, target := target, distance := distance }.n - 1)).getLsbD idx
case mk.isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w0 : Nat
aig : AIG α
lhs : BitVec w0
assign : α → Bool
idx : Nat
hidx : idx < w0
res : RefVecEntry α w0
n : Nat
target : aig.RefVec w0
distance : aig.RefVec n
rhs : BitVec { n := n, target := target, distance := distance }.n
hleft :
∀ (idx : Nat) (hidx : idx < w0),
⟦assign, { aig := aig, ref := { n := n, target := target, distance := distance }.target.get idx hidx }⟧ =
lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < { n := n, target := target, distance := distance }.n),
⟦assign, { aig := aig, ref := { n := n, target := target, distance := distance }.distance.get idx hidx }⟧ =
rhs.getLsbD idx
h✝ : ¬n = 0
hres :
blastShiftRight.go (blastShiftRight.twoPowShift aig { n := n, lhs := target, rhs := distance, pow := 0 }).aig
(distance.cast ⋯) 0 (blastShiftRight.twoPowShift aig { n := n, lhs := target, rhs := distance, pow := 0 }).vec =
res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx }⟧ =
(lhs.ushiftRightRec rhs ({ n := n, target := target, distance := distance }.n - 1)).getLsbD idx
|
e96a4b8802385854
|
SpectrumRestricts.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 ∧ SpectrumRestricts a f) :
ContinuousFunctionalCalculus R p where
predicate_zero := h0
spectrum_nonempty a ha := ((h a).mp ha).2.image ▸
(ContinuousFunctionalCalculus.spectrum_nonempty a ((h a).mp ha).1 |>.image f)
compactSpace_spectrum a
|
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : IsTopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : IsTopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : Algebra S A
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus S q
inst✝ : CompleteSpace R
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(spectrum R a), R)
⊢ ⇑(algebraMap R S) ⁻¹'
range
⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
(g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) =
range ⇑g
|
ext x
|
case h
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : IsTopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : IsTopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : Algebra S A
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus S q
inst✝ : CompleteSpace R
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈
⇑(algebraMap R S) ⁻¹'
range
⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
(g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) ↔
x ∈ range ⇑g
|
ea9b650e25b4334d
|
Ideal.map_includeRight_eq
|
Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
|
/-- The ideal of `A ⊗[R] B` generated by `I` is the image of `A ⊗[R] I` -/
lemma Ideal.map_includeRight_eq (I : Ideal B) :
(I.map (Algebra.TensorProduct.includeRight : B →ₐ[R] A ⊗[R] B)).restrictScalars R
= LinearMap.range (LinearMap.lTensor A (Submodule.subtype (I.restrictScalars R)))
|
case h
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
x : A ⊗[R] ↥(Submodule.restrictScalars R I)
⊢ (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =
0 • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x
|
simp only [map_zero, smul_eq_mul, zero_mul]
|
no goals
|
433dfbb2c63fd4cd
|
LinearMap.trace_one
|
Mathlib/LinearAlgebra/Trace.lean
|
theorem trace_one : trace R M 1 = (finrank R M : R)
|
case inr
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
h✝ : Nontrivial R
⊢ (trace R M) 1 = ↑(finrank R M)
|
have b := Module.Free.chooseBasis R M
|
case inr
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
h✝ : Nontrivial R
b : Basis (Free.ChooseBasisIndex R M) R M
⊢ (trace R M) 1 = ↑(finrank R M)
|
4959d10d80e34ef9
|
Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul
|
Mathlib/RingTheory/Finiteness/Nakayama.lean
|
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M)
|
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
hn : N.FG
hin : N ≤ I • N
⊢ ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
|
rw [fg_def] at hn
|
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
hn : ∃ S, S.Finite ∧ span R S = N
hin : N ≤ I • N
⊢ ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
|
ddde3c87abdd1812
|
SimpleGraph.Walk.dropLast_concat
|
Mathlib/Combinatorics/SimpleGraph/Walk.lean
|
@[simp]
lemma dropLast_concat {t u v} (p : G.Walk u v) (h : G.Adj v t) :
(p.concat h).dropLast = p.copy rfl (by simp)
|
case cons
V : Type u
G : SimpleGraph V
t u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (h : G.Adj w✝ t), (p✝.concat h).dropLast = p✝.copy ⋯ ⋯
h : G.Adj w✝ t
⊢ cons h✝ ((p✝.concat h).dropLast.copy ⋯ ⋯) = (cons h✝ p✝).copy ⋯ ⋯
|
simp [*]
|
no goals
|
d5401d0415d66680
|
MeasureTheory.Measure.map_smul
|
Mathlib/MeasureTheory/Measure/Map.lean
|
theorem map_smul {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
(c : R) (μ : Measure α) (f : α → β) : (c • μ).map f = c • μ.map f
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : AEMeasurable f μ
hfc : AEMeasurable f (c • μ)
⊢ map f (c • μ) = c • map f μ
|
simp only [← mapₗ_mk_apply_of_aemeasurable hf, ← mapₗ_mk_apply_of_aemeasurable hfc,
LinearMap.map_smulₛₗ, RingHom.id_apply]
|
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : Measure α
f : α → β
c : ℝ≥0∞
hc : c ≠ 0
hf : AEMeasurable f μ
hfc : AEMeasurable f (c • μ)
⊢ c • (mapₗ (AEMeasurable.mk f hfc)) μ = c • (mapₗ (AEMeasurable.mk f hf)) μ
|
6b7fed798cea7599
|
dimH_range_le_of_locally_lipschitzOn
|
Mathlib/Topology/MetricSpace/HausdorffDimension.lean
|
theorem dimH_range_le_of_locally_lipschitzOn [SecondCountableTopology X] {f : X → Y}
(hf : ∀ x : X, ∃ C : ℝ≥0, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s) :
dimH (range f) ≤ dimH (univ : Set X)
|
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
f : X → Y
hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s
⊢ dimH (f '' univ) ≤ dimH univ
|
refine dimH_image_le_of_locally_lipschitzOn fun x _ => ?_
|
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
f : X → Y
hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s
x : X
x✝ : x ∈ univ
⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, LipschitzOnWith C f t
|
2d461ca5b4f2951f
|
rothNumberNat_le_ruzsaSzemerediNumberNat
|
Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean
|
lemma rothNumberNat_le_ruzsaSzemerediNumberNat (n : ℕ) :
(2 * n + 1) * rothNumberNat n ≤ ruzsaSzemerediNumberNat (6 * n + 3)
|
n : ℕ
α : Type := Fin (2 * n + 1)
this✝ : Coprime 2 (2 * n + 1)
this : Fact (IsUnit 2)
⊢ ruzsaSzemerediNumber (α ⊕ α ⊕ α) = ruzsaSzemerediNumberNat (6 * n + 3)
|
simp_rw [← ruzsaSzemerediNumberNat_card, Fintype.card_sum, α, Fintype.card_fin]
|
n : ℕ
α : Type := Fin (2 * n + 1)
this✝ : Coprime 2 (2 * n + 1)
this : Fact (IsUnit 2)
⊢ ruzsaSzemerediNumberNat (2 * n + 1 + (2 * n + 1 + (2 * n + 1))) = ruzsaSzemerediNumberNat (6 * n + 3)
|
f10db59dd468e568
|
Submodule.isInternal_prime_power_torsion_of_is_torsion_by_ideal
|
Mathlib/Algebra/Module/DedekindDomain.lean
|
theorem isInternal_prime_power_torsion_of_is_torsion_by_ideal [DecidableEq (Ideal R)]
{I : Ideal R} (hI : I ≠ ⊥) (hM : Module.IsTorsionBySet R M I) :
DirectSum.IsInternal fun p : (factors I).toFinset =>
torsionBySet R M (p ^ (factors I).count ↑p : Ideal R)
|
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDedekindDomain R
inst✝ : DecidableEq (Ideal R)
I : Ideal R
hI : I ≠ ⊥
hM : Module.IsTorsionBySet R M ↑I
P : Multiset (Ideal R) := factors I
prime_of_mem : ∀ p ∈ P.toFinset, Prime p
p : Ideal R
hp : p ∈ ↑(factors I).toFinset
q : Ideal R
hq : q ∈ ↑(factors I).toFinset
pq : p ≠ q
⊢ ¬(Multiset.count q P ≠ 0 ∧ p = normalize q)
