name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
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|---|---|---|---|---|---|---|
Filter.limsup_top_eq_iSup
|
Mathlib/Order/LiminfLimsup.lean
|
@[simp] lemma limsup_top_eq_iSup (u : β → α) : limsup u ⊤ = ⨆ i, u i
|
α : Type u_1
β : Type u_2
inst✝ : CompleteLattice α
u : β → α
⊢ limsup u ⊤ = ⨆ i, u i
|
rw [limsup, map_top, limsSup_principal_eq_sSup, sSup_range]
|
no goals
|
6caebe39043690ac
|
MeasureTheory.IntegrableOn.smul_continuousOn
|
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
(hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
IntegrableOn (fun x => f x • g x) K μ
|
X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K μ
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ IntegrableOn (fun x => f x • g x) K μ
|
rw [IntegrableOn, ← integrable_norm_iff]
|
X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K μ
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ Integrable (fun a => ‖f a • g a‖) (μ.restrict K)
X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K μ
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ AEStronglyMeasurable (fun x => f x • g x) (μ.restrict K)
|
f776dc909af1d266
|
ContinuousMultilinearMap.norm_map_snoc_le
|
Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean
|
theorem ContinuousMultilinearMap.norm_map_snoc_le (f : ContinuousMultilinearMap 𝕜 Ei G)
(m : ∀ i : Fin n, Ei <| castSucc i) (x : Ei (last n)) :
‖f (snoc m x)‖ ≤ (‖f‖ * ∏ i, ‖m i‖) * ‖x‖ :=
calc
‖f (snoc m x)‖ ≤ ‖f‖ * ∏ i, ‖snoc m x i‖ := f.le_opNorm _
_ = (‖f‖ * ∏ i, ‖m i‖) * ‖x‖
|
𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ContinuousMultilinearMap 𝕜 Ei G
m : (i : Fin n) → Ei i.castSucc
x : Ei (last n)
⊢ ‖f‖ * ((∏ i : Fin n, ‖snoc m x i.castSucc‖) * ‖snoc m x (last n)‖) = (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖
|
simp [mul_assoc]
|
no goals
|
4ce129524315d14f
|
BitVec.slt_eq_not_carry
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean
|
theorem slt_eq_not_carry (x y : BitVec w) :
x.slt y = (x.msb == y.msb).xor (carry w x (~~~y) true)
|
w : Nat
x y : BitVec w
⊢ x.slt y = (x.msb == y.msb ^^ carry w x (~~~y) true)
|
simp only [slt_eq_ult, bne, ult_eq_not_carry]
|
w : Nat
x y : BitVec w
⊢ (!(!x.msb == y.msb) == !carry w x (~~~y) true) = !(x.msb == y.msb) == carry w x (~~~y) true
|
6fcae454a0b69725
|
Compactum.cl_cl
|
Mathlib/Topology/Category/Compactum.lean
|
theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A
|
case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
⊢ X.str F ∈ Compactum.cl A
|
suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
use X.join G
have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
rw [join_distrib, this]
exact ⟨h1 (Or.inl rfl), rfl⟩
|
case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
⊢ ∀ (T : fsu), ι T ⊆ C1 → (⋂₀ ι T).Nonempty
|
1c30fb7238f2120d
|
List.sorted_merge
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean
|
theorem sorted_merge
(trans : ∀ (a b c : α), le a b → le b c → le a c)
(total : ∀ (a b : α), le a b || le b a)
(l₁ l₂ : List α) (h₁ : l₁.Pairwise le) (h₂ : l₂.Pairwise le) : (merge l₁ l₂ le).Pairwise le
|
case cons.cons.isTrue
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
x : α
l₁ : List α
ih₁ :
∀ (l₂ : List α),
Pairwise (fun a b => le a b = true) l₁ →
Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) (l₁.merge l₂ le)
h₁ : Pairwise (fun a b => le a b = true) (x :: l₁)
y : α
l₂ : List α
ih₂ : Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) ((x :: l₁).merge l₂ le)
h₂ : Pairwise (fun a b => le a b = true) (y :: l₂)
h : le x y = true
⊢ Pairwise (fun a b => le a b = true) (x :: l₁.merge (y :: l₂) le)
|
apply Pairwise.cons
|
case cons.cons.isTrue.a
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
x : α
l₁ : List α
ih₁ :
∀ (l₂ : List α),
Pairwise (fun a b => le a b = true) l₁ →
Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) (l₁.merge l₂ le)
h₁ : Pairwise (fun a b => le a b = true) (x :: l₁)
y : α
l₂ : List α
ih₂ : Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) ((x :: l₁).merge l₂ le)
h₂ : Pairwise (fun a b => le a b = true) (y :: l₂)
h : le x y = true
⊢ ∀ (a' : α), a' ∈ l₁.merge (y :: l₂) le → le x a' = true
case cons.cons.isTrue.a
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
x : α
l₁ : List α
ih₁ :
∀ (l₂ : List α),
Pairwise (fun a b => le a b = true) l₁ →
Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) (l₁.merge l₂ le)
h₁ : Pairwise (fun a b => le a b = true) (x :: l₁)
y : α
l₂ : List α
ih₂ : Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) ((x :: l₁).merge l₂ le)
h₂ : Pairwise (fun a b => le a b = true) (y :: l₂)
h : le x y = true
⊢ Pairwise (fun a b => le a b = true) (l₁.merge (y :: l₂) le)
|
8efa7192a6a8e376
|
Vector.getElem_unattach
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Attach.lean
|
theorem getElem_unattach
{p : α → Prop} {l : Vector { x // p x } n} (i : Nat) (h : i < n) :
l.unattach[i] = (l[i]'(by simpa using h)).1
|
α : Type u_1
n : Nat
p : α → Prop
l : Vector { x // p x } n
i : Nat
h : i < n
⊢ l.unattach[i] = l[i].val
|
simp [unattach]
|
no goals
|
e32f050b9313d981
|
CategoryTheory.Localization.essSurj_mapArrow
|
Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean
|
lemma Localization.essSurj_mapArrow :
L.mapArrow.EssSurj where
mem_essImage f
|
case intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
f : Arrow D
this : L.EssSurj
X : C
eX : L.obj X ≅ f.left
Y : C
eY : L.obj Y ≅ f.right
φ : W.LeftFraction X Y
hφ : eX.hom ≫ f.hom ≫ eY.inv = φ.map L ⋯
⊢ eX.inv ≫ L.map φ.f = f.hom ≫ eY.inv ≫ L.map φ.s
|
simp only [← cancel_epi eX.hom, Iso.hom_inv_id_assoc, reassoc_of% hφ,
MorphismProperty.LeftFraction.map_comp_map_s]
|
no goals
|
164023e816364abf
|
sum_Ico_pow
|
Mathlib/NumberTheory/Bernoulli.lean
|
theorem sum_Ico_pow (n p : ℕ) :
(∑ k ∈ Ico 1 (n + 1), (k : ℚ) ^ p) =
∑ i ∈ range (p + 1), bernoulli' i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1)
|
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
h2 : f 1 + ↑n ^ p.succ = 1 / 2 * ↑n ^ p.succ
this :
∑ i ∈ range p, bernoulli (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2) =
∑ i ∈ range p, bernoulli' (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2)
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ k ∈ range n.succ, ↑k ^ p.succ
|
simp [sum_Ico_eq_sub _ hle, succ_ne_zero]
|
no goals
|
0112bf36c164f141
|
CategoryTheory.IsCofiltered.of_exists_of_isCofiltered_of_fullyFaithful
|
Mathlib/CategoryTheory/Filtered/Final.lean
|
theorem IsCofiltered.of_exists_of_isCofiltered_of_fullyFaithful [IsCofiltered D] [F.Full]
[F.Faithful] (h : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) : IsCofiltered C :=
{ IsCofilteredOrEmpty.of_exists_of_isCofiltered_of_fullyFaithful F h with
nonempty
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : IsCofiltered D
inst✝¹ : F.Full
inst✝ : F.Faithful
h : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)
this : Nonempty D
⊢ Nonempty C
|
obtain ⟨c, -⟩ := h (Classical.arbitrary D)
|
case intro
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : IsCofiltered D
inst✝¹ : F.Full
inst✝ : F.Faithful
h : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)
this : Nonempty D
c : C
⊢ Nonempty C
|
7e79f8bc2912c77e
|
FaithfulSMul.ker_algebraMap_eq_bot
|
Mathlib/RingTheory/Ideal/Maps.lean
|
theorem FaithfulSMul.ker_algebraMap_eq_bot (R A : Type*) [CommSemiring R] [Semiring A]
[Algebra R A] [FaithfulSMul R A] : RingHom.ker (algebraMap R A) = ⊥
|
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : Algebra R A
inst✝ : FaithfulSMul R A
⊢ RingHom.ker (algebraMap R A) = ⊥
|
ext
|
case h
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : Algebra R A
inst✝ : FaithfulSMul R A
x✝ : R
⊢ x✝ ∈ RingHom.ker (algebraMap R A) ↔ x✝ ∈ ⊥
|
64629293753ef70e
|
AddCircle.addWellApproximable_ae_empty_or_univ
|
Mathlib/NumberTheory/WellApproximable.lean
|
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
(∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x
|
case neg.h
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n => approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᶠ[ae μ] ∅ ∨ B ↑p =ᶠ[ae μ] univ
hC : ∀ (p : Nat.Primes), u p +ᵥ C ↑p = C ↑p
h : ∃ x, ¬A ↑x =ᶠ[ae μ] ∅ ∨ ¬B ↑x =ᶠ[ae μ] ∅
⊢ E =ᶠ[ae volume] univ
|
obtain ⟨p, hp⟩ := h
|
case neg.h.intro
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n => approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᶠ[ae μ] ∅ ∨ B ↑p =ᶠ[ae μ] univ
hC : ∀ (p : Nat.Primes), u p +ᵥ C ↑p = C ↑p
p : Nat.Primes
hp : ¬A ↑p =ᶠ[ae μ] ∅ ∨ ¬B ↑p =ᶠ[ae μ] ∅
⊢ E =ᶠ[ae volume] univ
|
aeca070d334a419e
|
MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
|
Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean
|
theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [TopologicalSpace β]
[OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β) :
(∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0
|
case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : FirstCountableTopology β
f : α → β
c : β
hc : ∀ b < c, μ {x | f x ≤ b} = 0
h : ¬∀ (b : β), c ≤ b
H : ¬¬IsLUB (Set.Iio c) c
⊢ μ {a | f a < c} = 0
|
push_neg at H h
|
case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : FirstCountableTopology β
f : α → β
c : β
hc : ∀ b < c, μ {x | f x ≤ b} = 0
H : IsLUB (Set.Iio c) c
h : ∃ b, b < c
⊢ μ {a | f a < c} = 0
|
094f21f52e958ba5
|
roth_3ap_theorem_nat