|
exact fun H => pq <| H.2.trans <| normalize_eq q
|
no goals
|
342bfde6ba181719
|
List.Forall₂.prod_le_prod'
|
Mathlib/Algebra/Order/BigOperators/Group/List.lean
|
@[to_additive sum_le_sum]
lemma Forall₂.prod_le_prod' [Preorder M] [MulRightMono M]
[MulLeftMono M] {l₁ l₂ : List M} (h : Forall₂ (· ≤ ·) l₁ l₂) :
l₁.prod ≤ l₂.prod
|
case cons
M : Type u_3
inst✝³ : Monoid M
inst✝² : Preorder M
inst✝¹ : MulRightMono M
inst✝ : MulLeftMono M
l₁ l₂ : List M
a b : M
la lb : List M
hab : a ≤ b
ih : Forall₂ (fun x1 x2 => x1 ≤ x2) la lb
ih' : la.prod ≤ lb.prod
⊢ (a :: la).prod ≤ (b :: lb).prod
|
simpa only [prod_cons] using mul_le_mul' hab ih'
|
no goals
|
1e4e14213ff126b1
|
Subgroup.IsComplement.equiv_snd_eq_iff_rightCosetEquivalence
|
Mathlib/GroupTheory/Complement.lean
|
theorem equiv_snd_eq_iff_rightCosetEquivalence {g₁ g₂ : G} :
(hHT.equiv g₁).snd = (hHT.equiv g₂).snd ↔ RightCosetEquivalence H g₁ g₂
|
case mpr.py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₁ * (↑(hHT.equiv g₂).2)⁻¹ ∈ ↑H
|
rw [SetLike.mem_coe, ← mul_mem_cancel_left h]
|
case mpr.py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₂ * g₁⁻¹ * (g₁ * (↑(hHT.equiv g₂).2)⁻¹) ∈ H
|
f30a129f3cb5b12f
|
Dynamics.coverMincard_le_netMaxcard
|
Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean
|
/-- Given an entourage `U` and a time `n`, a minimal dynamical cover by `U ○ U` has a smaller
cardinality than a maximal dynamical net by `U`. This lemma is the second of two key results to
compare two versions topological entropy: with cover and with nets. -/
lemma coverMincard_le_netMaxcard (T : X → X) (F : Set X) {U : Set (X × X)} (U_rfl : idRel ⊆ U)
(U_symm : SymmetricRel U) (n : ℕ) :
coverMincard T F (U ○ U) n ≤ netMaxcard T F U n
|
case inr.intro.intro.h.intro.intro
X : Type u_1
T : X → X
F : Set X
U : Set (X × X)
U_rfl : idRel ⊆ U
U_symm : SymmetricRel U
n : ℕ
h✝ : netMaxcard T F U n < ⊤
s : Finset X
s_net : IsDynNetIn T F U n ↑s
s_netMaxcard : ↑s.card = netMaxcard T F U n
h : ¬IsDynCoverOf T F (U ○ U) n ↑s
x : X
x_F : x ∈ F
x_uncov : ∀ x_1 ∈ s, x ∉ ball x_1 (dynEntourage T (U ○ U) n)
larger_net : IsDynNetIn T F U n ↑(insert x s)
⊢ netMaxcard T F U n < ↑(insert x s).card
|
rw [← s_netMaxcard, Nat.cast_lt]
|
case inr.intro.intro.h.intro.intro
X : Type u_1
T : X → X
F : Set X
U : Set (X × X)
U_rfl : idRel ⊆ U
U_symm : SymmetricRel U
n : ℕ
h✝ : netMaxcard T F U n < ⊤
s : Finset X
s_net : IsDynNetIn T F U n ↑s
s_netMaxcard : ↑s.card = netMaxcard T F U n
h : ¬IsDynCoverOf T F (U ○ U) n ↑s
x : X
x_F : x ∈ F
x_uncov : ∀ x_1 ∈ s, x ∉ ball x_1 (dynEntourage T (U ○ U) n)
larger_net : IsDynNetIn T F U n ↑(insert x s)
⊢ s.card < (insert x s).card
|
78a05381e650bf91
|
TopCat.Presheaf.app_surjective_of_stalkFunctor_map_bijective
|
Mathlib/Topology/Sheaves/Stalks.lean
|
theorem app_surjective_of_stalkFunctor_map_bijective {F G : Sheaf C X} (f : F ⟶ G) (U : Opens X)
(h : ∀ x ∈ U, Function.Bijective ((stalkFunctor C x).map f.1)) :
Function.Surjective (f.1.app (op U))
|
case intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
h : ∀ x ∈ U, Function.Bijective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map f.val))
t : CC (G.val.obj (op U))
x : ↑X
hx : x ∈ U
V₁ : Opens ↑X
hxV₁ : x ∈ V₁
s₁ : ToType (F.presheaf.obj (op V₁))
hs₁ :
(ConcreteCategory.hom (germ G.val V₁ x hxV₁)) ((ConcreteCategory.hom (f.val.app (op V₁))) s₁) =
(ConcreteCategory.hom (G.presheaf.germ U x hx)) t
V₂ : Opens ↑X
hxV₂ : x ∈ V₂
iV₂V₁ : V₂ ⟶ V₁
iV₂U : V₂ ⟶ U
heq :
(ConcreteCategory.hom (G.presheaf.map iV₂V₁.op)) ((ConcreteCategory.hom (f.val.app (op V₁))) s₁) =
(ConcreteCategory.hom (G.presheaf.map iV₂U.op)) t
⊢ ∃ V, ∃ (_ : x ∈ V), ∃ iVU s, (ConcreteCategory.hom (f.val.app (op V))) s = (ConcreteCategory.hom (G.val.map iVU.op)) t
|
use V₂, hxV₂, iV₂U, F.1.map iV₂V₁.op s₁
|
case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
h : ∀ x ∈ U, Function.Bijective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map f.val))
t : CC (G.val.obj (op U))
x : ↑X
hx : x ∈ U
V₁ : Opens ↑X
hxV₁ : x ∈ V₁
s₁ : ToType (F.presheaf.obj (op V₁))
hs₁ :
(ConcreteCategory.hom (germ G.val V₁ x hxV₁)) ((ConcreteCategory.hom (f.val.app (op V₁))) s₁) =
(ConcreteCategory.hom (G.presheaf.germ U x hx)) t
V₂ : Opens ↑X
hxV₂ : x ∈ V₂
iV₂V₁ : V₂ ⟶ V₁
iV₂U : V₂ ⟶ U
heq :
(ConcreteCategory.hom (G.presheaf.map iV₂V₁.op)) ((ConcreteCategory.hom (f.val.app (op V₁))) s₁) =
(ConcreteCategory.hom (G.presheaf.map iV₂U.op)) t
⊢ (ConcreteCategory.hom (f.val.app (op V₂))) ((ConcreteCategory.hom (F.val.map iV₂V₁.op)) s₁) =
(ConcreteCategory.hom (G.val.map iV₂U.op)) t
|
6bc960e8b7229d89
|
Matrix.vecMul_injective_iff
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
theorem Matrix.vecMul_injective_iff {R : Type*} [CommRing R] {M : Matrix m n R} :
Function.Injective M.vecMul ↔ LinearIndependent R (fun i ↦ M i)
|
case refine_1.h
m : Type u_3
n : Type u_4
inst✝¹ : Fintype m
R : Type u_5
inst✝ : CommRing R
M : Matrix m n R
h : ∀ (x : m → R), x ᵥ* M = 0 → x = 0
c : m → R
h0 : ∑ i : m, c i • M i = 0
i : n
⊢ (c ᵥ* M) i = (∑ i : m, c i • M i) i
|
simp [vecMul, dotProduct]
|
no goals
|
bce4056b31b0bb97
|
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod
|
Mathlib/LinearAlgebra/FreeModule/Int.lean
|
/-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
(snf : Basis.SmithNormalForm N ι n) :
N.toAddSubgroup.index = Nat.card R ^ (Fintype.card ι - n) *
∏ i : Fin n, (Ideal.span {snf.a i}).toAddSubgroup.index
|
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN' : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
c : Fin n → R
i : ι
h : ∃ j, f j = i
j : Fin n
hj : j ≠ h.choose
hinj : f j ≠ f h.choose
⊢ (c j • if i = f j then a j else 0) = 0
|
rw [h.choose_spec] at hinj
|
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN' : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
c : Fin n → R
i : ι
h : ∃ j, f j = i
j : Fin n
hj : j ≠ h.choose
hinj : f j ≠ i
⊢ (c j • if i = f j then a j else 0) = 0
|
3d7a9cb7455a5c0f
|
Nat.repr_length
|
Mathlib/Data/Nat/Digits.lean
|
/-- The core implementation of `Nat.repr` returns a String with length less than or equal to the
number of digits in the decimal number (represented by `e`). For example, the decimal string
representation of any number less than 1000 (10 ^ 3) has a length less than or equal to 3. -/
lemma repr_length (n e : Nat) : 0 < e → n < 10 ^ e → (Nat.repr n).length <= e
|
e n : ℕ
e0 : 0 < e
he : n + 1 < 10 ^ e
hterm : ¬n.succ / 10 = 0
⊢ (toDigitsCore 10 (n + 1 + 1) (n + 1) []).length ≤ e
|
exact toDigitsCore_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0
|
no goals
|
d511f3017bbee59f
|
MvPolynomial.mem_image_comap_C_basicOpen
|
Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean
|
lemma mem_image_comap_C_basicOpen (f : MvPolynomial σ R) (x : PrimeSpectrum R) :
x ∈ comap (C (σ := σ)) '' basicOpen f ↔ ∃ i, f.coeff i ∉ x.asIdeal
|
R : Type u_2
inst✝ : CommRing R
σ : Type u_1
f : MvPolynomial σ R
x : PrimeSpectrum R
⊢ x ∈ ⇑(comap C) '' ↑(basicOpen f) ↔ (map (algebraMap R x.asIdeal.ResidueField)) f ≠ 0
|
refine (mem_image_comap_basicOpen _ _).trans (not_iff_not.mpr ?_)
|
R : Type u_2
inst✝ : CommRing R
σ : Type u_1
f : MvPolynomial σ R
x : PrimeSpectrum R
⊢ IsNilpotent ((algebraMap (MvPolynomial σ R) (MvPolynomial σ R ⊗[R] x.asIdeal.ResidueField)) f) ↔
(map (algebraMap R x.asIdeal.ResidueField)) f = 0
|
8f5871a9394f4ea4
|
Rat.toNNRat_inv
|
Mathlib/Data/NNRat/Lemmas.lean
|
lemma toNNRat_inv (q : ℚ) : toNNRat q⁻¹ = (toNNRat q)⁻¹
|
case inr
q : ℚ
hq : 0 ≤ q
⊢ (↑q.toNNRat)⁻¹.toNNRat = q.toNNRat⁻¹
|
rw [← coe_inv, toNNRat_coe]
|
no goals
|
69eeea9e2e5af2bb
|
Fin.snoc_update
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem snoc_update : snoc (update p i y) x = update (snoc p x) i.castSucc y
|
case pos
n : ℕ
α : Fin (n + 1) → Sort u_1
x : α (last n)
p : (i : Fin n) → α i.castSucc
i : Fin n
y : α i.castSucc
j : Fin (n + 1)
h : ↑j < n
⊢ (if h : ↑j < n then cast ⋯ (update p i y (j.castLT h)) else cast ⋯ x) = update (snoc p x) i.castSucc y j
|
simp only [h]
|
case pos
n : ℕ
α : Fin (n + 1) → Sort u_1
x : α (last n)
p : (i : Fin n) → α i.castSucc
i : Fin n
y : α i.castSucc
j : Fin (n + 1)
h : ↑j < n
⊢ (if h_1 : True then cast ⋯ (update p i y (j.castLT ⋯)) else cast ⋯ x) = update (snoc p x) i.castSucc y j
|
4eef9dec140e05c3
|
AbsoluteValue.exists_partition_int
|
Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean
|
theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) :
∃ t : Fin n → Fin ⌈1 / ε⌉₊,
∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε
|
case refine_1
n : ℕ
ε : ℝ
hε : 0 < ε
b : ℤ
hb : b ≠ 0
A : Fin n → ℤ
hb' : 0 < ↑|b|
hbε : 0 < |b| • ε
hfloor : ∀ (i : Fin n), 0 ≤ ⌊↑(A i % b) / |b| • ε⌋
i : Fin n
⊢ ↑(A i % b) / ↑|b| / ε < ↑↑⌈1 / ε⌉₊
|
apply lt_of_lt_of_le _ (Nat.le_ceil _)
|
n : ℕ
ε : ℝ
hε : 0 < ε
b : ℤ
hb : b ≠ 0
A : Fin n → ℤ
hb' : 0 < ↑|b|
hbε : 0 < |b| • ε
hfloor : ∀ (i : Fin n), 0 ≤ ⌊↑(A i % b) / |b| • ε⌋
i : Fin n
⊢ ↑(A i % b) / ↑|b| / ε < 1 / ε
|
b91f00c7ed7d0496
|
PowerSeries.maximalIdeal_eq_span_X
|
Mathlib/RingTheory/PowerSeries/Inverse.lean
|
theorem maximalIdeal_eq_span_X : IsLocalRing.maximalIdeal (k⟦X⟧) = Ideal.span {X}
|
case left
k : Type u_2
inst✝ : Field k
⊢ ¬X ∣ 1
|
exact Prime.not_dvd_one X_prime
|
no goals
|
e0962a486b118068
|
Fin.zero_eq_one_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean
|
theorem zero_eq_one_iff {n : Nat} [NeZero n] : (0 : Fin n) = 1 ↔ n = 1
|
case mpr
n : Nat
inst✝ : NeZero n
⊢ n = 1 → 0 = 1
|
rintro rfl
|
case mpr
inst✝ : NeZero 1
⊢ 0 = 1
|
1b6c9cdc3a95a9f5
|
Vector.all_bne
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean
|
theorem all_bne [BEq α] {xs : Vector α n} : (xs.all fun x => a != x) = !xs.contains a
|
case mk
α : Type u_1
a : α
inst✝ : BEq α
xs : Array α
⊢ ({ toArray := xs, size_toArray := ⋯ }.all fun x => a != x) = !{ toArray := xs, size_toArray := ⋯ }.contains a
|
simp [Array.all_bne]
|
no goals
|
2ecfac75e4394e13
|
Polynomial.natDegree_sum_eq_of_disjoint
|
Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
|
theorem natDegree_sum_eq_of_disjoint (f : S → R[X]) (s : Finset S)
(h : Set.Pairwise { i | i ∈ s ∧ f i ≠ 0 } (Ne on natDegree ∘ f)) :
natDegree (s.sum f) = s.sup fun i => natDegree (f i)
|
case pos.intro.intro
R : Type u
S : Type v
inst✝ : Semiring R
f : S → R[X]
s : Finset S
h : {i | i ∈ s ∧ f i ≠ 0}.Pairwise (Ne on natDegree ∘ f)
x : S
hx : x ∈ s
hx' : f x ≠ 0
⊢ (s.sum f).natDegree = s.sup fun i => (f i).natDegree
|
have hs : s.Nonempty := ⟨x, hx⟩
|
case pos.intro.intro
R : Type u
S : Type v
inst✝ : Semiring R
f : S → R[X]
s : Finset S
h : {i | i ∈ s ∧ f i ≠ 0}.Pairwise (Ne on natDegree ∘ f)
x : S
hx : x ∈ s
hx' : f x ≠ 0
hs : s.Nonempty
⊢ (s.sum f).natDegree = s.sup fun i => (f i).natDegree
|
daf835299b38da62
|
Ordinal.add_le_of_limit
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem add_le_of_limit {a b c : Ordinal} (h : IsLimit b) :
a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c :=
⟨fun h _ l => (add_le_add_left l.le _).trans h, fun H =>
le_of_not_lt <| by
-- Porting note: `induction` tactics are required because of the parser bug.
induction a using inductionOn with
| H α r =>
induction b using inductionOn with
| H β s =>
intro l
suffices ∀ x : β, Sum.Lex r s (Sum.inr x) (enum _ ⟨_, l⟩) by
-- Porting note: `revert` & `intro` is required because `cases'` doesn't replace
-- `enum _ _ l` in `this`.
revert this; rcases enum _ ⟨_, l⟩ with x | x <;> intro this
· cases this (enum s ⟨0, h.pos⟩)
· exact irrefl _ (this _)
intro x
rw [← typein_lt_typein (Sum.Lex r s), typein_enum]
have := H _ (h.succ_lt (typein_lt_type s x))
rw [add_succ, succ_le_iff] at this
refine
(RelEmbedding.ofMonotone (fun a => ?_) fun a b => ?_).ordinal_type_le.trans_lt this
· rcases a with ⟨a | b, h⟩
· exact Sum.inl a
· exact Sum.inr ⟨b, by cases h; assumption⟩
· rcases a with ⟨a | a, h₁⟩ <;> rcases b with ⟨b | b, h₂⟩ <;> cases h₁ <;> cases h₂ <;>
rintro ⟨⟩ <;> constructor <;> assumption⟩
|
case inr
c : Ordinal.{u_4}
α : Type u_4
r : α → α → Prop
inst✝¹ : IsWellOrder α r
β : Type u_4
s : β → β → Prop
inst✝ : IsWellOrder β s
h : (type s).IsLimit
H : ∀ b' < type s, type r + b' ≤ c
l : c < type r + type s
x : β
this : type r + (typein s).toRelEmbedding x < c
b : β
h✝ : s b x
⊢ s b x
|
assumption
|
no goals
|
51d16ae877a4fc6c
|
zpow_neg_mul_zpow_self
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
lemma zpow_neg_mul_zpow_self (n : ℤ) (ha : a ≠ 0) : a ^ (-n) * a ^ n = 1
|
G₀ : Type u_3
inst✝ : GroupWithZero G₀
a : G₀
n : ℤ
ha : a ≠ 0
⊢ a ^ (-n) * a ^ n = 1
|
rw [zpow_neg]
|
G₀ : Type u_3
inst✝ : GroupWithZero G₀
a : G₀
n : ℤ
ha : a ≠ 0
⊢ (a ^ n)⁻¹ * a ^ n = 1
|
71024ef797df0d81
|
strictConvexOn_exp
|
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
theorem strictConvexOn_exp : StrictConvexOn ℝ univ exp
|
x y z : ℝ
hxy : x < y
hyz : y < z
h1 : 0 < y - x
h2 : x - y < 0
⊢ rexp y - rexp x = rexp y - rexp (y + (x - y))
|
ring_nf
|
no goals
|
12b8c11111e0e8d6
|
ProbabilityTheory.strong_law_aux2
|
Mathlib/Probability/StrongLaw.lean
|
theorem strong_law_aux2 {c : ℝ} (c_one : 1 < c) :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω -
𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]) =o[atTop] fun n : ℕ => (⌊c ^ n⌋₊ : ℝ)
|
case intro.intro.intro
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
⊢ ∀ᵐ (ω : Ω),
(fun n =>
∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
fun n => ↑⌊c ^ n⌋₊
|
have := fun i => strong_law_aux1 X hint hindep hident hnonneg c_one (v_pos i)
|
case intro.intro.intro
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
this :
∀ (i : ℕ),
∀ᵐ (ω : Ω),
∀ᶠ (n : ℕ) in atTop,
|∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
v i * ↑⌊c ^ n⌋₊
⊢ ∀ᵐ (ω : Ω),
(fun n =>
∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
fun n => ↑⌊c ^ n⌋₊
|
0a014d4dcbee42df
|
MeasureTheory.ProbabilityMeasure.map_fst_prod
|
Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
|
/-- The first marginal of a product probability measure is the first probability measure. -/
@[simp] lemma map_fst_prod : (μ.prod ν).map measurable_fst.aemeasurable = μ
|
case a
α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : ProbabilityMeasure α
ν : ProbabilityMeasure β
⊢ ↑((μ.prod ν).map ⋯) = ↑μ
|
simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_fst_prod,
measure_univ, one_smul]
|
no goals
|
99d5022a084cc581
|
Complex.arg_mem_Ioc
|
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π
|
z : ℂ
⊢ z.arg ∈ Ioc (-π) π
|
have hπ : 0 < π := Real.pi_pos
|
z : ℂ
hπ : 0 < π
⊢ z.arg ∈ Ioc (-π) π
|
5bfcd13fb0005b53
|
HasFDerivWithinAt.abs_of_pos
|
Mathlib/Analysis/Calculus/Deriv/Abs.lean
|
theorem HasFDerivWithinAt.abs_of_pos (hf : HasFDerivWithinAt f f' s x)
(h₀ : 0 < f x) : HasFDerivWithinAt (fun x ↦ |f x|) f' s x
|
case h.e'_12
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : E →L[ℝ] ℝ
s : Set E
x : E
hf : HasFDerivWithinAt f f' s x
h₀ : 0 < f x
⊢ f' = 1 • f'
|
simp
|
no goals
|
fa607aa50b983aec
|
MeasureTheory.Measure.tendsto_IicSnd_atBot
|
Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
|
theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0)
|
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
q r : ℚ
hqr : q ≤ r
x : ℝ
hx : x ∈ Iic ↑(-r)
⊢ x ∈ Iic ↑(-q)
|
simp only [Rat.cast_neg, mem_Iic] at hx ⊢
|
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
q r : ℚ
hqr : q ≤ r
x : ℝ
hx : x ≤ -↑r
⊢ x ≤ -↑q
|
cd38db034d7af3e8
|
MeasureTheory.measurable_stoppedValue
|
Mathlib/Probability/Process/Stopping.lean
|
theorem measurable_stoppedValue [MetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
(hf_prog : ProgMeasurable f u) (hτ : IsStoppingTime f τ) :
Measurable[hτ.measurableSpace] (stoppedValue u τ)
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝² : MetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : ProgMeasurable f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => τ ω ⊓ i)
t : Set β
ht : MeasurableSet t
i : ι
this : stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} = (stoppedValue u fun ω => τ ω ⊓ i) ⁻¹' t ∩ {ω | τ ω ≤ i}
⊢ MeasurableSet (stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i})
|
rw [this]
|
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝² : MetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : ProgMeasurable f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => τ ω ⊓ i)
t : Set β
ht : MeasurableSet t
i : ι
this : stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} = (stoppedValue u fun ω => τ ω ⊓ i) ⁻¹' t ∩ {ω | τ ω ≤ i}
⊢ MeasurableSet ((stoppedValue u fun ω => τ ω ⊓ i) ⁻¹' t ∩ {ω | τ ω ≤ i})
|
1bd0ad42e2bec67b
|
HasFPowerSeriesAt.eventually_hasSum_of_comp
|
Mathlib/Analysis/Analytic/Inverse.lean
|
lemma HasFPowerSeriesAt.eventually_hasSum_of_comp {f : E → F} {g : F → G}
{q : FormalMultilinearSeries 𝕜 F G} {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x) (hf : HasFPowerSeriesAt f p x)
(hq : 0 < q.radius) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => q n fun _ : Fin n => (f (x + y) - f x)) (g (f (x + y)))
|
case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
g : F → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x
hf : HasFPowerSeriesAt f p x
hq : 0 < q.radius
this : ∀ᶠ (y : E) in 𝓝 0, f (x + y) - f x ∈ EMetric.ball 0 q.radius
y : E
hy : Tendsto (fun a => q.partialSum a.1 (p.partialSum a.2 y - (p 0) fun x => 0)) atTop (𝓝 ((g ∘ f) (x + y)))
h'y : Tendsto (fun n => p.partialSum n y) atTop (𝓝 (f (x + y)))
h''y : f (x + y) - f x ∈ EMetric.ball 0 q.radius
L : Tendsto (fun n => q.partialSum n (f (x + y) - f x)) atTop (𝓝 (g (f (x + y))))
C : CauchySeq fun s => ∑ n ∈ s, (q n) fun x_1 => f (x + y) - f x
⊢ HasSum (fun n => (q n) fun x_1 => f (x + y) - f x) (g (f (x + y)))
|
exact tendsto_nhds_of_cauchySeq_of_subseq C tendsto_finset_range L
|
no goals
|
879ec15fea67d40e
|
RingHom.finite_ofLocalizationSpan
|
Mathlib/RingTheory/RingHom/Finite.lean
|
theorem RingHom.finite_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.Finite
|
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) :=
fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ :
∀ (r : { x // x ∈ s }),
IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
⊢ (algebraMap S (Localization.Away (f ↑r))) x ∈ Submodule.span (Localization.Away ↑r) ↑(s₁ r)
|
rw [s₂ r]
|
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) :=
fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ :
∀ (r : { x // x ∈ s }),
IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
⊢ (algebraMap S (Localization.Away (f ↑r))) x ∈ ⊤
|
9687fc93cf781ddc
|
Polynomial.degree_div_lt
|
Mathlib/Algebra/Polynomial/FieldDivision.lean
|
theorem degree_div_lt (hp : p ≠ 0) (hq : 0 < degree q) : degree (p / q) < degree p
|
R : Type u
inst✝ : Field R
p q : R[X]
hp : p ≠ 0
hq : 0 < q.degree
hq0 : q = 0
⊢ False
|
simp [hq0] at hq
|
no goals
|
8175cb88ff4d388e
|
FormalMultilinearSeries.compPartialSumTargetSet_image_compPartialSumSource
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem compPartialSumTargetSet_image_compPartialSumSource (m M N : ℕ)
(i : Σ n, Composition n) (hi : i ∈ compPartialSumTargetSet m M N) :
∃ (j : _) (hj : j ∈ compPartialSumSource m M N), compChangeOfVariables m M N j hj = i
|
case mk.refine_2
m M N n : ℕ
c : Composition n
hi : ⟨n, c⟩ ∈ compPartialSumTargetSet m M N
⊢ (ofFn fun a => c.blocks.get a) = c.blocks
|
conv_rhs => rw [← List.ofFn_get c.blocks]
|
no goals
|
ef44e0f46a80a4f8
|
MeasureTheory.continuousOn_convolution_right_with_param_comp
|
Mathlib/Analysis/Convolution.lean
|
theorem continuousOn_convolution_right_with_param_comp {s : Set P} {v : P → G}
(hv : ContinuousOn v s) {g : P → G → E'} {k : Set G} (hk : IsCompact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
(hg : ContinuousOn (↿g) (s ×ˢ univ)) : ContinuousOn (fun x => (f ⋆[L, μ] g x) (v x)) s
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
s : Set P
v : P → G
hv : ContinuousOn v s
g : P → G → E'
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
⊢ MapsTo (fun x => (_root_.id x, v x)) s (s ×ˢ univ)
|
intro x hx
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
s : Set P
v : P → G
hv : ContinuousOn v s
g : P → G → E'
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
x : P
hx : x ∈ s
⊢ (fun x => (_root_.id x, v x)) x ∈ s ×ˢ univ
|
8a59b4459ff1f975
|
contDiffWithinAt_localInvariantProp_of_le
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
(contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
(ContDiffWithinAtProp I I' m) where
is_local {s x u f} u_open xu
|
case h₁
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f g : H → H'
h : ∀ y ∈ s, f y = g y
hx : f x = g x
hf : ContDiffWithinAtProp I I' m f s x
y : E
hy : ↑I.symm y ∈ s ∧ y ∈ range ↑I
⊢ (↑I' ∘ g ∘ ↑I.symm) y = (↑I' ∘ f ∘ ↑I.symm) y
|
simp only [h, hy, mfld_simps]
|
no goals
|
841928255da4a913
|
Set.image_sigmaMk_preimage_sigmaMap
|
Mathlib/Data/Set/Sigma.lean
|
theorem image_sigmaMk_preimage_sigmaMap {β : ι' → Type*} {f : ι → ι'} (hf : Function.Injective f)
(g : ∀ i, α i → β (f i)) (i : ι) (s : Set (β (f i))) :
Sigma.mk i '' (g i ⁻¹' s) = Sigma.map f g ⁻¹' (Sigma.mk (f i) '' s)
|
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
β : ι' → Type u_4
f : ι → ι'
hf : Function.Injective f
g : (i : ι) → α i → β (f i)
i : ι
s : Set (β (f i))
⊢ Sigma.map f g ⁻¹' (Sigma.mk (f i) '' s) ⊆ Sigma.mk i '' (g i ⁻¹' s)
|
rintro ⟨j, x⟩ ⟨y, hys, hxy⟩
|
case mk.intro.intro
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
β : ι' → Type u_4
f : ι → ι'
hf : Function.Injective f
g : (i : ι) → α i → β (f i)
i : ι
s : Set (β (f i))
j : ι
x : α j
y : β (f i)
hys : y ∈ s
hxy : ⟨f i, y⟩ = Sigma.map f g ⟨j, x⟩
⊢ ⟨j, x⟩ ∈ Sigma.mk i '' (g i ⁻¹' s)
|
81b64c1b0c0d75fb
|
QuotientGroup.leftRel_eq
|
Mathlib/GroupTheory/Coset/Defs.lean
|
theorem leftRel_eq : ⇑(leftRel s) = fun x y => x⁻¹ * y ∈ s :=
funext₂ <| by
simp only [eq_iff_iff]
apply leftRel_apply
|
α : Type u_1
inst✝ : Group α
s : Subgroup α
⊢ ∀ (a b : α), (leftRel s) a b ↔ a⁻¹ * b ∈ s
|
apply leftRel_apply
|
no goals
|
4c518f60fe2e6d5d
|
Set.PartiallyWellOrderedOn.union
|
Mathlib/Order/WellFoundedSet.lean
|
theorem PartiallyWellOrderedOn.union (hs : s.PartiallyWellOrderedOn r)
(ht : t.PartiallyWellOrderedOn r) : (s ∪ t).PartiallyWellOrderedOn r
|
case intro.inl
α : Type u_2
r : α → α → Prop
s t : Set α
hs : s.PartiallyWellOrderedOn r
ht : t.PartiallyWellOrderedOn r
f : ℕ → α
hf : ∀ (n : ℕ), f n ∈ s ∪ t
g : ℕ ↪o ℕ
hgs : ∀ (n : ℕ), f (g n) ∈ s
⊢ ∃ m n, m < n ∧ r (f m) (f n)
|
rcases hs _ hgs with ⟨m, n, hlt, hr⟩
|
case intro.inl.intro.intro.intro
α : Type u_2
r : α → α → Prop
s t : Set α
hs : s.PartiallyWellOrderedOn r