|
Mathlib/Combinatorics/Additive/Corner/Roth.lean
|
theorem roth_3ap_theorem_nat (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound (ε / 3) ≤ n)
(A : Finset ℕ) (hAn : A ⊆ range n) (hAε : ε * n ≤ #A) : ¬ ThreeAPFree (A : Set ℕ)
|
n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' (Nat.cast '' ↑A))
this✝¹ : ↑A = Fin.val '' (Nat.cast '' ↑A)
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x => ↑x) A)
⊢ False
|
refine roth_3ap_theorem (ε / 3) (by positivity) (by simp; omega) _ ?_ this
|
n : ℕ
ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound (ε / 3) ≤ n
A : Finset ℕ
hAn : ↑A ⊆ Set.Iio n
hAε : ε * ↑n ≤ ↑(#A)
hA : ThreeAPFree (Fin.val '' (Nat.cast '' ↑A))
this✝¹ : ↑A = Fin.val '' (Nat.cast '' ↑A)
this✝ : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val
this : ThreeAPFree ↑(image (fun x => ↑x) A)
⊢ ε / 3 * ↑(Fintype.card (Fin (2 * n + 1))) ≤ ↑(#(image (fun x => ↑x) A))
|
c707b5b13a459d8d
|
LieAlgebra.isEngelian_of_isNoetherian
|
Mathlib/Algebra/Lie/Engel.lean
|
theorem LieAlgebra.isEngelian_of_isNoetherian [IsNoetherian R L] : LieAlgebra.IsEngelian R L
|
R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
_i4 : LieModule R L M
L' : LieSubalgebra R (Module.End R M) := (toEnd R L M).range
h : ∀ (y : ↥L'), IsNilpotent ↑y
s : Set (LieSubalgebra R ↥L') := {K | IsEngelian R ↥K}
hs : s.Nonempty
this : ∀ K ∈ s, K ≠ ⊤ → ∃ K' ∈ s, K < K'
_i5 : IsNoetherian R ↥L'
⊢ ⊤ ∈ s
|
obtain ⟨K, hK₁, hK₂⟩ := (LieSubalgebra.wellFoundedGT_of_noetherian R L').wf.has_min s hs
|
case intro.intro
R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
_i4 : LieModule R L M
L' : LieSubalgebra R (Module.End R M) := (toEnd R L M).range
h : ∀ (y : ↥L'), IsNilpotent ↑y
s : Set (LieSubalgebra R ↥L') := {K | IsEngelian R ↥K}
hs : s.Nonempty
this : ∀ K ∈ s, K ≠ ⊤ → ∃ K' ∈ s, K < K'
_i5 : IsNoetherian R ↥L'
K : LieSubalgebra R ↥L'
hK₁ : K ∈ s
hK₂ : ∀ x ∈ s, ¬x > K
⊢ ⊤ ∈ s
|
12098dbf4ba9f91c
|
IntermediateField.induction_on_adjoin_finset
|
Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean
|
theorem induction_on_adjoin_finset (S : Finset E) (P : IntermediateField F E → Prop) (base : P ⊥)
(ih : ∀ (K : IntermediateField F E), ∀ x ∈ S, P K → P (K⟮x⟯.restrictScalars F)) :
P (adjoin F S)
|
case refine_2
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Finset E
P : IntermediateField F E → Prop
base : P ⊥
ih : ∀ (K : IntermediateField F E), ∀ x ∈ S, P K → P (restrictScalars F (↥K)⟮x⟯)
a✝ : E
s✝ : Finset E
ha : a✝ ∈ S
x✝¹ : s✝ ⊆ S
x✝ : a✝ ∉ s✝
h : P (adjoin F ↑s✝)
⊢ P (restrictScalars F (↥(adjoin F ↑s✝))⟮a✝⟯)
|
exact ih (adjoin F _) _ ha h
|
no goals
|
d2a5f16bfb45070f
|
Batteries.HashMap.Imp.mem_replaceF
|
Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean
|
theorem mem_replaceF {l : List (α × β)} {x : α × β} {p : α × β → Bool} {f : α × β → β} :
x ∈ (l.replaceF fun a => bif p a then some (k, f a) else none) → x.1 = k ∨ x ∈ l
|
case cons
α : Type u_1
β : Type u_2
k : α
x : α × β
p : α × β → Bool
f : α × β → β
a : α × β
l : List (α × β)
ih : x ∈ List.replaceF (fun a => bif p a then some (k, f a) else none) l → x.fst = k ∨ x ∈ l
z : Option (α × β)
⊢ (match p a with
| true => some (k, f a)
| false => none) =
z →
(x ∈
match z with
| none =>
a ::
List.replaceF
(fun a =>
match p a with
| true => some (k, f a)
| false => none)
l
| some a => a :: l) →
x.fst = k ∨ x = a ∨ x ∈ l
|
split <;> (intro h; subst h; simp)
|
case cons.h_1
α : Type u_1
β : Type u_2
k : α
x : α × β
p : α × β → Bool
f : α × β → β
a : α × β
l : List (α × β)
ih : x ∈ List.replaceF (fun a => bif p a then some (k, f a) else none) l → x.fst = k ∨ x ∈ l
c✝ : Bool
heq✝ : p a = true
⊢ x = (k, f a) ∨ x ∈ l → x.fst = k ∨ x = a ∨ x ∈ l
case cons.h_2
α : Type u_1
β : Type u_2
k : α
x : α × β
p : α × β → Bool
f : α × β → β
a : α × β
l : List (α × β)
ih : x ∈ List.replaceF (fun a => bif p a then some (k, f a) else none) l → x.fst = k ∨ x ∈ l
c✝ : Bool
heq✝ : p a = false
⊢ x = a ∨
x ∈
List.replaceF
(fun a =>
match p a with
| true => some (k, f a)
| false => none)
l →
x.fst = k ∨ x = a ∨ x ∈ l
|
9d1c6914f67596a9
|
Polynomial.isRoot_cyclotomic_prime_pow_mul_iff_of_charP
|
Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean
|
theorem isRoot_cyclotomic_prime_pow_mul_iff_of_charP {m k p : ℕ} {R : Type*} [CommRing R]
[IsDomain R] [hp : Fact (Nat.Prime p)] [hchar : CharP R p] {μ : R} [NeZero (m : R)] :
(Polynomial.cyclotomic (p ^ k * m) R).IsRoot μ ↔ IsPrimitiveRoot μ m
|
case inr.refine_2
m k p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
hp : Fact (Nat.Prime p)
hchar : CharP R p
μ : R
inst✝ : NeZero ↑m
hk : k > 0
h : IsPrimitiveRoot μ m
⊢ (cyclotomic (p ^ k * m) R).IsRoot μ
|
rw [← isRoot_cyclotomic_iff, IsRoot.def] at h
|
case inr.refine_2
m k p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
hp : Fact (Nat.Prime p)
hchar : CharP R p
μ : R
inst✝ : NeZero ↑m
hk : k > 0
h : eval μ (cyclotomic m R) = 0
⊢ (cyclotomic (p ^ k * m) R).IsRoot μ
|
27af3270e81a0c49
|
SzemerediRegularity.a_add_one_le_four_pow_parts_card
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean
|
theorem a_add_one_le_four_pow_parts_card : a + 1 ≤ 4 ^ #P.parts
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
h : 1 ≤ 4 ^ #P.parts
⊢ Fintype.card α / #P.parts ≤ Fintype.card α / #P.parts / 4 ^ #P.parts * 4 ^ #P.parts + 4 ^ #P.parts - 1
|
exact Nat.le_sub_one_of_lt (Nat.lt_div_mul_add h)
|
no goals
|
f1ed2fd6d3c75a66
|
List.min?_replicate
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean
|
theorem min?_replicate [Min α] {n : Nat} {a : α} (w : min a a = a) :
(replicate n a).min? = if n = 0 then none else some a
|
α : Type u_1
inst✝ : Min α
n : Nat
a : α
w : min a a = a
⊢ (replicate n a).min? = if n = 0 then none else some a
|
induction n with
| zero => rfl
| succ n ih => cases n <;> simp_all [replicate_succ, min?_cons']
|
no goals
|
d350d5e15666a16d
|
Complex.IsExpCmpFilter.isLittleO_cpow_exp
|
Mathlib/Analysis/SpecialFunctions/CompareExp.lean
|
theorem isLittleO_cpow_exp (hl : IsExpCmpFilter l) (a : ℂ) {b : ℝ} (hb : 0 < b) :
(fun z => z ^ a) =o[l] fun z => exp (b * z) :=
calc
(fun z => z ^ a) =Θ[l] fun z => Real.exp (re a * Real.log ‖z‖) :=
hl.isTheta_cpow_exp_re_mul_log a
_ =o[l] fun z => exp (b * z) :=
IsLittleO.of_norm_right <| by
simp only [norm_exp, re_ofReal_mul, Real.isLittleO_exp_comp_exp_comp]
refine (IsEquivalent.refl.sub_isLittleO ?_).symm.tendsto_atTop
(hl.tendsto_re.const_mul_atTop hb)
exact (hl.isLittleO_log_norm_re.const_mul_left _).const_mul_right hb.ne'
|
l : Filter ℂ
hl : IsExpCmpFilter l
a : ℂ
b : ℝ
hb : 0 < b
⊢ (fun x => a.re * Real.log ‖x‖) =o[l] fun x => b * x.re
|
exact (hl.isLittleO_log_norm_re.const_mul_left _).const_mul_right hb.ne'
|
no goals
|
08b0face6825c488
|
CompositionAsSet.toComposition_boundaries
|
Mathlib/Combinatorics/Enumerative/Composition.lean
|
theorem CompositionAsSet.toComposition_boundaries (c : CompositionAsSet n) :
c.toComposition.boundaries = c.boundaries
|
n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
i : ℕ
i_lt : i < c.boundaries.card
hi : (take i c.blocks).sum = ↑j
⊢ ↑i ∈ Finset.univ
|
simp
|
no goals
|
8036986d8b3b60a7
|
AlgebraicGeometry.pointsPi_surjective
|
Mathlib/AlgebraicGeometry/PointsPi.lean
|
lemma pointsPi_surjective [CompactSpace X] [∀ i, IsLocalRing (R i)] :
Function.Surjective (pointsPi R X)
|
ι : Type u
R : ι → CommRingCat
X : Scheme
inst✝¹ : CompactSpace ↑↑X.toPresheafedSpace
inst✝ : ∀ (i : ι), IsLocalRing ↑(R i)
f : (i : ι) → Spec (R i) ⟶ X
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
this✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
this : ∀ (i : ι), ∃ j, Set.range ⇑(ConcreteCategory.hom (f i).base) ⊆ ↑(Scheme.Hom.opensRange (𝒰.map j))
⊢ ∃ a, pointsPi R X a = f
|
choose j hj using this
|
ι : Type u
R : ι → CommRingCat
X : Scheme
inst✝¹ : CompactSpace ↑↑X.toPresheafedSpace
inst✝ : ∀ (i : ι), IsLocalRing ↑(R i)
f : (i : ι) → Spec (R i) ⟶ X
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
this : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
j : ι → 𝒰.J
hj : ∀ (i : ι), Set.range ⇑(ConcreteCategory.hom (f i).base) ⊆ ↑(Scheme.Hom.opensRange (𝒰.map (j i)))
⊢ ∃ a, pointsPi R X a = f
|
35d67c613e419cbd
|
ExteriorAlgebra.ι_ne_one
|
Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean
|
theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1
|
R : Type u1
inst✝³ : CommRing R
M : Type u2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Nontrivial R
x : M
⊢ ¬(x = 0 ∧ 1 = 0)
|
exact one_ne_zero ∘ And.right
|
no goals
|
4e5df1de1d0d0c4e
|
IsLocalization.Away.sec_spec
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
lemma sec_spec (s : S) : s * (algebraMap R S) (x ^ (IsLocalization.Away.sec x s).2) =
algebraMap R S (IsLocalization.Away.sec x s).1
|
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
x : R
inst✝ : Away x S
s : S
⊢ s * (algebraMap R S) (x ^ (sec x s).2) = (algebraMap R S) (sec x s).1
|
simp only [IsLocalization.Away.sec, ← IsLocalization.sec_spec]
|
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
x : R
inst✝ : Away x S
s : S
⊢ s * (algebraMap R S) (x ^ Exists.choose ⋯) = s * (algebraMap R S) ↑(IsLocalization.sec (Submonoid.powers x) s).2
|
ca7b7b572824d46a
|
Ordinal.bmex_lt_ord_succ_card
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem bmex_lt_ord_succ_card {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{u}) :
bmex.{_, u} o f < (succ o.card).ord
|
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
⊢ o.bmex f < (succ #o.toType).ord
|
exact mex_lt_ord_succ_mk (familyOfBFamily o f)
|
no goals
|
af4b42b286bbd3a5
|
HasDerivAt.rpow_const
|
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
theorem HasDerivAt.rpow_const (hf : HasDerivAt f f' x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
HasDerivAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) x
|
f : ℝ → ℝ
f' x p : ℝ
hf : HasDerivAt f f' x
hx : f x ≠ 0 ∨ 1 ≤ p
⊢ HasDerivAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) x
|
rw [← hasDerivWithinAt_univ] at *
|
f : ℝ → ℝ
f' x p : ℝ
hf : HasDerivWithinAt f f' Set.univ x
hx : f x ≠ 0 ∨ 1 ≤ p
⊢ HasDerivWithinAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) Set.univ x
|
1a2dcf95dedfff6e
|
Equiv.Perm.support_noncommProd
|
Mathlib/GroupTheory/Perm/Support.lean
|
theorem support_noncommProd {ι : Type*} {k : ι → Perm α} {s : Finset ι}
(hs : Set.Pairwise s fun i j ↦ Disjoint (k i) (k j)) :
(s.noncommProd k (hs.imp (fun _ _ ↦ Perm.Disjoint.commute))).support =
s.biUnion fun i ↦ (k i).support
|
case insert.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ι : Type u_2
k : ι → Perm α
i : ι
s : Finset ι
hi : i ∉ s
hrec :
∀ (hs : (↑s).Pairwise fun i j => (k i).Disjoint (k j)), (s.noncommProd k ⋯).support = s.biUnion fun i => (k i).support
hs : (↑(insert i s)).Pairwise fun i j => (k i).Disjoint (k j)
hs' : (↑s).Pairwise fun i j => (k i).Disjoint (k j)
j : ι
hj : j ∈ s
⊢ (k i).Disjoint (k j)
|
apply hs _ _ (ne_of_mem_of_not_mem hj hi).symm <;>
simp only [Finset.coe_insert, Set.mem_insert_iff, Finset.mem_coe, hj, or_true, true_or]
|
no goals
|
219e81871813f296
|
hasDerivAt_update
|
Mathlib/Analysis/Calculus/Deriv/Pi.lean
|
theorem hasDerivAt_update (x : ι → 𝕜) (i : ι) (y : 𝕜) :
HasDerivAt (Function.update x i) (Pi.single i (1 : 𝕜)) y
|
case h.e'_9.h.e.h.h
𝕜 : Type u_1
ι : Type u_2
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : NontriviallyNormedField 𝕜
x : ι → 𝕜
i : ι
y z : 𝕜
j : ι
⊢ Pi.single i z j = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 𝕜))) z j
|
rw [Pi.single, Function.update_apply]
|
case h.e'_9.h.e.h.h
𝕜 : Type u_1
ι : Type u_2
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : NontriviallyNormedField 𝕜
x : ι → 𝕜
i : ι
y z : 𝕜
j : ι
⊢ (if j = i then z else 0 j) = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 𝕜))) z j
|
ceec33df69319b6c
|
CoxeterSystem.prod_leftInvSeq
|
Mathlib/GroupTheory/Coxeter/Inversion.lean
|
theorem prod_leftInvSeq (ω : List B) : prod (lis ω) = (π ω)⁻¹
|
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
⊢ (List.map (fun x => x⁻¹) (cs.rightInvSeq ω.reverse)).prod = cs.wordProd ω
|
have : List.map (fun x ↦ x⁻¹) (ris ω.reverse) = ris ω.reverse := calc
List.map (fun x ↦ x⁻¹) (ris ω.reverse)
_ = List.map id (ris ω.reverse) := by
apply List.map_congr_left
intro t ht
exact (cs.isReflection_of_mem_rightInvSeq _ ht).inv
_ = ris ω.reverse := map_id _
|
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
this : List.map (fun x => x⁻¹) (cs.rightInvSeq ω.reverse) = cs.rightInvSeq ω.reverse
⊢ (List.map (fun x => x⁻¹) (cs.rightInvSeq ω.reverse)).prod = cs.wordProd ω
|
761854a10de60b7d
|
Real.doublingGamma_log_convex_Ioi
|
Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
|
theorem doublingGamma_log_convex_Ioi : ConvexOn ℝ (Ioi (0 : ℝ)) (log ∘ doublingGamma)
|
case h.e'_10.h
x : ℝ
⊢ log (Gamma (x / 2 + 1 / 2)) =
((log ∘ Gamma) ∘ ⇑((DistribMulAction.toLinearMap ℝ ℝ (1 / 2)).toAffineMap + AffineMap.const ℝ ℝ (1 / 2))) x
|
change log (Gamma (x / 2 + 1 / 2)) = log (Gamma ((1 / 2 : ℝ) • x + 1 / 2))
|
case h.e'_10.h
x : ℝ
⊢ log (Gamma (x / 2 + 1 / 2)) = log (Gamma ((1 / 2) • x + 1 / 2))
|
ddad181789e3d6af
|
integral_log_from_zero
|
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
/-- Helper lemma for `integral_log`: case where `a = 0`. -/
lemma integral_log_from_zero {b : ℝ} : ∫ s in (0)..b, log s = b * log b - b
|
case inr.inr
b : ℝ
h : 0 < b
⊢ ∫ (s : ℝ) in 0 ..b, log s = b * log b - b
|
exact integral_log_from_zero_of_pos h
|
no goals
|
9782902149e09d34
|
OreLocalization.add_smul
|
Mathlib/RingTheory/OreLocalization/Ring.lean
|
theorem add_smul (y z : R[S⁻¹]) (x : X[S⁻¹]) :
(y + z) • x = y • x + z • x
|
case c.c.c.mk.mk.intro.mk.mk.intro
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra * ↑s₃ ∈ S
⊢ ((sa • r₂ + ra • r₃) /ₒ (sa * s₂)) • (r₁ /ₒ s₁) =
rb • r₁ /ₒ (sb * (sa * s₂)) + ((↑sb * ra) • r₃ /ₒ ⟨↑sb * ra * ↑s₃, hs₃rasb⟩) • (r₁ /ₒ s₁)
|
have ha' : ↑((sb * sa) * s₂) = sb * ra * s₃ := by simp [ha, mul_assoc]
|
case c.c.c.mk.mk.intro.mk.mk.intro
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra * ↑s₃ ∈ S
ha' : ↑(sb * sa * s₂) = ↑sb * ra * ↑s₃
⊢ ((sa • r₂ + ra • r₃) /ₒ (sa * s₂)) • (r₁ /ₒ s₁) =
rb • r₁ /ₒ (sb * (sa * s₂)) + ((↑sb * ra) • r₃ /ₒ ⟨↑sb * ra * ↑s₃, hs₃rasb⟩) • (r₁ /ₒ s₁)
|
79dc4eb2119c8abc
|
ZMod.ringHom_eq_of_ker_eq
|
Mathlib/RingTheory/ZMod.lean
|
theorem ZMod.ringHom_eq_of_ker_eq {n : ℕ} {R : Type*} [CommRing R] (f g : R →+* ZMod n)
(h : RingHom.ker f = RingHom.ker g) : f = g
|
n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R →+* ZMod n
h : RingHom.ker f = RingHom.ker g
this : ((f.liftOfRightInverse cast ⋯) ⟨g, ⋯⟩).comp f = ↑⟨g, ⋯⟩
⊢ f = g
|
rw [Subtype.coe_mk] at this
|
n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R →+* ZMod n
h : RingHom.ker f = RingHom.ker g
this : ((f.liftOfRightInverse cast ⋯) ⟨g, ⋯⟩).comp f = g
⊢ f = g
|
f630274845f4ad52
|
LinearMap.BilinForm.apply_smul_sub_smul_sub_eq
|
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
lemma apply_smul_sub_smul_sub_eq [CommRing R] [AddCommGroup M] [Module R M]
(B : LinearMap.BilinForm R M) (x y : M) :
B ((B x y) • x - (B x x) • y) ((B x y) • x - (B x x) • y) =
(B x x) * ((B x x) * (B y y) - (B x y) * (B y x))
|
R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
x y : M
⊢ (B ((B x) y • x - (B x) x • y)) ((B x) y • x - (B x) x • y) = (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)
|
simp only [map_sub, map_smul, sub_apply, smul_apply, smul_eq_mul, mul_sub,
mul_comm (B x y) (B x x), mul_left_comm (B x y) (B x x)]
|
R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
x y : M
⊢ (B x) x * ((B x) y * (B x) y) - (B x) x * ((B x) y * (B y) x) -
((B x) x * ((B x) y * (B x) y) - (B x) x * ((B x) x * (B y) y)) =
(B x) x * ((B x) x * (B y) y) - (B x) x * ((B x) y * (B y) x)
|
eecf1a97014a5559
|
Vector.mk_lex_mk
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lex.lean
|
theorem mk_lex_mk [BEq α] (lt : α → α → Bool) {l₁ l₂ : Array α} {n₁ : l₁.size = n} {n₂ : l₂.size = n} :
(Vector.mk l₁ n₁).lex (Vector.mk l₂ n₂) lt = l₁.lex l₂ lt
|
α : Type u_1
n : Nat
inst✝ : BEq α
lt : α → α → Bool
l₁ l₂ : Array α
n₁ : l₁.size = n
n₂ : l₂.size = n
⊢ { toArray := l₁, size_toArray := n₁ }.lex { toArray := l₂, size_toArray := n₂ } lt = l₁.lex l₂ lt
|
simp [Vector.lex, Array.lex, n₁, n₂]
|
α : Type u_1
n : Nat
inst✝ : BEq α
lt : α → α → Bool
l₁ l₂ : Array α
n₁ : l₁.size = n
n₂ : l₂.size = n
⊢ (match
(forIn' (List.range' 0 n 1) ⟨none, PUnit.unit⟩ fun a m b =>
if lt l₁[a] l₂[a] = true then ForInStep.done ⟨some true, PUnit.unit⟩
else
if (l₁[a] != l₂[a]) = true then ForInStep.done ⟨some false, PUnit.unit⟩
else ForInStep.yield ⟨none, PUnit.unit⟩).fst with
| none => false
| some a => a).run =
(match
(forIn' (List.range' 0 n 1) ⟨none, PUnit.unit⟩ fun a m b =>
if lt l₁[a] l₂[a] = true then ForInStep.done ⟨some true, PUnit.unit⟩
else
if (l₁[a] != l₂[a]) = true then ForInStep.done ⟨some false, PUnit.unit⟩
else ForInStep.yield ⟨none, PUnit.unit⟩).fst with
| none => false
| some a => a).run
|
4bf189d34ce1841c
|
Matrix.adjugate_conjTranspose
|
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
theorem adjugate_conjTranspose [StarRing α] (A : Matrix n n α) : A.adjugateᴴ = adjugate Aᴴ
|
n : Type v
α : Type w
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : CommRing α
inst✝ : StarRing α
A : Matrix n n α
⊢ A.adjugateᴴ = Aᴴ.adjugate
|
dsimp only [conjTranspose]
|
n : Type v
α : Type w
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : CommRing α
inst✝ : StarRing α
A : Matrix n n α
⊢ A.adjugateᵀ.map star = (Aᵀ.map star).adjugate
|
e4c2626f38a168f5
|
GenContFract.IntFractPair.stream_nth_fr_num_le_fr_num_sub_n_rat
|
Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean
|
theorem stream_nth_fr_num_le_fr_num_sub_n_rat :
∀ {ifp_n : IntFractPair ℚ},
IntFractPair.stream q n = some ifp_n → ifp_n.fr.num ≤ (IntFractPair.of q).fr.num - n
|
q : ℚ
n : ℕ
ifp_zero : IntFractPair ℚ
stream_zero_eq : IntFractPair.stream q 0 = some ifp_zero
⊢ IntFractPair.of q = ifp_zero
|
injection stream_zero_eq
|
no goals
|
9def6df4c64f53d0
|
Submodule.basis_of_pid_aux
|
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0) ∧
∀ (n') (bN' : Basis (Fin n') R N'),
∃ bN : Basis (Fin (n' + 1)) R N,
∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
∀ as : Fin n' → R,
(∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)) →
∃ as' : Fin (n' + 1) → R,
∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O)
|
case neg.intro.intro.intro.refine_1.refine_1
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c✝ : ι → R
hc✝ : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c✝ i
val✝ : Fintype ι
y' : O := ∑ i : ι, c✝ i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c✝ i • b'M i
a_smul_y' : a • y' = y
ϕy'_eq : ϕ ⟨y', y'M⟩ = 1
ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0
M' : Submodule R O := map M.subtype (LinearMap.ker ϕ)