ht : t.PartiallyWellOrderedOn r
f : ℕ → α
hf : ∀ (n : ℕ), f n ∈ s ∪ t
g : ℕ ↪o ℕ
hgs : ∀ (n : ℕ), f (g n) ∈ s
m n : ℕ
hlt : m < n
hr : r (f (g m)) (f (g n))
⊢ ∃ m n, m < n ∧ r (f m) (f n)
|
2f912f07888fb86c
|
Nat.two_pow_pred_mod_two_pow
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean
|
theorem two_pow_pred_mod_two_pow (h : 0 < w) :
2 ^ (w - 1) % 2 ^ w = 2 ^ (w - 1)
|
w : Nat
h : 0 < w
⊢ 1 < 2
|
omega
|
no goals
|
4b63a19c5ed33e08
|
MeasureTheory.Measure.haar.chaar_empty
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
theorem chaar_empty (K₀ : PositiveCompacts G) : chaar K₀ ⊥ = 0
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
⊢ chaar K₀ ⊥ = 0
|
let eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
⊢ chaar K₀ ⊥ = 0
|
c7a9478c0abc9e00
|
exists_continuous_nonneg_pos
|
Mathlib/Topology/UrysohnsLemma.lean
|
theorem exists_continuous_nonneg_pos [RegularSpace X] [LocallyCompactSpace X] (x : X) :
∃ f : C(X, ℝ), HasCompactSupport f ∧ 0 ≤ (f : X → ℝ) ∧ f x ≠ 0
|
case intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
x : X
k : Set X
hk : IsCompact k
k_mem : k ∈ 𝓝 x
f : C(X, ℝ)
fk : EqOn (⇑f) 1 k
f_comp : HasCompactSupport ⇑f
hf : ∀ (x : X), f x ∈ Icc 0 1
⊢ f x ≠ 0
|
have := fk (mem_of_mem_nhds k_mem)
|
case intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
x : X
k : Set X
hk : IsCompact k
k_mem : k ∈ 𝓝 x
f : C(X, ℝ)
fk : EqOn (⇑f) 1 k
f_comp : HasCompactSupport ⇑f
hf : ∀ (x : X), f x ∈ Icc 0 1
this : f x = 1 x
⊢ f x ≠ 0
|
d3cde429bf91b0e7
|
finprod_mem_iUnion
|
Mathlib/Algebra/BigOperators/Finprod.lean
|
theorem finprod_mem_iUnion [Finite ι] {t : ι → Set α} (h : Pairwise (Disjoint on t))
(ht : ∀ i, (t i).Finite) : ∏ᶠ a ∈ ⋃ i : ι, t i, f a = ∏ᶠ i, ∏ᶠ a ∈ t i, f a
|
case intro.intro
α : Type u_1
ι : Type u_3
M : Type u_5
inst✝¹ : CommMonoid M
f : α → M
inst✝ : Finite ι
val✝ : Fintype ι
t : ι → Finset α
h : Pairwise (Disjoint on fun i => ↑(t i))
⊢ ∏ᶠ (a : α) (_ : a ∈ ⋃ i, (fun i => ↑(t i)) i), f a = ∏ᶠ (i : ι) (a : α) (_ : a ∈ (fun i => ↑(t i)) i), f a
|
rw [← biUnion_univ, ← Finset.coe_univ, ← Finset.coe_biUnion, finprod_mem_coe_finset,
Finset.prod_biUnion]
|
case intro.intro
α : Type u_1
ι : Type u_3
M : Type u_5
inst✝¹ : CommMonoid M
f : α → M
inst✝ : Finite ι
val✝ : Fintype ι
t : ι → Finset α
h : Pairwise (Disjoint on fun i => ↑(t i))
⊢ ∏ x : ι, ∏ i ∈ t x, f i = ∏ᶠ (i : ι) (a : α) (_ : a ∈ (fun i => ↑(t i)) i), f a
case intro.intro
α : Type u_1
ι : Type u_3
M : Type u_5
inst✝¹ : CommMonoid M
f : α → M
inst✝ : Finite ι
val✝ : Fintype ι
t : ι → Finset α
h : Pairwise (Disjoint on fun i => ↑(t i))
⊢ (↑Finset.univ).PairwiseDisjoint t
|
df5991dc52740a08
|
strictConvexOn_of_slope_strict_mono_adjacent
|
Mathlib/Analysis/Convex/Slope.lean
|
theorem strictConvexOn_of_slope_strict_mono_adjacent (hs : Convex 𝕜 s)
(hf :
∀ {x y z : 𝕜},
x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)) :
StrictConvexOn 𝕜 s f :=
LinearOrder.strictConvexOn_of_lt hs fun x hx z hz hxz a b ha hb hab => by
let y := a * x + b * z
have hxy : x < y
|
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hs : Convex 𝕜 s
hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)
x : 𝕜
hx : x ∈ s
z : 𝕜
hz : z ∈ s
hxz : x < z
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
y : 𝕜 := a * x + b * z
⊢ f (a • x + b • z) < a • f x + b • f z
|
have hxy : x < y := by
rw [← one_mul x, ← hab, add_mul]
exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _
|
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hs : Convex 𝕜 s
hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)
x : 𝕜
hx : x ∈ s
z : 𝕜
hz : z ∈ s
hxz : x < z
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
y : 𝕜 := a * x + b * z
hxy : x < y
⊢ f (a • x + b • z) < a • f x + b • f z
|
0f2988c10bcecabb
|
IncidenceAlgebra.zeta_prod_apply
|
Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.lean
|
lemma zeta_prod_apply (a b : α × β) : zeta 𝕜 a b = zeta 𝕜 a.1 b.1 * zeta 𝕜 a.2 b.2
|
𝕜 : Type u_2
α : Type u_5
β : Type u_6
inst✝⁴ : Ring 𝕜
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
a b : α × β
⊢ (zeta 𝕜) a b = (zeta 𝕜) a.1 b.1 * (zeta 𝕜) a.2 b.2
|
simp [← ite_and, Prod.le_def, and_comm]
|
no goals
|
c7b77d79335a807b
|
ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq
|
Mathlib/NumberTheory/ArithmeticFunction.lean
|
theorem sum_eq_iff_sum_smul_moebius_eq [AddCommGroup R] {f g : ℕ → R} :
(∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔
∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, μ x.fst • g x.snd = f n
|
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun x => if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun x => if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
|
trans (ζ : ArithmeticFunction ℤ) • f' = g'
|
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun x => if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun x => if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ (∀ n > 0, ∑ i ∈ n.divisors, f i = g n) ↔ ↑ζ • f' = g'
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
f' : ArithmeticFunction R := { toFun := fun x => if x = 0 then 0 else f x, map_zero' := ⋯ }
g' : ArithmeticFunction R := { toFun := fun x => if x = 0 then 0 else g x, map_zero' := ⋯ }
⊢ ↑ζ • f' = g' ↔ ∀ n > 0, ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
|
2f30d3e992a2cea6
|
IsRelPrime.mul_right
|
Mathlib/Algebra/Divisibility/Units.lean
|
theorem IsRelPrime.mul_right (H1 : IsRelPrime x y) (H2 : IsRelPrime x z) :
IsRelPrime x (y * z)
|
α : Type u_1
inst✝¹ : CommMonoid α
x y z : α
inst✝ : DecompositionMonoid α
H1 : IsRelPrime y x
H2 : IsRelPrime z x
⊢ IsRelPrime (y * z) x
|
exact H1.mul_left H2
|
no goals
|
ac436550352dc02b
|
CategoryTheory.Monoidal.Reflective.isIso_tfae
|
Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean
|
theorem isIso_tfae : List.TFAE
[ ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
, ∀ (c : C) (d : D), IsIso ((pre (adj.unit.app d)).app (R.obj c))
, ∀ (d d' : D), IsIso (L.map ((adj.unit.app d) ▷ d'))
, ∀ (d d' : D), IsIso (L.map ((adj.unit.app d) ⊗ (adj.unit.app d')))]
|
case h.h
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))
tfae_4_to_1 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))) →
∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
tfae_1_to_3 :
(∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))
d d' : D
c : C
w₁ :
(coyoneda.map (L.map (adj.unit.app d ▷ d')).op).app c =
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d'))) c).symm ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d'))) c)
w₂ :
(yoneda.map ((pre (adj.unit.app d)).app (R.obj c))).app (Opposite.op d') =
⇑((ihom.adjunction d).homEquiv d' (R.obj c)) ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑((ihom.adjunction ((L ⋙ R).obj d)).homEquiv d' (R.obj c)).symm
⊢ IsIso ((yoneda.map ((pre (adj.unit.app d)).app (R.obj c))).app (Opposite.op d')) ↔
IsIso ((coyoneda.map (L.map (adj.unit.app d ▷ d')).op).app c)
|
rw [w₂, w₁, isIso_iff_bijective, isIso_iff_bijective]
|
case h.h
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))
tfae_4_to_1 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))) →
∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
tfae_1_to_3 :
(∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))
d d' : D
c : C
w₁ :
(coyoneda.map (L.map (adj.unit.app d ▷ d')).op).app c =
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d'))) c).symm ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d'))) c)
w₂ :
(yoneda.map ((pre (adj.unit.app d)).app (R.obj c))).app (Opposite.op d') =