N' : Submodule R O := map N.subtype (LinearMap.ker (ϕ ∘ₗ inclusion N_le_M))
M'_le_M : M' ≤ M
N'_le_M' : N' ≤ M'
N'_le_N : N' ≤ N
y'_ortho_M' : ∀ (c : R), ∀ z ∈ M', c • y' + z = 0 → c = 0
ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0
n' : ℕ
bN' : Basis (Fin n') R ↥N'
c : R
z : O
zN' : z ∈ N'
hc : c • y + z = 0
⊢ c = 0
|
refine ay'_ortho_N' c z zN' ?_
|
case neg.intro.intro.intro.refine_1.refine_1
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c✝ : ι → R
hc✝ : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c✝ i
val✝ : Fintype ι
y' : O := ∑ i : ι, c✝ i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c✝ i • b'M i
a_smul_y' : a • y' = y
ϕy'_eq : ϕ ⟨y', y'M⟩ = 1
ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0
M' : Submodule R O := map M.subtype (LinearMap.ker ϕ)
N' : Submodule R O := map N.subtype (LinearMap.ker (ϕ ∘ₗ inclusion N_le_M))
M'_le_M : M' ≤ M
N'_le_M' : N' ≤ M'
N'_le_N : N' ≤ N
y'_ortho_M' : ∀ (c : R), ∀ z ∈ M', c • y' + z = 0 → c = 0
ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0
n' : ℕ
bN' : Basis (Fin n') R ↥N'
c : R
z : O
zN' : z ∈ N'
hc : c • y + z = 0
⊢ c • a • y' + z = 0
|
48ee91e21f4317c3
|
Pell.matiyasevic
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
theorem matiyasevic {a k x y} :
(∃ a1 : 1 < a, xn a1 k = x ∧ yn a1 k = y) ↔
1 < a ∧ k ≤ y ∧ (x = 1 ∧ y = 0 ∨
∃ u v s t b : ℕ,
x * x - (a * a - 1) * y * y = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧
s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * y] ∧
b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]) :=
⟨fun ⟨a1, hx, hy⟩ => by
rw [← hx, ← hy]
refine ⟨a1,
(Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact ⟨le_rfl, Or.inl ⟨rfl, rfl⟩⟩)
fun kpos => ?_⟩
exact
let x := xn a1 k
let y := yn a1 k
let m := 2 * (k * y)
let u := xn a1 m
let v := yn a1 m
have ky : k ≤ y := yn_ge_n a1 k
have yv : y * y ∣ v := (ysq_dvd_yy a1 k).trans <| (y_dvd_iff _ _ _).2 <| dvd_mul_left _ _
have uco : Nat.Coprime u (4 * y) :=
have : 2 ∣ v :=
modEq_zero_iff_dvd.1 <| (yn_modEq_two _ _).trans (dvd_mul_right _ _).modEq_zero_nat
have : Nat.Coprime u 2 := (xy_coprime a1 m).coprime_dvd_right this
(this.mul_right this).mul_right <|
(xy_coprime _ _).coprime_dvd_right (dvd_of_mul_left_dvd yv)
let ⟨b, ba, bm1⟩ := chineseRemainder uco a 1
have m1 : 1 < m :=
have : 0 < k * y := mul_pos kpos (strictMono_y a1 kpos)
Nat.mul_le_mul_left 2 this
have vp : 0 < v := strictMono_y a1 (lt_trans zero_lt_one m1)
have b1 : 1 < b :=
have : xn a1 1 < u := strictMono_x a1 m1
have : a < u
|
a k x✝¹ y✝ : ℕ
x✝ : ∃ (a1 : 1 < a), xn a1 k = x✝¹ ∧ yn a1 k = y✝
a1 : 1 < a
hx : xn a1 k = x✝¹
hy : yn a1 k = y✝
kpos : k > 0
x : ℕ := xn a1 k
y : ℕ := yn a1 k
m : ℕ := 2 * (k * y)
u : ℕ := xn a1 m
v : ℕ := yn a1 m
ky : k ≤ y
yv : y * y ∣ v
uco : u.Coprime (4 * y)
b : ℕ
ba : b % u = a
bm1 : b ≡ 1 [MOD 4 * y]
m1 : 1 < m
vp : 0 < v
this✝ : xn a1 1 < u
this : a < u
⊢ b % u ≤ b
|
apply Nat.mod_le
|
no goals
|
86da2519f71702c3
|
Fin.eq_none_of_findSome?_eq_none
|
Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean
|
theorem eq_none_of_findSome?_eq_none {f : Fin n → Option α} (h : findSome? f = none) (i) :
f i = none
|
α : Type u_1
n : Nat
ih : ∀ {f : Fin n → Option α}, findSome? f = none → ∀ (i : Fin n), f i = none
f : Fin (n + 1) → Option α
h : (f 0 <|> findSome? fun i => f i.succ) = none
i : Fin (n + 1)
heq : f 0 = none
⊢ f i = none
|
rw [heq, Option.none_orElse] at h
|
α : Type u_1
n : Nat
ih : ∀ {f : Fin n → Option α}, findSome? f = none → ∀ (i : Fin n), f i = none
f : Fin (n + 1) → Option α
h : (findSome? fun i => f i.succ) = none
i : Fin (n + 1)
heq : f 0 = none
⊢ f i = none
|
a08d3d7f0b0c00f6
|
SetTheory.PGame.inv_one
|
Mathlib/SetTheory/Game/Basic.lean
|
/-- `1⁻¹` has exactly the same moves as `1`. -/
lemma inv_one : 1⁻¹ ≡ 1
|
⊢ inv' 1 ≡ 1
|
exact inv'_one
|
no goals
|
60a3d871b690ff8b
|
LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional
|
Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
|
theorem hasEigenvalue_iSup_of_finiteDimensional [Nontrivial E] (hT : T.IsSymmetric) :
HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ)
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : Nontrivial E
hT : T.IsSymmetric
this✝¹ : ProperSpace E
T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint
x : E
hx : x ≠ 0
H₁ : IsCompact (sphere 0 ‖x‖)
H₂ : (sphere 0 ‖x‖).Nonempty
x₀ : E
hx₀' : x₀ ∈ sphere 0 ‖x‖
hTx₀ : IsMaxOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀
hx₀ : ‖x₀‖ = ‖x‖
this✝ : IsMaxOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
this : ‖x₀‖ ≠ 0
⊢ x₀ ≠ 0
|
simpa [← norm_eq_zero, Ne]
|
no goals
|
313e798d747e5e61
|
MeasureTheory.norm_stoppedValue_leastGE_le
|
Mathlib/Probability/Martingale/BorelCantelli.lean
|
theorem norm_stoppedValue_leastGE_le (hr : 0 ≤ r) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
∀ᵐ ω ∂μ, stoppedValue f (leastGE f r i) ω ≤ r + R
|
case neg.intro
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
r : ℝ
R : ℝ≥0
hr : 0 ≤ r
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
i : ℕ
ω : Ω
hbddω : ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
heq : ¬leastGE f r i ω = 0
k : ℕ
hk : leastGE f r i ω = k.succ
⊢ f (leastGE f r i ω) ω ≤ r + ↑R
|
rw [hk, add_comm, ← sub_le_iff_le_add]
|
case neg.intro
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
r : ℝ
R : ℝ≥0
hr : 0 ≤ r
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
i : ℕ
ω : Ω
hbddω : ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
heq : ¬leastGE f r i ω = 0
k : ℕ
hk : leastGE f r i ω = k.succ
⊢ f k.succ ω - r ≤ ↑R
|
464e9208e86698f5
|
Real.sqrt_eq_iff_mul_self_eq_of_pos
|
Mathlib/Data/Real/Sqrt.lean
|
theorem sqrt_eq_iff_mul_self_eq_of_pos (h : 0 < y) : √x = y ↔ y * y = x
|
x y : ℝ
h : 0 < y
⊢ √x = y ↔ y * y = x
|
simp [sqrt_eq_cases, h.ne', h.le]
|
no goals
|
8df00457bd88bba7
|
List.Vector.zipWith_get
|
Mathlib/Data/Vector/Zip.lean
|
theorem zipWith_get (x : Vector α n) (y : Vector β n) (i) :
(Vector.zipWith f x y).get i = f (x.get i) (y.get i)
|
α : Type u_1
β : Type u_2
γ : Type u_3
n : ℕ
f : α → β → γ
x : Vector α n
y : Vector β n
i : Fin n
⊢ (zipWith f x y).get i = f (x.get i) (y.get i)
|
dsimp only [Vector.zipWith, Vector.get]
|
α : Type u_1
β : Type u_2
γ : Type u_3
n : ℕ
f : α → β → γ
x : Vector α n
y : Vector β n
i : Fin n
⊢ (List.zipWith f ↑x ↑y).get (Fin.cast ⋯ i) = f ((↑x).get (Fin.cast ⋯ i)) ((↑y).get (Fin.cast ⋯ i))
|
6572de3857333d1c
|
Matrix.fromRows_row0_isTotallyUnimodular_iff
|
Mathlib/LinearAlgebra/Matrix/Determinant/TotallyUnimodular.lean
|
lemma fromRows_row0_isTotallyUnimodular_iff (A : Matrix m n R) :
(fromRows A (row m' 0)).IsTotallyUnimodular ↔ A.IsTotallyUnimodular
|
case h
m : Type u_1
m' : Type u_2
n : Type u_3
R : Type u_5
inst✝ : CommRing R
A : Matrix m n R
x✝¹ : Nonempty n
x✝ : m'
inhabited_h : Inhabited n
x : n
⊢ row m' 0 x✝ x = Pi.single default (↑0) x
|
simp [Pi.single_apply]
|
no goals
|
3fd2b268cdb07152
|
Finset.weightedVSubOfPoint_eq_of_weights_eq
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
(hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂
|
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
⊢ (s.weightedVSubOfPoint p (p j)) w₁ = (s.weightedVSubOfPoint p (p j)) w₂
|
simp only [Finset.weightedVSubOfPoint_apply]
|
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
⊢ ∑ i ∈ s, w₁ i • (p i -ᵥ p j) = ∑ i ∈ s, w₂ i • (p i -ᵥ p j)
|
8a064655adca0142
|
MulAction.IsBlock.of_subset
|
Mathlib/GroupTheory/GroupAction/Blocks.lean
|
theorem of_subset (a : X) (hfB : B.Finite) :
IsBlock G (⋂ (k : G) (_ : a ∈ k • B), k • B)
|
case inr
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
⊢ IsBlock G (⋂ k, ⋂ (_ : a ∈ k • B), k • B)
|
have hB'₀ : ∀ (k : G) (_ : a ∈ k • B), B' ≤ k • B := by
intro k hk
exact Set.biInter_subset_of_mem hk
|
case inr
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
a : X
hfB : B.Finite
B' : Set X := ⋂ k, ⋂ (_ : a ∈ k • B), k • B
hfB_ne : B.Nonempty
hB'₀ : ∀ (k : G), a ∈ k • B → B' ≤ k • B
⊢ IsBlock G (⋂ k, ⋂ (_ : a ∈ k • B), k • B)
|
9d1f232c7990ecc0
|
CategoryTheory.ShortComplex.epi_τ₂_of_exact_of_epi
|
Mathlib/Algebra/Homology/ShortComplex/Exact.lean
|
lemma epi_τ₂_of_exact_of_epi {S₁ S₂ : ShortComplex C} (φ : S₁ ⟶ S₂)
(h₂ : S₂.Exact) [Epi S₁.g] [Epi S₂.g] [Epi φ.τ₁] [Epi φ.τ₃] : Epi φ.τ₂
|
C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂ : S₂.Exact
inst✝³ : Epi S₁.g
inst✝² : Epi S₂.g
inst✝¹ : Epi φ.τ₁
inst✝ : Epi φ.τ₃
this✝³ : Mono S₁.op.f
this✝² : Mono S₂.op.f
this✝¹ : Mono (opMap φ).τ₁
this✝ : Mono (opMap φ).τ₃
this : Mono (opMap φ).τ₂
⊢ Epi φ.τ₂
|
exact unop_epi_of_mono (opMap φ).τ₂
|
no goals
|
73523644f772ac25
|
FormalMultilinearSeries.changeOrigin_eval
|
Mathlib/Analysis/Analytic/ChangeOrigin.lean
|
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y)
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
radius_pos : 0 < p.radius
x_mem_ball : x ∈ EMetric.ball 0 p.radius
⊢ ‖y‖ₑ < p.radius - ↑‖x‖₊
|
rwa [lt_tsub_iff_right, add_comm]
|
no goals
|
ccc324cf6a7a6ae5
|
Equiv.Perm.sign_surjective
|
Mathlib/GroupTheory/Perm/Sign.lean
|
theorem sign_surjective [Nontrivial α] : Function.Surjective (sign : Perm α → ℤˣ) := fun a =>
(Int.units_eq_one_or a).elim (fun h => ⟨1, by simp [h]⟩) fun h =>
let ⟨x, y, hxy⟩ := exists_pair_ne α
⟨swap x y, by rw [sign_swap hxy, h]⟩
|
α : Type u
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : Nontrivial α
a : ℤˣ
h : a = -1
x y : α
hxy : x ≠ y
⊢ sign (swap x y) = a
|
rw [sign_swap hxy, h]
|
no goals
|
06d44a68218be9ca
|
FreeGroup.Red.cons_nil_iff_singleton
|
Mathlib/GroupTheory/FreeGroup/Basic.lean
|
theorem cons_nil_iff_singleton {x b} : Red ((x, b) :: L) [] ↔ Red L [(x, not b)] :=
Iff.intro
(fun h => by
have h₁ : Red ((x, not b) :: (x, b) :: L) [(x, not b)] := cons_cons h
have h₂ : Red ((x, not b) :: (x, b) :: L) L := ReflTransGen.single Step.cons_not_rev
let ⟨L', h₁, h₂⟩ := church_rosser h₁ h₂
rw [singleton_iff] at h₁
subst L'
assumption)
fun h => (cons_cons h).tail Step.cons_not
|
α : Type u
L : List (α × Bool)
x : α
b : Bool
h : Red ((x, b) :: L) []
⊢ Red L [(x, !b)]
|
have h₁ : Red ((x, not b) :: (x, b) :: L) [(x, not b)] := cons_cons h
|
α : Type u
L : List (α × Bool)
x : α
b : Bool
h : Red ((x, b) :: L) []
h₁ : Red ((x, !b) :: (x, b) :: L) [(x, !b)]
⊢ Red L [(x, !b)]
|
76d3bdec0b2e0f0e
|
MeasurableSet.exists_isOpen_symmDiff_lt
|
Mathlib/MeasureTheory/Measure/Regular.lean
|
theorem _root_.MeasurableSet.exists_isOpen_symmDiff_lt [InnerRegularCompactLTTop μ]
[IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
{s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, IsOpen U ∧ μ U < ∞ ∧ μ (U ∆ s) < ε
|
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
this : ε / 2 ≠ 0
K : Set α
hKs : K ⊆ s
hKco : IsCompact K
hKcl : IsClosed K
hμK : μ (s \ K) < ε / 2
U : Set α
hKU : K ⊆ U
hUo : IsOpen U
hμU : μ U < μ K + ε / 2
⊢ μ K ≤ μ s
|
gcongr
|
no goals
|
498e42d44efdd46a
|
Zlattice.FG
|
Mathlib/Algebra/Module/ZLattice/Basic.lean
|
theorem Zlattice.FG [hs : IsZLattice K L] : L.FG
|
case intro.intro.intro.refine_1
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
⊢ (Subtype.val ∘ ⇑(quotientEquiv b) '' ↑(Submodule.map (span ℤ (Set.range ⇑b)).mkQ L)).Finite
|
refine Set.Finite.subset this ?_
|
case intro.intro.intro.refine_1
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
⊢ Subtype.val ∘ ⇑(quotientEquiv b) '' ↑(Submodule.map (span ℤ (Set.range ⇑b)).mkQ L) ⊆ fundamentalDomain b ∩ ↑L
|
68cb8b310e25c5ae
|
ContinuousMap.isUnit_iff_forall_isUnit
|
Mathlib/Topology/ContinuousMap/Units.lean
|
theorem isUnit_iff_forall_isUnit (f : C(X, R)) : IsUnit f ↔ ∀ x, IsUnit (f x) :=
Iff.intro (fun h => fun x => ⟨unitsLift.symm h.unit x, rfl⟩) fun h =>
⟨ContinuousMap.unitsLift (unitsOfForallIsUnit h), by ext; rfl⟩
|
case h
X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
a✝ : X
⊢ ↑(unitsLift (unitsOfForallIsUnit h)) a✝ = f a✝
|
rfl
|
no goals
|
31684b75ee155730
|
RingOfIntegers.isUnit_norm
|
Mathlib/NumberTheory/NumberField/Norm.lean
|
theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x
|
K : Type u_2
inst✝⁵ : Field K
F : Type u_3
inst✝⁴ : Field F
inst✝³ : Algebra K F
inst✝² : Algebra.IsSeparable K F
inst✝¹ : FiniteDimensional K F
inst✝ : CharZero K
x : 𝓞 F
this : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
⊢ IsUnit ((norm K) x) ↔ IsUnit x
|
let L := normalClosure K F (AlgebraicClosure F)
|
K : Type u_2
inst✝⁵ : Field K
F : Type u_3
inst✝⁴ : Field F
inst✝³ : Algebra K F
inst✝² : Algebra.IsSeparable K F
inst✝¹ : FiniteDimensional K F
inst✝ : CharZero K
x : 𝓞 F
this : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
L : IntermediateField K (AlgebraicClosure F) := normalClosure K F (AlgebraicClosure F)
⊢ IsUnit ((norm K) x) ↔ IsUnit x
|
81c4016c5a7e9693
|
List.nodup_permutations
|
Mathlib/Data/List/Permutation.lean
|
theorem nodup_permutations (s : List α) (hs : Nodup s) : Nodup s.permutations
|
α : Type u_1
s : List α
x : α
l : List α
h : ∀ a' ∈ l, x ≠ a'
h' : Pairwise (fun x1 x2 => x1 ≠ x2) l
IH : l.permutations'.Nodup
as : List α
ha : as ~ l
bs : List α
hb : bs ~ l
H : as ≠ bs
a : List α
ha' : a ∈ permutations'Aux x as
hb' : a ∈ permutations'Aux x bs
n : ℕ
hn✝ : n < (permutations'Aux x as).length
hn' : insertIdx n x as = a
m : ℕ
hm✝ : m < (permutations'Aux x bs).length
hm' : insertIdx m x bs = a
hl : as.length = bs.length
hn : n ≤ as.length
hm : m ≤ bs.length
hx : (insertIdx n x as)[m] = x
hx' : (insertIdx m x bs)[n] = x
ht : m < n
this : x ∈ as
⊢ False
|
exact h x (ha.subset this) rfl
|
no goals
|
25ee05be5844251a
|
Set.chainHeight_insert_of_forall_gt
|
Mathlib/Order/Height.lean
|
theorem chainHeight_insert_of_forall_gt (a : α) (hx : ∀ b ∈ s, a < b) :
(insert a s).chainHeight = s.chainHeight + 1
|
case a.cons
α : Type u_1
s : Set α
inst✝ : Preorder α
a : α
hx : ∀ b ∈ s, a < b
y : α
ys : List α
h : y :: ys ∈ (insert a s).subchain
⊢ ∃ l' ∈ s.subchain, (y :: ys).length + 0 ≤ l'.length + 1
|
have h' := cons_mem_subchain_iff.mp h
|
case a.cons
α : Type u_1
s : Set α
inst✝ : Preorder α
a : α
hx : ∀ b ∈ s, a < b
y : α
ys : List α
h : y :: ys ∈ (insert a s).subchain
h' : y ∈ insert a s ∧ ys ∈ (insert a s).subchain ∧ ∀ b ∈ ys.head?, y < b
⊢ ∃ l' ∈ s.subchain, (y :: ys).length + 0 ≤ l'.length + 1
|
abd1373dde588775
|
KuratowskiEmbedding.embeddingOfSubset_isometry
|
Mathlib/Topology/MetricSpace/Kuratowski.lean
|
theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x)
|
case intro
α : Type u
inst✝ : MetricSpace α
x : ℕ → α
H : DenseRange x
a b : α
e : ℝ
epos : 0 < e
n : ℕ
hn : dist a (x n) < e / 2
C : dist b (x n) - dist a (x n) = ↑(embeddingOfSubset x b) n - ↑(embeddingOfSubset x a) n
⊢ dist a b ≤ dist (embeddingOfSubset x a) (embeddingOfSubset x b) + e
|
have :=
calc
dist a b ≤ dist a (x n) + dist (x n) b := dist_triangle _ _ _
_ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) := by simp [dist_comm]; ring
_ ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| := by
apply_rules [add_le_add_left, le_abs_self]
_ ≤ 2 * (e / 2) + |embeddingOfSubset x b n - embeddingOfSubset x a n| := by
rw [C]
gcongr
_ ≤ 2 * (e / 2) + dist (embeddingOfSubset x b) (embeddingOfSubset x a) := by
gcongr
simp only [dist_eq_norm]
exact lp.norm_apply_le_norm ENNReal.top_ne_zero
(embeddingOfSubset x b - embeddingOfSubset x a) n
_ = dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e := by ring
|
case intro
α : Type u
inst✝ : MetricSpace α
x : ℕ → α
H : DenseRange x
a b : α
e : ℝ
epos : 0 < e
n : ℕ
hn : dist a (x n) < e / 2
C : dist b (x n) - dist a (x n) = ↑(embeddingOfSubset x b) n - ↑(embeddingOfSubset x a) n
this : dist a b ≤ dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e
⊢ dist a b ≤ dist (embeddingOfSubset x a) (embeddingOfSubset x b) + e
|
a61e80bcf82ba877
|
RelSeries.length_eq_zero
|
Mathlib/Order/RelSeries.lean
|
lemma length_eq_zero (irrefl : Irreflexive r) : s.length = 0 ↔ {x | x ∈ s}.Subsingleton
|
α : Type u_1
r : Rel α α
s : RelSeries r
irrefl : Irreflexive r
⊢ s.length = 0 ↔ {x | x ∈ s}.Subsingleton
|
rw [← not_ne_iff, length_ne_zero irrefl, Set.not_nontrivial_iff]
|
no goals
|
860153b78e780aba
|
EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq
|
Mathlib/Geometry/Euclidean/Circumcenter.lean
|
theorem dist_eq_iff_dist_orthogonalProjection_eq {s : AffineSubspace ℝ P} [Nonempty s]
[HasOrthogonalProjection s.direction] {p₁ p₂ : P} (p₃ : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) :
dist p₁ p₃ = dist p₂ p₃ ↔
dist p₁ (orthogonalProjection s p₃) = dist p₂ (orthogonalProjection s p₃)
|
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥s
inst✝ : HasOrthogonalProjection s.direction
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
⊢ dist p₁ ↑((orthogonalProjection s) p₃) * dist p₁ ↑((orthogonalProjection s) p₃) +
dist p₃ ↑((orthogonalProjection s) p₃) * dist p₃ ↑((orthogonalProjection s) p₃) =
dist p₂ ↑((orthogonalProjection s) p₃) * dist p₂ ↑((orthogonalProjection s) p₃) +
dist p₃ ↑((orthogonalProjection s) p₃) * dist p₃ ↑((orthogonalProjection s) p₃) ↔
dist p₁ ↑((orthogonalProjection s) p₃) * dist p₁ ↑((orthogonalProjection s) p₃) =
dist p₂ ↑((orthogonalProjection s) p₃) * dist p₂ ↑((orthogonalProjection s) p₃)
|
simp
|
no goals
|
5ba325a87b66032f
|
Array.eq_or_ne_mem_of_mem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem eq_or_ne_mem_of_mem {a b : α} {l : Array α} (h' : a ∈ l.push b) :
a = b ∨ (a ≠ b ∧ a ∈ l)
|
α : Type u_1
a b : α
l : Array α
h' : a ∈ l.push b
h : a = b
⊢ a = b ∨ a ≠ b ∧ a ∈ l
|
exact .inl h
|
no goals
|
e9078e8e28b0d7b0
|
SimpleGraph.Walk.takeUntil_cons
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.lean
|
lemma takeUntil_cons {v' : V} {p : G.Walk v' v} (hwp : w ∈ p.support) (h : u ≠ w)
(hadj : G.Adj u v') :
(p.cons hadj).takeUntil w (List.mem_of_mem_tail hwp) = (p.takeUntil w hwp).cons hadj
|
V : Type u
G : SimpleGraph V
v w u : V
inst✝ : DecidableEq V
v' : V
p : G.Walk v' v
hwp : w ∈ p.support
h : u ≠ w
hadj : G.Adj u v'
⊢ (cons hadj p).takeUntil w ⋯ = cons hadj (p.takeUntil w hwp)
|
simp [Walk.takeUntil, h]
|
no goals
|
1c50405372a9780c
|
Set.ncard_eq_one
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_one : s.ncard = 1 ↔ ∃ a, s = {a}
|
α : Type u_1
s : Set α
hft : Fintype ↑s
h : ∃ a, s.toFinset = {a}
a : α
ha : ∀ (a_1 : α), a_1 ∈ s ↔ a_1 = a
⊢ ∀ (x : α), x ∈ s ↔ x = a
|
exact ha
|
no goals
|
6b099bb5edde6c64
|
Polynomial.not_isUnit_X_pow_sub_one
|
Mathlib/Algebra/Polynomial/Monic.lean
|
theorem not_isUnit_X_pow_sub_one (R : Type*) [CommRing R] [Nontrivial R] (n : ℕ) :
¬IsUnit (X ^ n - 1 : R[X])
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Nontrivial R
n : ℕ
⊢ ¬IsUnit (X ^ n - 1)
|
intro h
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Nontrivial R
n : ℕ
h : IsUnit (X ^ n - 1)
⊢ False
|
f9d4950f92455e15
|
CFC.nnrpow_zero
|
Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Basic.lean