⇑((ihom.adjunction d).homEquiv d' (R.obj c)) ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑((ihom.adjunction ((L ⋙ R).obj d)).homEquiv d' (R.obj c)).symm
⊢ Function.Bijective
(⇑((ihom.adjunction d).homEquiv d' (R.obj c)) ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑((ihom.adjunction ((L ⋙ R).obj d)).homEquiv d' (R.obj c)).symm) ↔
Function.Bijective
(⇑(adj.homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d'))) c).symm ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d'))) c))
|
8ccd82e9c49cc7ac
|
Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub
|
Mathlib/Geometry/Euclidean/MongePoint.lean
|
theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2))
{i₁ i₂ : Fin (n + 3)} :
⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points,
s.points i₁ -ᵥ s.points i₂⟫ =
0
|
case neg
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -(∑ x : Fin (n + 2 + 1),
∑ x_1 : PointsWithCircumcenterIndex (n + 2),
(if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
(pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
(dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) +
∑ x : PointsWithCircumcenterIndex (n + 2),
-2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
(dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
2 =
0
|
rw [← sum_subset fs.subset_univ _]
|
case neg
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -(∑ x ∈ fs,
∑ x_1 : PointsWithCircumcenterIndex (n + 2),
(if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
(pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
(dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) +
∑ x : PointsWithCircumcenterIndex (n + 2),
-2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x *
(dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) /
2 =
0
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ ∀ x ∈ univ,
x ∉ fs →
∑ x_1 : PointsWithCircumcenterIndex (n + 2),
(if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
(pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 *
(dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) =
0
|
4dd62aff1c506de0
|
PFunctor.M.agree_iff_agree'
|
Mathlib/Data/PFunctor/Univariate/M.lean
|
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y
|
case mpr.succ.step.f.f.refl.refl.a
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x y : F.M), Agree' n✝ x y → Agree (x.approx n✝) (y.approx (n✝ + 1))
a : F.A
x' y' : F.B a → F.M
a✝² : ∀ (i : F.B a), Agree' n✝ (x' i) (y' i)
a✝¹ : M.mk ⟨a, x'⟩ = M.mk ⟨a, x'⟩
a✝ : M.mk ⟨a, y'⟩ = M.mk ⟨a, y'⟩
i : F.B a
⊢ Agree ((x' i).approx n✝) ((y' i).approx (n✝ + 1))
|
apply n_ih
|
case mpr.succ.step.f.f.refl.refl.a.h
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x y : F.M), Agree' n✝ x y → Agree (x.approx n✝) (y.approx (n✝ + 1))
a : F.A
x' y' : F.B a → F.M
a✝² : ∀ (i : F.B a), Agree' n✝ (x' i) (y' i)
a✝¹ : M.mk ⟨a, x'⟩ = M.mk ⟨a, x'⟩
a✝ : M.mk ⟨a, y'⟩ = M.mk ⟨a, y'⟩
i : F.B a
⊢ Agree' n✝ (x' i) (y' i)
|
74e48f06950cd29d
|
ZetaAsymptotics.continuousOn_term
|
Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean
|
lemma continuousOn_term (n : ℕ) :
ContinuousOn (fun x ↦ term (n + 1) x) (Ici 1)
|
case h_cont.h.refine_2
n : ℕ
x : ℝ
hx : x ∈ Ioc (↑(n + 1)) (↑(n + 1) + 1)
s : ℝ
hs : 1 ≤ s
⊢ x ^ (s + 1) ≠ 0
|
exact (rpow_pos_of_pos ((Nat.cast_pos.mpr (by simp)).trans hx.1) _).ne'
|
no goals
|
bbd328e1b624f3ab
|
NumberField.house.ξ_ne_0
|
Mathlib/NumberTheory/NumberField/House.lean
|
theorem ξ_ne_0 : ξ K x ≠ 0
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
β : Type u_3
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
H : NumberField.house.ξ K x = 0
⊢ False
|
apply hxl
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
β : Type u_3
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
H : NumberField.house.ξ K x = 0
⊢ x = 0
|
d09ff3eeaba5df9f
|
ProbabilityTheory.iteratedDeriv_two_cgf_eq_integral
|
Mathlib/Probability/Moments/MGFAnalytic.lean
|
lemma iteratedDeriv_two_cgf_eq_integral (h : v ∈ interior (integrableExpSet X μ)) :
iteratedDeriv 2 (cgf X μ) v
= μ[fun ω ↦ (X ω - deriv (cgf X μ) v)^2 * exp (v * X ω)] / mgf X μ v
|
case e_a.hb
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ mgf X μ v ≠ 0
|
exact (mgf_pos' hμ (interior_subset (s := integrableExpSet X μ) h)).ne'
|
no goals
|
f81a041fb4b6f1f5
|
CategoryTheory.IsFiltered.iff_nonempty_limit
|
Mathlib/CategoryTheory/Limits/Filtered.lean
|
theorem IsFiltered.iff_nonempty_limit : IsFiltered C ↔
∀ {J : Type v} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
∃ (X : C), Nonempty (limit (F.op ⋙ yoneda.obj X))
|
case refine_1.intro
C : Type u
inst✝ : Category.{v, u} C
h : ∀ {J : Type v} [inst : SmallCategory J] [inst_1 : FinCategory J] (F : J ⥤ C), Nonempty (Cocone F)
J : Type v
x✝¹ : SmallCategory J
x✝ : FinCategory J
F : J ⥤ C
c : Cocone F
⊢ ∃ X, Nonempty (limit (F.op ⋙ yoneda.obj X))
|
exact ⟨c.pt, ⟨(limitCompYonedaIsoCocone F c.pt).inv c.ι⟩⟩
|
no goals
|
c0b0888232b0e657
|
minSmoothness_add
|
Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean
|
lemma minSmoothness_add {n m : WithTop ℕ∞} : minSmoothness 𝕜 (n + m) = minSmoothness 𝕜 n + m
|
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n m : WithTop ℕ∞
⊢ minSmoothness 𝕜 (n + m) = minSmoothness 𝕜 n + m
|
simp only [minSmoothness]
|
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n m : WithTop ℕ∞
⊢ (if IsRCLikeNormedField 𝕜 then n + m else ω) = (if IsRCLikeNormedField 𝕜 then n else ω) + m
|
6dbf34c7a3eb9f1b
|
Subgroup.isCoatom_comap_of_surjective
|
Mathlib/Algebra/Group/Subgroup/Order.lean
|
lemma isCoatom_comap_of_surjective
{H : Type*} [Group H] {φ : G →* H} (hφ : Function.Surjective φ)
{M : Subgroup H} (hM : IsCoatom M) : IsCoatom (M.comap φ)
|
case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM✝ : IsCoatom M
K : Subgroup G
hK : comap φ M < K
hM : comap φ M < comap φ (map φ K) → comap φ (map φ K) = comap φ ⊤
⊢ K = ⊤
|
rw [comap_map_eq_self ((M.ker_le_comap φ).trans hK.le), comap_top] at hM
|
case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM✝ : IsCoatom M
K : Subgroup G
hK : comap φ M < K
hM : comap φ M < K → K = ⊤
⊢ K = ⊤
|
1eac4c9273ad2e88
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_fold_fn_preserves_induction_motive
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem reduce_fold_fn_preserves_induction_motive {c_arr : Array (Literal (PosFin n))}
{assignment : Array Assignment}
(idx : Fin c_arr.size) (res : ReduceResult (PosFin n))
(ih : ReducePostconditionInductionMotive c_arr assignment idx.1 res) :
ReducePostconditionInductionMotive c_arr assignment (idx.1 + 1) (reduce_fold_fn assignment res c_arr[idx])
|
n : Nat
c_arr : Array (Literal (PosFin n))
assignment : Array Assignment
idx : Fin c_arr.size
i : PosFin n
b : Bool
p : PosFin n → Bool
j : Fin c_arr.size
j_lt_idx_add_one : ↑j < ↑idx + 1
p_entails_c_arr_j : p ⊨ c_arr[↑j]
acc✝ : ReduceResult (PosFin n)
ih : ReducePostconditionInductionMotive c_arr assignment (↑idx) reducedToEmpty
x✝ : Assignment
heq : assignment[c_arr[↑idx].fst.val]! = pos
h : c_arr[↑idx] = (i, b)
c_arr_idx_eq_true : b = true
hp : hasAssignment (decide (p c_arr[↑idx].fst = false)) pos = false
⊢ p ⊨ (i, b)
|
by_cases p c_arr[idx.val].1
|
case pos
n : Nat
c_arr : Array (Literal (PosFin n))
assignment : Array Assignment
idx : Fin c_arr.size
i : PosFin n
b : Bool
p : PosFin n → Bool
j : Fin c_arr.size
j_lt_idx_add_one : ↑j < ↑idx + 1
p_entails_c_arr_j : p ⊨ c_arr[↑j]
acc✝ : ReduceResult (PosFin n)
ih : ReducePostconditionInductionMotive c_arr assignment (↑idx) reducedToEmpty