|
@[simp]
lemma nnrpow_zero {a : A} : a ^ (0 : ℝ≥0) = 0
|
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : IsScalarTower ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ≥0 fun a => 0 ≤ a
a : A
⊢ a ^ 0 = 0
|
simp [nnrpow_def, cfcₙ_apply_of_not_map_zero]
|
no goals
|
c8debcd2cafb6ab1
|
SemidirectProduct.map_comp_inr
|
Mathlib/GroupTheory/SemidirectProduct.lean
|
theorem map_comp_inr : (map fn fg h).comp inr = inr.comp fg
|
N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
⊢ (map fn fg h).comp inr = inr.comp fg
|
ext <;> simp [map]
|
no goals
|
14e7a054c227ee16
|
List.Perm.inter_append
|
Mathlib/Data/List/Perm/Lattice.lean
|
theorem Perm.inter_append {l t₁ t₂ : List α} (h : Disjoint t₁ t₂) :
l ∩ (t₁ ++ t₂) ~ l ∩ t₁ ++ l ∩ t₂
|
case pos
α : Type u_1
inst✝ : DecidableEq α
t₁ t₂ : List α
h : t₁.Disjoint t₂
x : α
xs : List α
l_ih : xs ∩ (t₁ ++ t₂) ~ xs ∩ t₁ ++ xs ∩ t₂
h₁ : x ∉ t₁
h₂ : x ∈ t₂
⊢ (x :: xs) ∩ (t₁ ++ t₂) ~ (x :: xs) ∩ t₁ ++ (x :: xs) ∩ t₂
|
simp only [*, inter_cons_of_not_mem, false_or, mem_append, inter_cons_of_mem,
not_false_iff]
|
case pos
α : Type u_1
inst✝ : DecidableEq α
t₁ t₂ : List α
h : t₁.Disjoint t₂
x : α
xs : List α
l_ih : xs ∩ (t₁ ++ t₂) ~ xs ∩ t₁ ++ xs ∩ t₂
h₁ : x ∉ t₁
h₂ : x ∈ t₂
⊢ x :: xs ∩ (t₁ ++ t₂) ~ xs ∩ t₁ ++ x :: xs ∩ t₂
|
e668c0ceba98db61
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastMul.denote_blast
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Mul.lean
|
theorem denote_blast (aig : AIG BVBit) (lhs rhs : BitVec w) (assign : Assignment)
(input : BinaryRefVec aig w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.lhs.get idx hidx, assign.toAIGAssignment⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.rhs.get idx hidx, assign.toAIGAssignment⟧ = rhs.getLsbD idx) :
∀ (idx : Nat) (hidx : idx < w),
⟦(blast aig input).aig, (blast aig input).vec.get idx hidx, assign.toAIGAssignment⟧
=
(lhs * rhs).getLsbD idx
|
case intro.hacc
aig : AIG BVBit
assign : Assignment
idx w : Nat
lhs rhs : BitVec w.succ
input : aig.BinaryRefVec w.succ
hleft :
∀ (idx : Nat) (hidx : idx < w.succ),
⟦assign.toAIGAssignment, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < w.succ),
⟦assign.toAIGAssignment, { aig := aig, ref := input.rhs.get idx hidx }⟧ = rhs.getLsbD idx
hidx : idx < w.succ
res : RefVecEntry BVBit w.succ
hne : ¬w.succ = 0
hb :
go
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs, rhs := (blastConst aig 0).vec }).aig
((input.cast ⋯).lhs.cast ⋯) ((input.cast ⋯).rhs.cast ⋯) 1
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs, rhs := (blastConst aig 0).vec }).vec =
res
⊢ ∀ (idx : Nat) (hidx : idx < w.succ),
⟦assign.toAIGAssignment,
{
aig :=
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs, rhs := (blastConst aig 0).vec }).aig,
ref :=
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs,
rhs := (blastConst aig 0).vec }).vec.get
idx hidx }⟧ =
(lhs.mulRec rhs 0).getLsbD idx
|
intro idx hidx
|
case intro.hacc
aig : AIG BVBit
assign : Assignment
idx✝ w : Nat
lhs rhs : BitVec w.succ
input : aig.BinaryRefVec w.succ
hleft :
∀ (idx : Nat) (hidx : idx < w.succ),
⟦assign.toAIGAssignment, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < w.succ),
⟦assign.toAIGAssignment, { aig := aig, ref := input.rhs.get idx hidx }⟧ = rhs.getLsbD idx
hidx✝ : idx✝ < w.succ
res : RefVecEntry BVBit w.succ
hne : ¬w.succ = 0
hb :
go
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs, rhs := (blastConst aig 0).vec }).aig
((input.cast ⋯).lhs.cast ⋯) ((input.cast ⋯).rhs.cast ⋯) 1
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs, rhs := (blastConst aig 0).vec }).vec =
res
idx : Nat
hidx : idx < w.succ
⊢ ⟦assign.toAIGAssignment,
{
aig :=
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs, rhs := (blastConst aig 0).vec }).aig,
ref :=
(RefVec.ite (blastConst aig 0).aig
{ discr := (input.cast ⋯).rhs.get 0 ⋯, lhs := (input.cast ⋯).lhs,
rhs := (blastConst aig 0).vec }).vec.get
idx hidx }⟧ =
(lhs.mulRec rhs 0).getLsbD idx
|
8c44f256f3a4efbd
|
MeasureTheory.SeparableSpace.exists_measurable_partition_diam_le
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
/-- In a separable pseudometric space, for any ε > 0 there exists a countable collection of
disjoint Borel measurable subsets of diameter at most ε that cover the whole space. -/
lemma SeparableSpace.exists_measurable_partition_diam_le {ε : ℝ} (ε_pos : 0 < ε) :
∃ (As : ℕ → Set Ω), (∀ n, MeasurableSet (As n)) ∧ (∀ n, Bornology.IsBounded (As n)) ∧
(∀ n, diam (As n) ≤ ε) ∧ (⋃ n, As n = univ) ∧
(Pairwise (fun (n m : ℕ) ↦ Disjoint (As n) (As m)))
|
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
⊢ ∃ As,
(∀ (n : ℕ), MeasurableSet (As n)) ∧
(∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
(∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m => Disjoint (As n) (As m)
|
by_cases X_emp : IsEmpty Ω
|
case pos
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
X_emp : IsEmpty Ω
⊢ ∃ As,
(∀ (n : ℕ), MeasurableSet (As n)) ∧
(∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
(∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m => Disjoint (As n) (As m)
case neg
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
X_emp : ¬IsEmpty Ω
⊢ ∃ As,
(∀ (n : ℕ), MeasurableSet (As n)) ∧
(∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
(∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m => Disjoint (As n) (As m)
|
237abb1cee85e7bf
|
LinearMap.charpoly_toMatrix
|
Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean
|
theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
(toMatrix b b f).charpoly = f.charpoly
|
R : Type u_1
M : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : Nontrivial R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
ι : Type w
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
A : Matrix ι ι R := (toMatrix b b) f
⊢ ((toMatrix b b) f).charpoly = f.charpoly
|
let b' := chooseBasis R M
|
R : Type u_1
M : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : Nontrivial R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
ι : Type w
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι R M
A : Matrix ι ι R := (toMatrix b b) f
b' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M
⊢ ((toMatrix b b) f).charpoly = f.charpoly
|
ff9c8772a8b3d3c4
|
AddCircle.coe_eq_zero_iff_of_mem_Ico
|
Mathlib/Topology/Instances/AddCircle.lean
|
lemma coe_eq_zero_iff_of_mem_Ico (ha : a ∈ Ico 0 p) :
(a : AddCircle p) = 0 ↔ a = 0
|
𝕜 : Type u_1
inst✝¹ : LinearOrderedAddCommGroup 𝕜
p : 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
h0 : 0 ∈ Ico 0 (0 + p)
ha' : a ∈ Ico 0 (0 + p)
⊢ ↑a = 0 ↔ a = 0
|
rw [← AddCircle.coe_eq_coe_iff_of_mem_Ico ha' h0, QuotientAddGroup.mk_zero]
|
no goals
|
4faacd3d488ea321
|
CategoryTheory.ChosenFiniteProducts.prodComparison_inv_natural
|
Mathlib/CategoryTheory/ChosenFiniteProducts.lean
|
theorem prodComparison_inv_natural (f : A ⟶ A') (g : B ⟶ B') [IsIso (prodComparison F A' B')] :
inv (prodComparison F A B) ≫ F.map (f ⊗ g) =
(F.map f ⊗ F.map g) ≫ inv (prodComparison F A' B')
|
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ChosenFiniteProducts C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : ChosenFiniteProducts D
F : C ⥤ D
A B A' B' : C
inst✝¹ : IsIso (prodComparison F A B)
f : A ⟶ A'
g : B ⟶ B'
inst✝ : IsIso (prodComparison F A' B')
⊢ inv (prodComparison F A B) ≫ F.map (f ⊗ g) = (F.map f ⊗ F.map g) ≫ inv (prodComparison F A' B')
|
rw [IsIso.eq_comp_inv, Category.assoc, IsIso.inv_comp_eq, prodComparison_natural]
|
no goals
|
4528d89595aa8c0e
|
CategoryTheory.Comma.inv_right
|
Mathlib/CategoryTheory/Comma/Basic.lean
|
@[simp]
lemma inv_right [IsIso e] : (inv e).right = inv e.right
|
case hom_inv_id
A : Type u₁
inst✝³ : Category.{v₁, u₁} A
B : Type u₂
inst✝² : Category.{v₂, u₂} B
T : Type u₃
inst✝¹ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
e : X ⟶ Y
inst✝ : IsIso e
⊢ e.right ≫ (inv e).right = 𝟙 X.right
|
rw [← Comma.comp_right, IsIso.hom_inv_id, id_right]
|
no goals
|
c0e969a93d7f9594
|
MeasureTheory.LpAddConst_lt_top
|
Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
|
theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞
|
p : ℝ≥0∞
h : p ∈ Set.Ioo 0 1
⊢ p⁻¹ ≠ ⊤
|
simpa using h.1.ne'
|
no goals
|
792486120083fd0f
|
HurwitzZeta.hasSum_int_completedHurwitzZetaEven
|
Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
|
/-- Formula for `completedHurwitzZetaEven` as a Dirichlet series in the convergence range. -/
lemma hasSum_int_completedHurwitzZetaEven (a : ℝ) {s : ℂ} (hs : 1 < re s) :
HasSum (fun n : ℤ ↦ Gammaℝ s / (↑|n + a| : ℂ) ^ s / 2) (completedHurwitzZetaEven a s)
|
case pos
a : ℝ
s : ℂ
hs : 1 < s.re
t : ℝ
ht : 0 < t
n : ℤ
h✝ : ↑n + a = 0
⊢ 0 = ↑0 / 2
|
rw [ofReal_zero, zero_div]
|
no goals
|
0299a5c4a49e4f15
|
convexOn_of_hasDerivWithinAt2_nonneg
|
Mathlib/Analysis/Convex/Deriv.lean
|
/-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
lemma convexOn_of_hasDerivWithinAt2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
(hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
(hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
(hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x) : ConvexOn ℝ D f
|
case refine_2
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
⊢ 0 ≤ deriv^[2] f x
|
convert hf''₀ _ hx using 1
|
case h.e'_4
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
x : ℝ
hx : x ∈ interior D
⊢ deriv^[2] f x = f'' x
|