x✝ : Assignment
heq : assignment[c_arr[↑idx].fst.val]! = pos
h : c_arr[↑idx] = (i, b)
c_arr_idx_eq_true : b = true
hp : hasAssignment (decide (p c_arr[↑idx].fst = false)) pos = false
h✝ : p c_arr[↑idx].fst = true
⊢ p ⊨ (i, b)
case neg
n : Nat
c_arr : Array (Literal (PosFin n))
assignment : Array Assignment
idx : Fin c_arr.size
i : PosFin n
b : Bool
p : PosFin n → Bool
j : Fin c_arr.size
j_lt_idx_add_one : ↑j < ↑idx + 1
p_entails_c_arr_j : p ⊨ c_arr[↑j]
acc✝ : ReduceResult (PosFin n)
ih : ReducePostconditionInductionMotive c_arr assignment (↑idx) reducedToEmpty
x✝ : Assignment
heq : assignment[c_arr[↑idx].fst.val]! = pos
h : c_arr[↑idx] = (i, b)
c_arr_idx_eq_true : b = true
hp : hasAssignment (decide (p c_arr[↑idx].fst = false)) pos = false
h✝ : ¬p c_arr[↑idx].fst = true
⊢ p ⊨ (i, b)
|
681a60500ce41fce
|
Real.binEntropy_neg_of_neg
|
Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean
|
/-- Outside the usual range of `binEntropy`, it is negative. This is due to `log p = log |p|`. -/
lemma binEntropy_neg_of_neg (hp : p < 0) : binEntropy p < 0
|
case pos
p : ℝ
hp : p < 0
hp' : p < -1
this : log p < log (1 - p)
⊢ -p * log p < (1 - p) * log (1 - p)
|
nlinarith [log_pos_of_lt_neg_one hp']
|
no goals
|
c3c271761677db38
|
HasFiniteFPowerSeriesOnBall.mk'
|
Mathlib/Analysis/Analytic/CPolynomialDef.lean
|
theorem HasFiniteFPowerSeriesOnBall.mk' {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E}
{n : ℕ} {r : ℝ≥0∞} (finite : ∀ (m : ℕ), n ≤ m → p m = 0) (pos : 0 < r)
(sum_eq : ∀ y ∈ EMetric.ball 0 r, (∑ i ∈ Finset.range n, p i fun _ ↦ y) = f (x + y)) :
HasFiniteFPowerSeriesOnBall f p x n r where
r_le := p.radius_eq_top_of_eventually_eq_zero (Filter.eventually_atTop.mpr ⟨n, finite⟩) ▸ le_top
r_pos := pos
hasSum hy := sum_eq _ hy ▸ hasSum_sum_of_ne_finset_zero fun m hm ↦ by
rw [Finset.mem_range, not_lt] at hm; rw [finite m hm]; rfl
finite := finite
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
r : ℝ≥0∞
finite : ∀ (m : ℕ), n ≤ m → p m = 0
pos : 0 < r
sum_eq : ∀ y ∈ EMetric.ball 0 r, (∑ i ∈ Finset.range n, (p i) fun x => y) = f (x + y)
y✝ : E
hy : y✝ ∈ EMetric.ball 0 r
m : ℕ
hm : m ∉ Finset.range n
⊢ ((p m) fun x => y✝) = 0
|
rw [Finset.mem_range, not_lt] at hm
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
r : ℝ≥0∞
finite : ∀ (m : ℕ), n ≤ m → p m = 0
pos : 0 < r
sum_eq : ∀ y ∈ EMetric.ball 0 r, (∑ i ∈ Finset.range n, (p i) fun x => y) = f (x + y)
y✝ : E
hy : y✝ ∈ EMetric.ball 0 r
m : ℕ
hm : n ≤ m
⊢ ((p m) fun x => y✝) = 0
|
65df954b712c494a
|
Array.mapIdx_eq_append_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/MapIdx.lean
|
theorem mapIdx_eq_append_iff {l : Array α} {f : Nat → α → β} {l₁ l₂ : Array β} :
mapIdx f l = l₁ ++ l₂ ↔
∃ (l₁' : Array α) (l₂' : Array α), l = l₁' ++ l₂' ∧
l₁'.mapIdx f = l₁ ∧
l₂'.mapIdx (fun i => f (i + l₁'.size)) = l₂
|
case mk.mk.mk
α : Type u_1
β : Type u_2
f : Nat → α → β
l : List α
l₁ l₂ : List β
⊢ mapIdx f { toList := l } = { toList := l₁ } ++ { toList := l₂ } ↔
∃ l₁' l₂',
{ toList := l } = l₁' ++ l₂' ∧
mapIdx f l₁' = { toList := l₁ } ∧ mapIdx (fun i => f (i + l₁'.size)) l₂' = { toList := l₂ }
|
simp only [List.mapIdx_toArray, List.append_toArray, mk.injEq, List.mapIdx_eq_append_iff,
toArray_eq_append_iff]
|
case mk.mk.mk
α : Type u_1
β : Type u_2
f : Nat → α → β
l : List α
l₁ l₂ : List β
⊢ (∃ l₁' l₂', l = l₁' ++ l₂' ∧ List.mapIdx f l₁' = l₁ ∧ List.mapIdx (fun i => f (i + l₁'.length)) l₂' = l₂) ↔
∃ l₁' l₂',
l = l₁'.toList ++ l₂'.toList ∧
mapIdx f l₁' = { toList := l₁ } ∧ mapIdx (fun i => f (i + l₁'.size)) l₂' = { toList := l₂ }
|
34328ac48040b85d
|
ProbabilityTheory.evariance_def'
|
Mathlib/Probability/Variance.lean
|
theorem evariance_def' [IsProbabilityMeasure μ] {X : Ω → ℝ} (hX : AEStronglyMeasurable X μ) :
evariance X μ = (∫⁻ ω, ‖X ω‖ₑ ^ 2 ∂μ) - ENNReal.ofReal (μ[X] ^ 2)
|
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
hℒ : ¬MemLp X 2 μ
⊢ ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ = ⊤ ∧ ENNReal.ofReal ((∫ (x : Ω), X x ∂μ) ^ 2) ≠ ⊤
|
refine ⟨?_, ENNReal.ofReal_ne_top⟩
|
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
hℒ : ¬MemLp X 2 μ
⊢ ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ = ⊤
|
eb83c290f9e735d8
|
MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates
|
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
theorem addHaar_eq_zero_of_disjoint_translates {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E)
[IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (hu : IsBounded (range u))
(hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s) : μ s = 0
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ μ (s ∩ closedBall 0 R) = 0
|
apply addHaar_eq_zero_of_disjoint_translates_aux μ u
(isBounded_closedBall.subset inter_subset_right) hu _ (h's.inter measurableSet_closedBall)
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ Pairwise (Disjoint on fun n => {u n} + s ∩ closedBall 0 R)
|
d6dc5eb0e45c9230
|
NeZero.of_map
|
Mathlib/Algebra/GroupWithZero/Hom.lean
|
lemma of_map (f : F) [neZero : NeZero (f a)] : NeZero a :=
⟨fun h ↦ ne (f a) <| by rw [h]; exact ZeroHomClass.map_zero f⟩
|
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroHomClass F α β
a : α
f : F
neZero : NeZero (f a)
h : a = 0
⊢ f a = 0
|
rw [h]
|
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroHomClass F α β
a : α
f : F
neZero : NeZero (f a)
h : a = 0
⊢ f 0 = 0
|
aa20fb6a33fbb663
|
smul_orthogonalProjection_singleton
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem smul_orthogonalProjection_singleton {v : E} (w : E) :
((‖v‖ ^ 2 : ℝ) : 𝕜) • (orthogonalProjection (𝕜 ∙ v) w : E) = ⟪v, w⟫ • v
|
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
this : ↑((orthogonalProjection (Submodule.span 𝕜 {v})) (↑‖v‖ ^ 2 • w)) = ⟪v, w⟫_𝕜 • v
⊢ ↑(‖v‖ ^ 2) • ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = ⟪v, w⟫_𝕜 • v
|
simpa using this
|
no goals
|
eea63064d21a7d29
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastZeroExtend.go_denote_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ZeroExtend.lean
|
theorem go_denote_eq (aig : AIG α) (w : Nat) (input : AIG.RefVec aig w) (newWidth curr : Nat)
(hcurr : curr ≤ newWidth) (s : AIG.RefVec aig curr) (assign : α → Bool) :
∀ (idx : Nat) (hidx1 : idx < newWidth),
curr ≤ idx
→
⟦
(go aig w input newWidth curr hcurr s).aig,
(go aig w input newWidth curr hcurr s).vec.get idx hidx1,
assign
⟧
=
if hidx : idx < w then
⟦aig, input.get idx hidx, assign⟧
else
false
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
w : Nat
input : aig.RefVec w
newWidth curr : Nat
hcurr : curr ≤ newWidth
s : aig.RefVec curr
assign : α → Bool
idx : Nat
hidx1 : idx < newWidth
hidx2 : curr ≤ idx
⊢ ⟦assign,
{ aig := (go aig w input newWidth curr hcurr s).aig,
ref := (go aig w input newWidth curr hcurr s).vec.get idx hidx1 }⟧ =
if hidx : idx < w then ⟦assign, { aig := aig, ref := input.get idx hidx }⟧ else false
|
generalize hgo : go aig w input newWidth curr hcurr s = res
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
w : Nat
input : aig.RefVec w
newWidth curr : Nat
hcurr : curr ≤ newWidth
s : aig.RefVec curr
assign : α → Bool
idx : Nat
hidx1 : idx < newWidth
hidx2 : curr ≤ idx
res : RefVecEntry α newWidth
hgo : go aig w input newWidth curr hcurr s = res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ =
if hidx : idx < w then ⟦assign, { aig := aig, ref := input.get idx hidx }⟧ else false
|
4f5d95c54c74e073
|
NonarchimedeanGroup.cauchySeq_prod_of_tendsto_cofinite_one
|
Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean
|
theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f cofinite (𝓝 1)) :
CauchySeq (fun s ↦ ∏ i ∈ s, f i)
|
case intro
α : Type u_1
G : Type u_2
inst✝³ : CommGroup G
inst✝² : UniformSpace G
inst✝¹ : UniformGroup G
inst✝ : NonarchimedeanGroup G
f : α → G
hf : Tendsto f cofinite (𝓝 1)
U : Set G
hU : U ∈ 𝓝 1
V : OpenSubgroup G
hV : ↑V ⊆ U
⊢ ∃ s, ∀ (t : Finset α), Disjoint t s → ∏ b ∈ t, f b ∈ U
|
use (tendsto_def.mp hf V V.mem_nhds_one).toFinset
|
case h
α : Type u_1
G : Type u_2
inst✝³ : CommGroup G
inst✝² : UniformSpace G
inst✝¹ : UniformGroup G
inst✝ : NonarchimedeanGroup G
f : α → G
hf : Tendsto f cofinite (𝓝 1)
U : Set G
hU : U ∈ 𝓝 1
V : OpenSubgroup G
hV : ↑V ⊆ U
⊢ ∀ (t : Finset α), Disjoint t (Set.Finite.toFinset ⋯) → ∏ b ∈ t, f b ∈ U
|
8faaca2207b959f4
|
PrimeMultiset.prod_ofNatList
|
Mathlib/Data/PNat/Factors.lean
|
theorem prod_ofNatList (l : List ℕ) (h) : ((ofNatList l h).prod : ℕ) = l.prod
|
l : List ℕ
h : ∀ p ∈ l, Nat.Prime p
⊢ ↑(ofNatList l h).prod = l.prod
|
have := prod_ofNatMultiset (l : Multiset ℕ) h
|
l : List ℕ
h : ∀ p ∈ l, Nat.Prime p
this : ↑(ofNatMultiset (↑l) h).prod = (↑l).prod
⊢ ↑(ofNatList l h).prod = l.prod
|
a0b6e678c5f86872
|
Ideal.injective_algebraMap_quotient_residueField
|
Mathlib/RingTheory/LocalRing/ResidueField/Ideal.lean
|
lemma Ideal.injective_algebraMap_quotient_residueField :
Function.Injective (algebraMap (R ⧸ I) I.ResidueField)
|
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ RingHom.ker (algebraMap (R ⧸ I) I.ResidueField) = ⊥
|
refine (Ideal.ker_quotient_lift _ _).trans ?_
|
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ map (Quotient.mk I) (RingHom.ker ↑(Algebra.ofId R I.ResidueField)) = ⊥
|
a3bc889e6f8e164d
|
Multiset.prod_map_neg
|
Mathlib/Algebra/BigOperators/Ring/Multiset.lean
|
@[simp] lemma prod_map_neg (s : Multiset α) : (s.map Neg.neg).prod = (-1) ^ card s * s.prod :=
Quotient.inductionOn s (by simp)
|
α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : HasDistribNeg α
s : Multiset α
⊢ ∀ (a : List α), (map Neg.neg ⟦a⟧).prod = (-1) ^ card ⟦a⟧ * prod ⟦a⟧
|
simp
|
no goals
|
428036c611cb1c3f
|
Nat.sqrt.lt_iter_succ_sq
|
Mathlib/Data/Nat/Sqrt.lean
|
lemma sqrt.lt_iter_succ_sq (n guess : ℕ) (hn : n < (guess + 1) * (guess + 1)) :
n < (sqrt.iter n guess + 1) * (sqrt.iter n guess + 1)
|
n guess : ℕ
hn : n < (guess + 1) * (guess + 1)
m : ℕ := (guess + n / guess) / 2
⊢ n <
((if (guess + n / guess) / 2 < guess then iter n ((guess + n / guess) / 2) else guess) + 1) *
((if (guess + n / guess) / 2 < guess then iter n ((guess + n / guess) / 2) else guess) + 1)
|
split_ifs with h
|
case pos
n guess : ℕ
hn : n < (guess + 1) * (guess + 1)
m : ℕ := (guess + n / guess) / 2
h : (guess + n / guess) / 2 < guess
⊢ n < (iter n ((guess + n / guess) / 2) + 1) * (iter n ((guess + n / guess) / 2) + 1)
case neg
n guess : ℕ
hn : n < (guess + 1) * (guess + 1)
m : ℕ := (guess + n / guess) / 2
h : ¬(guess + n / guess) / 2 < guess
⊢ n < (guess + 1) * (guess + 1)
|
d01ade80ccfefae8
|
List.mem_of_elem_eq_true
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean
|
theorem mem_of_elem_eq_true [BEq α] [LawfulBEq α] {a : α} {as : List α} : elem a as = true → a ∈ as
|
α : Type u
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
as : List α
⊢ elem a nil = true → a ∈ nil
|
simp [elem]
|
no goals
|
f8b48be2038fd24e
|
IsLocallyConstant.apply_eq_of_isPreconnected
|
Mathlib/Topology/LocallyConstant/Basic.lean
|
theorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}
(hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y
|
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
hf : IsLocallyConstant f
s : Set X
hs : IsPreconnected s
x y : X
hx : x ∈ s
hy : y ∈ s
⊢ f x = f y
|
let U := f ⁻¹' {f y}
|
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
hf : IsLocallyConstant f
s : Set X
hs : IsPreconnected s
x y : X
hx : x ∈ s
hy : y ∈ s
U : Set X := f ⁻¹' {f y}
⊢ f x = f y
|
29d497e5ddecc234
|
TensorProduct.tmul_sum
|
Mathlib/LinearAlgebra/TensorProduct/Basic.lean
|
theorem tmul_sum (m : M) {α : Type*} (s : Finset α) (n : α → N) :
(m ⊗ₜ[R] ∑ a ∈ s, n a) = ∑ a ∈ s, m ⊗ₜ[R] n a
|
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_5
N : Type u_6
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
m : M
α : Type u_11
s : Finset α
n : α → N
⊢ m ⊗ₜ[R] ∑ a ∈ s, n a = ∑ a ∈ s, m ⊗ₜ[R] n a
|
classical
induction s using Finset.induction with
| empty => simp
| insert has ih => simp [Finset.sum_insert has, tmul_add, ih]
|
no goals
|
b1511420f934e3d1
|
RootPairing.linearIndependent_iff_coxeterWeight_ne_four
|
Mathlib/LinearAlgebra/RootSystem/Reduced.lean
|
/-- See also `RootPairing.linearIndependent_iff_coxeterWeightIn_ne_four`. -/
lemma linearIndependent_iff_coxeterWeight_ne_four :
LinearIndependent R ![P.root i, P.root j] ↔ P.coxeterWeight i j ≠ 4
|
ι : Type u_1
R : Type u_2
M : 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
P : RootPairing ι R M N
i j : ι
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
this : NoZeroSMulDivisors ℤ M
h : ¬LinearIndependent R ![P.root j, P.root i]
h₁ : P.pairing j i • P.root i = 2 • P.root j
h₂ : P.pairing i j • P.root j = 2 • P.root i
⊢ P.coxeterWeight i j = 4
|
suffices P.coxeterWeight i j • P.root i = (4 : R) • P.root i from
smul_left_injective R (P.ne_zero i) this
|
ι : Type u_1
R : Type u_2
M : 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
P : RootPairing ι R M N
i j : ι
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
this : NoZeroSMulDivisors ℤ M
h : ¬LinearIndependent R ![P.root j, P.root i]
h₁ : P.pairing j i • P.root i = 2 • P.root j
h₂ : P.pairing i j • P.root j = 2 • P.root i
⊢ P.coxeterWeight i j • P.root i = 4 • P.root i
|
3e389464a4fe8362
|
MeasureTheory.Measure.haveLebesgueDecomposition_spec
|
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
theorem haveLebesgueDecomposition_spec (μ ν : Measure α) [h : HaveLebesgueDecomposition μ ν] :
Measurable (μ.rnDeriv ν) ∧
μ.singularPart ν ⟂ₘ ν ∧ μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
|
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (μ.rnDeriv ν) ∧ μ.singularPart ν ⟂ₘ ν ∧ μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
|
rw [singularPart, rnDeriv, dif_pos h, dif_pos h]
|
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (Classical.choose ⋯).2 ∧
(Classical.choose ⋯).1 ⟂ₘ ν ∧ μ = (Classical.choose ⋯).1 + ν.withDensity (Classical.choose ⋯).2
|
d00ff8c00e79ec10
|
ArithmeticFunction.LSeries_zeta_eq_riemannZeta
|
Mathlib/NumberTheory/LSeries/Dirichlet.lean
|
/-- The L-series of the arithmetic function `ζ` equals the Riemann Zeta Function on its
domain of convergence `1 < re s`. -/
lemma LSeries_zeta_eq_riemannZeta {s : ℂ} (hs : 1 < s.re) : L ↗ζ s = riemannZeta s
|
s : ℂ
hs : 1 < s.re
⊢ L (fun n => ↑(ζ n)) s = riemannZeta s
|
suffices ∑' n, term (fun n ↦ if n = 0 then 0 else 1) s n = ∑' n : ℕ, 1 / (n : ℂ) ^ s by
simpa [LSeries, zeta_eq_tsum_one_div_nat_cpow hs]
|
s : ℂ
hs : 1 < s.re
⊢ ∑' (n : ℕ), term (fun n => if n = 0 then 0 else 1) s n = ∑' (n : ℕ), 1 / ↑n ^ s
|
326702a7da7dacde
|
Ordinal.op_eq_self_of_principal
|
Mathlib/SetTheory/Ordinal/Principal.lean
|
theorem op_eq_self_of_principal (hao : a < o) (H : IsNormal (op a))
(ho : Principal op o) (ho' : IsLimit o) : op a o = o
|
a o : Ordinal.{u}
op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
hao : a < o
H : IsNormal (op a)
ho : Principal op o
ho' : o.IsLimit
⊢ ∀ i < o, op a i ≤ o
|
exact fun b hbo => (ho hao hbo).le
|
no goals
|
3c41dd1f09f9fc53
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.