a62663d4b3c4bbdb
|
Std.DHashMap.Internal.Raw₀.Const.insertManyIfNewUnit_empty_list_nil
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem insertManyIfNewUnit_empty_list_nil :
insertManyIfNewUnit (empty : Raw₀ α (fun _ => Unit)) ([] : List α) =
(empty : Raw₀ α (fun _ => Unit))
|
α : Type u
inst✝¹ : BEq α
inst✝ : Hashable α
⊢ (insertManyIfNewUnit empty []).val = empty
|
simp
|
no goals
|
456a19fa6b2defe4
|
Turing.TM1to1.tr_respects
|
Mathlib/Computability/PostTuringMachine.lean
|
theorem tr_respects :
Respects (step M) (step (tr enc dec M)) fun c₁ c₂ ↦ trCfg enc enc0 c₁ = c₂ :=
fun_respects.2 fun ⟨l₁, v, T⟩ ↦ by
obtain ⟨L, R, rfl⟩ := T.exists_mk'
rcases l₁ with - | l₁
· exact rfl
suffices ∀ q R, Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))
(trCfg enc enc0 (stepAux q v (Tape.mk' L R))) by
refine TransGen.head' rfl ?_
rw [trTape_mk']
exact this _ R
clear R l₁
intro q R
induction q generalizing v L R with
| move d q IH =>
cases d <;>
simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons,
Tape.move_left_mk', ListBlank.cons_head_tail, ListBlank.tail_cons,
trTape'_move_left enc0, trTape'_move_right enc0] <;>
apply IH
| write f q IH =>
simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]
refine ReflTransGen.head rfl ?_
obtain ⟨a, R, rfl⟩ := R.exists_cons
rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move,
trTape'_move_left enc0, ListBlank.head_cons, ListBlank.tail_cons, Tape.write_mk']
apply IH
| load a q IH =>
simp only [trNormal, stepAux_read dec enc0 encdec]
apply IH
| branch p q₁ q₂ IH₁ IH₂ =>
simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, Tape.mk'_head]
cases p R.head v <;> [apply IH₂; apply IH₁]
| goto l =>
simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']
apply ReflTransGen.refl
| halt =>
simp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0,
trTape'_move_right enc0, trTape_mk']
apply ReflTransGen.refl
|
case intro.intro.some.goto
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝ : Inhabited Γ
enc0 : enc default = Vector.replicate n false
encdec : ∀ (a : Γ), dec (enc a) = a
x✝ : Cfg Γ Λ σ
l : Γ → σ → Λ
v : σ
L R : ListBlank Γ
⊢ Reaches (step (tr enc dec M)) { l := some (Λ'.normal (l R.head v)), var := v, Tape := trTape' enc0 L R }
{ l := Option.map Λ'.normal (some (l (Tape.mk' L R).head v)), var := v, Tape := trTape' enc0 L R }
|
apply ReflTransGen.refl
|
no goals
|
e86c623541388a9d
|
Basis.orientation_neg_single
|
Mathlib/LinearAlgebra/Orientation.lean
|
theorem orientation_neg_single (e : Basis ι R M) (i : ι) :
(e.unitsSMul (Function.update 1 i (-1))).orientation = -e.orientation
|
R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
i : ι
⊢ (-1 * ∏ x ∈ Finset.univ \ {i}, 1 x)⁻¹ • e.orientation = -e.orientation
|
simp
|
no goals
|
f65e024ec01068e5
|
NNRat.cast_div_of_ne_zero
|
Mathlib/Data/Rat/Cast/Defs.lean
|
@[norm_cast]
lemma cast_div_of_ne_zero (hq : (q.den : α) ≠ 0) (hr : (r.num : α) ≠ 0) :
↑(q / r) = (q / r : α)
|
α : Type u_3
inst✝ : DivisionSemiring α
q r : ℚ≥0
hq : ↑q.den ≠ 0
hr : ↑r.num ≠ 0
⊢ ↑q.num * ↑r.den / (↑q.den * ↑r.num) = ↑q.num * ↑r.den / (↑q.den * ↑r.num)
|
rfl
|
no goals
|
3456e21c9d9c01c8
|
CategoryTheory.HasShift.Induced.add_hom_app_obj
|
Mathlib/CategoryTheory/Shift/Induced.lean
|
@[simp]
lemma add_hom_app_obj (a b : A) (X : C) :
(add F s i a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X)
|
C : Type u_5
D : Type u_2
inst✝⁵ : Category.{u_4, u_5} C
inst✝⁴ : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
a b : A
X : C
h :
whiskerLeft F (add F s i a b).hom =
(i (a + b) ≪≫
isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫
(shiftFunctor C a).associator (shiftFunctor C b) F ≪≫
isoWhiskerLeft (shiftFunctor C a) (i b).symm ≪≫
((shiftFunctor C a).associator F (s b)).symm ≪≫
isoWhiskerRight (i a).symm (s b) ≪≫ F.associator (s a) (s b)).hom
⊢ (add F s i a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫
F.map ((shiftFunctorAdd C a b).hom.app X) ≫ (i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X)
|
exact (NatTrans.congr_app h X).trans (by simp)
|
no goals
|
3c48812c93cc085b
|
Array.lawfulBEq_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem lawfulBEq_iff [BEq α] : LawfulBEq (Array α) ↔ LawfulBEq α
|
case mp.rfl
α : Type u_1
inst✝ : BEq α
h : LawfulBEq (Array α)
a : α
⊢ (a == a) = true
|
apply beq_of_beq_singleton
|
case mp.rfl.a
α : Type u_1
inst✝ : BEq α
h : LawfulBEq (Array α)
a : α
⊢ (#[a] == #[a]) = true
|
2e5ca1ebe152af97
|
PhragmenLindelof.quadrant_I
|
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
theorem quadrant_I (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0))
(hB : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * ‖z‖ ^ c))
(hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, 0 ≤ x → ‖f (x * I)‖ ≤ C) (hz_re : 0 ≤ z.re)
(hz_im : 0 ≤ z.im) : ‖f z‖ ≤ C
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
hzne : z ≠ 0
⊢ (log z).im ∈ Icc 0 (π / 2)
|
rw [log_im]
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
hzne : z ≠ 0
⊢ z.arg ∈ Icc 0 (π / 2)
|
a9f7cc9a241d3219
|
MvPowerSeries.map_X
|
Mathlib/RingTheory/MvPowerSeries/Basic.lean
|
theorem map_X (s : σ) : map σ f (X s) = X s
|
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
s : σ
⊢ (map σ f) (X s) = X s
|
simp [MvPowerSeries.X]
|
no goals
|
b1f131d8b00f775b
|
MeasureTheory.AddContent.isCaratheodory_ofFunction_of_mem
|
Mathlib/MeasureTheory/OuterMeasure/OfAddContent.lean
|
theorem isCaratheodory_ofFunction_of_mem (hC : IsSetSemiring C) (m : AddContent C)
(m_top : ∀ s ∉ C, m s = ∞) (hs : s ∈ C) :
(OuterMeasure.ofFunction m addContent_empty).IsCaratheodory s
|
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent C
m_top : ∀ s ∉ C, m s = ⊤
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i => hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)
|
have h_m_eq i : m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u :=
eq_add_disjointOfDiff_of_subset hC (hC.inter_mem (f i) (hf i) s hs) (hf i) inter_subset_left
|
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent C
m_top : ∀ s ∉ C, m s = ⊤
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i => hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)
|
150b27f94c399a97
|
MeasureTheory.integral_condExpL2_eq
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
|
theorem integral_condExpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
(hμs : μ s ≠ ∞) : ∫ x in s, (condExpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ
|
case refine_1
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp E' 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
⊢ Integrable (↑↑(↑((condExpL2 E' 𝕜 hm) f) - f)) (μ.restrict s)
|
exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
|
no goals
|
f34acd4f2cd16da9
|
ProbabilityTheory.strong_law_aux1
|
Mathlib/Probability/StrongLaw.lean
|
theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) : ∀ᵐ ω, ∀ᶠ n : ℕ in atTop,
|∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω - 𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]| <
ε * ⌊c ^ n⌋₊
|
case inl
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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
hj : 0 ∈ range (u (N - 1))
⊢ (∑ i ∈ filter (fun i => 0 < u i) (range N), (↑(u i) ^ 2)⁻¹) * Var[Y 0; ℙ] ≤
c ^ 5 * (c - 1)⁻¹ ^ 3 / ↑0 ^ 2 * Var[Y 0; ℙ]
|
simp only [Nat.cast_zero, zero_pow, Ne, Nat.one_ne_zero,
not_false_iff, div_zero, zero_mul]
|
case inl
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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
hj : 0 ∈ range (u (N - 1))
⊢ (∑ i ∈ filter (fun i => 0 < u i) (range N), (↑(u i) ^ 2)⁻¹) * Var[Y 0; ℙ] ≤
c ^ 5 * (c - 1)⁻¹ ^ 3 / 0 ^ 2 * Var[Y 0; ℙ]
|
e1bd35d16fc7ec49
|
Complex.uniformContinuous_ringHom_eq_id_or_conj
|
Mathlib/Topology/Instances/Complex.lean
|
theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype
|
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
⊢ (↑↑ψ).toFun = ⇑K.subtype ∨ (↑↑ψ).toFun = ⇑(starRingEnd ℂ) ∘ ⇑K.subtype
|
letI : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
|
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
⊢ (↑↑ψ).toFun = ⇑K.subtype ∨ (↑↑ψ).toFun = ⇑(starRingEnd ℂ) ∘ ⇑K.subtype
|
f78bcdf9d891573b
|
LinearMap.BilinForm.exists_orthogonal_basis
|
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
theorem exists_orthogonal_basis [hK : Invertible (2 : K)] {B : LinearMap.BilinForm K V}
(hB₂ : B.IsSymm) : ∃ v : Basis (Fin (finrank K V)) K V, B.IsOrthoᵢ v
|
K : Type v
inst✝³ : Field K
hK : Invertible 2
d : ℕ
ih :
∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] {B : BilinForm K V},
IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v
V : Type u
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
B : BilinForm K V
hB₂ : IsSymm B
this✝ : Nontrivial V
hB₁ : B ≠ 0
x : V
hd : finrank K ↥((Submodule.span K {x}).orthogonalBilin B) + 1 = d + 1
hx : ¬IsOrtho B x x
B' : ↥((Submodule.span K {x}).orthogonalBilin B) →ₗ[K] ↥((Submodule.span K {x}).orthogonalBilin B) →ₗ[K] K :=
domRestrict₁₂ B ((Submodule.span K {x}).orthogonalBilin B) ((Submodule.span K {x}).orthogonalBilin B)
v' : Basis (Fin d) K ↥((Submodule.span K {x}).orthogonalBilin B)
hv₁ :
(domRestrict₁₂ B ((Submodule.span K {x}).orthogonalBilin B) ((Submodule.span K {x}).orthogonalBilin B)).IsOrthoᵢ ⇑v'
c : K
y : V
hy : c • x ∈ (Submodule.span K {x}).orthogonalBilin B
hc : -(c • x) = y
this : Disjoint (Submodule.span K {x}) ((Submodule.span K {x}).orthogonalBilin B)
⊢ c = 0
|
rw [Submodule.disjoint_def] at this
|
K : Type v
inst✝³ : Field K
hK : Invertible 2
d : ℕ
ih :
∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] {B : BilinForm K V},
IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v
V : Type u
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
B : BilinForm K V
hB₂ : IsSymm B
this✝ : Nontrivial V
hB₁ : B ≠ 0
x : V
hd : finrank K ↥((Submodule.span K {x}).orthogonalBilin B) + 1 = d + 1
hx : ¬IsOrtho B x x
B' : ↥((Submodule.span K {x}).orthogonalBilin B) →ₗ[K] ↥((Submodule.span K {x}).orthogonalBilin B) →ₗ[K] K :=
domRestrict₁₂ B ((Submodule.span K {x}).orthogonalBilin B) ((Submodule.span K {x}).orthogonalBilin B)
v' : Basis (Fin d) K ↥((Submodule.span K {x}).orthogonalBilin B)
hv₁ :
(domRestrict₁₂ B ((Submodule.span K {x}).orthogonalBilin B) ((Submodule.span K {x}).orthogonalBilin B)).IsOrthoᵢ ⇑v'
c : K
y : V
hy : c • x ∈ (Submodule.span K {x}).orthogonalBilin B
hc : -(c • x) = y
this : ∀ x_1 ∈ Submodule.span K {x}, x_1 ∈ (Submodule.span K {x}).orthogonalBilin B → x_1 = 0
⊢ c = 0
|
f15d20b415f151ca
|
RingQuot.lift_mkRingHom_apply
|
Mathlib/Algebra/RingQuot.lean
|
theorem lift_mkRingHom_apply (f : R →+* T) {r : R → R → Prop} (w : ∀ ⦃x y⦄, r x y → f x = f y) (x) :
lift ⟨f, w⟩ (mkRingHom r x) = f x
|
R : Type uR
inst✝¹ : Semiring R
T : Type uT
inst✝ : Semiring T
f : R →+* T
r : R → R → Prop
w : ∀ ⦃x y : R⦄, r x y → f x = f y
x : R
⊢ ({
toFun := fun f =>
{ toFun := fun x => Quot.lift ⇑↑f ⋯ x.toQuot, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ },
invFun := fun F =>
⟨F.comp
{ toFun := fun x => { toQuot := Quot.mk (Rel r) x }, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,
map_add' := ⋯ },
⋯⟩,
left_inv := ⋯, right_inv := ⋯ }
⟨f, w⟩)
({ toFun := fun x => { toQuot := Quot.mk (Rel r) x }, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,
map_add' := ⋯ }
x) =
f x
|
rfl
|
no goals
|
a9e31936f2bb55cf
|
AlgebraicGeometry.AffineSpace.reindex_id
|
Mathlib/AlgebraicGeometry/AffineSpace.lean
|
@[simp]
lemma reindex_id : reindex id S = 𝟙 𝔸(n; S)
|
n : Type v
S : Scheme
⊢ reindex id S = 𝟙 𝔸(n; S)
|
ext1 <;> simp
|
no goals
|
77f8196d90b13ea8
|
ENNReal.le_of_forall_nnreal_lt
|
Mathlib/Data/ENNReal/Inv.lean
|
theorem le_of_forall_nnreal_lt {x y : ℝ≥0∞} (h : ∀ r : ℝ≥0, ↑r < x → ↑r ≤ y) : x ≤ y
|
x y : ℝ≥0∞
h : ∀ (r : ℝ≥0), ↑r < x → ↑r ≤ y
⊢ x ≤ y
|
refine le_of_forall_lt_imp_le_of_dense fun r hr => ?_
|
x y : ℝ≥0∞
h : ∀ (r : ℝ≥0), ↑r < x → ↑r ≤ y
r : ℝ≥0∞
hr : r < x
⊢ r ≤ y
|
841714a7f2825562
|
ContinuousAffineMap.to_continuousMap_injective
|
Mathlib/Topology/Algebra/ContinuousAffineMap.lean
|
theorem to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
f = g
|
case h
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module R W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f g : P →ᴬ[R] Q
h : ↑f = ↑g
a : P
⊢ f a = g a
|
exact ContinuousMap.congr_fun h a
|
no goals
|
71afe41dac370e91
|
MeromorphicOn.eventually_codiscreteWithin_analyticAt
|
Mathlib/Analysis/Meromorphic/Basic.lean
|
theorem eventually_codiscreteWithin_analyticAt
[CompleteSpace E] (f : 𝕜 → E) (h : MeromorphicOn f U) :
∀ᶠ (y : 𝕜) in codiscreteWithin U, AnalyticAt 𝕜 f y
|
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x : 𝕜
hx : x ∈ U
⊢ {x | (fun y => AnalyticAt 𝕜 f y) x} ⊆ (U \ {x | AnalyticAt 𝕜 f x})ᶜ
|
intro x hx
|
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x✝ : 𝕜
hx✝ : x✝ ∈ U
x : 𝕜
hx : x ∈ {x | (fun y => AnalyticAt 𝕜 f y) x}
⊢ x ∈ (U \ {x | AnalyticAt 𝕜 f x})ᶜ
|
a35b18b98ec6aed2
|
WittVector.poly_eq_of_wittPolynomial_bind_eq'
|
Mathlib/RingTheory/WittVector/IsPoly.lean
|
theorem poly_eq_of_wittPolynomial_bind_eq' [Fact p.Prime] (f g : ℕ → MvPolynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g
|
case h
p : ℕ
idx : Type u_1
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial (idx × ℕ) ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ f n = g n
|
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
|
case h.a
p : ℕ
idx : Type u_1
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial (idx × ℕ) ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ (MvPolynomial.map (Int.castRingHom ℚ)) (f n) = (MvPolynomial.map (Int.castRingHom ℚ)) (g n)
|
d7abeb200abb0ee9
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastUmod.denote_go_eq_divRec_r
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean
|
theorem denote_go_eq_divRec_r (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, d.get idx hidx, assign⟧ = rhs.getLsbD idx)
(hq : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, q.get idx hidx, assign⟧ = qbv.getLsbD idx)
(hr : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, r.get idx hidx, assign⟧ = rbv.getLsbD idx)
(hfalse : ⟦aig, falseRef, assign⟧ = false)
(htrue : ⟦aig, trueRef, assign⟧ = true)
:
∀ (idx : Nat) (hidx : idx < w),
⟦
(go aig curr falseRef trueRef n d wn wr q r).aig,
(go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx,
assign
⟧
=
(BitVec.divRec curr { n := lhs, d := rhs} { wn, wr, q := qbv, r := rbv }).r.getLsbD idx
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ⟦assign,
{
aig :=
{
aig :=
(go (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig curr (falseRef.cast ⋯)
(trueRef.cast ⋯) (n.cast ⋯) (d.cast ⋯) (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r).aig,
q :=
(go (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig curr (falseRef.cast ⋯)
(trueRef.cast ⋯) (n.cast ⋯) (d.cast ⋯) (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r).q,
r :=
(go (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig curr (falseRef.cast ⋯)
(trueRef.cast ⋯) (n.cast ⋯) (d.cast ⋯) (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r).r,
hle := ⋯ }.aig,
ref :=
{
aig :=
(go (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig curr (falseRef.cast ⋯)
(trueRef.cast ⋯) (n.cast ⋯) (d.cast ⋯)
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r).aig,
q :=
(go (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig curr (falseRef.cast ⋯)
(trueRef.cast ⋯) (n.cast ⋯) (d.cast ⋯)
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r).q,
r :=
(go (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig curr (falseRef.cast ⋯)
(trueRef.cast ⋯) (n.cast ⋯) (d.cast ⋯)
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r).r,
hle := ⋯ }.r.get
idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs }
{ wn := { wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1,
wr := { wn := wn, wr := wr, q := qbv, r := rbv }.wr + 1,
q := { wn := wn, wr := wr, q := qbv, r := rbv }.q.shiftConcat false,
r :=
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) }).r.getLsbD
idx
|
rw [ih]
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ (BitVec.divRec curr { n := lhs, d := rhs }
{ wn := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wn,
wr := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).wr, q := ?qbv, r := ?rbv }).r.getLsbD
idx =
(BitVec.divRec curr { n := lhs, d := rhs }
{ wn := { wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1,
wr := { wn := wn, wr := wr, q := qbv, r := rbv }.wr + 1,
q := { wn := wn, wr := wr, q := qbv, r := rbv }.q.shiftConcat false,
r :=
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) }).r.getLsbD
idx
case rbv
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ BitVec w
case qbv
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ BitVec w
case hleft
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig, ref := (n.cast ⋯).get idx hidx }⟧ =
lhs.getLsbD idx
case hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig, ref := (d.cast ⋯).get idx hidx }⟧ =
rhs.getLsbD idx
case hq
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig,
ref := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q.get idx hidx }⟧ =
BitVec.getLsbD ?qbv idx
case hr
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig,
ref := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
BitVec.getLsbD ?rbv idx
case hfalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ⟦assign, { aig := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig, ref := falseRef.cast ⋯ }⟧ = false
case htrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx : Nat
hidx : idx < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
⊢ ⟦assign, { aig := (blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig, ref := trueRef.cast ⋯ }⟧ = true
|
123e2e165ae62131
|
Nat.ordProj_dvd
|
Mathlib/Data/Nat/Factorization/Defs.lean
|
theorem ordProj_dvd (n p : ℕ) : ordProj[p] n ∣ n
|
n p : ℕ
hp : Prime p
⊢ (p ^ count p n.primeFactorsList).primeFactorsList <+~ n.primeFactorsList
|
rw [hp.primeFactorsList_pow, List.subperm_ext_iff]
|
n p : ℕ
hp : Prime p
⊢ ∀ x ∈ replicate (count p n.primeFactorsList) p,
count x (replicate (count p n.primeFactorsList) p) ≤ count x n.primeFactorsList
|
bd7926bf85841618
|
IsCoprime.wronskian_eq_zero_iff
|
Mathlib/RingTheory/Polynomial/Wronskian.lean
|
theorem _root_.IsCoprime.wronskian_eq_zero_iff
[NoZeroDivisors R] {a b : R[X]} (hc : IsCoprime a b) :
wronskian a b = 0 ↔ derivative a = 0 ∧ derivative b = 0 where
mp hw
|
case intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
a b : R[X]
hc : IsCoprime a b
hda : derivative a = 0
hdb : derivative b = 0
⊢ a * 0 - 0 * b = 0
|
simp only [MulZeroClass.mul_zero, MulZeroClass.zero_mul, sub_self]
|
no goals
|
f3fde22d9abb12de
|
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