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
|
|---|---|---|---|---|---|---|
CategoryTheory.StrongEpi.of_arrow_iso
|
Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
|
theorem StrongEpi.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) [h : StrongEpi f] : StrongEpi g :=
{ epi
|
C : Type u
inst✝ : Category.{v, u} C
A B A' B' : C
f : A ⟶ B
g : A' ⟶ B'
e : Arrow.mk f ≅ Arrow.mk g
h : StrongEpi f
⊢ Epi (e.inv.left ≫ f ≫ e.hom.right)
|
infer_instance
|
no goals
|
cad0b0aefe5bf042
|
CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃'
|
Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean
|
/-- Alternative formulation of `contravariant_sequence_exact₃` -/
lemma contravariant_sequence_exact₃' :
(ShortComplex.mk (AddCommGrp.ofHom (hS.extClass.precomp Y h))
(AddCommGrp.ofHom (((mk₀ S.g).precomp Y (zero_add n₁)))) (by
ext
dsimp
simp only [ShortComplex.ShortExact.comp_extClass_assoc])).Exact
|
case hf.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
x✝ : ↑(AddCommGrp.of (Ext S.X₁ Y n₀))
⊢ (mk₀ S.g).comp (hS.extClass.comp x✝ h) ⋯ = 0
|
simp only [ShortComplex.ShortExact.comp_extClass_assoc]
|
no goals
|
3b5361343b948c2c
|
Set.ncard_insert_of_mem
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_insert_of_mem {a : α} (h : a ∈ s) : ncard (insert a s) = s.ncard
|
α : Type u_1
s : Set α
a : α
h : a ∈ s
⊢ (insert a s).ncard = s.ncard
|
rw [insert_eq_of_mem h]
|
no goals
|
f442188566650fe7
|
solvableByRad.induction2
|
Mathlib/FieldTheory/AbelRuffini.lean
|
theorem induction2 {α β γ : solvableByRad F E} (hγ : γ ∈ F⟮α, β⟯) (hα : P α) (hβ : P β) : P γ
|
case inr
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
x : ↥(solvableByRad F E)
hx : x = β
⊢ IsIntegral F β ∧ Splits (algebraMap F (p * q).SplittingField) (minpoly F β)
|
exact ⟨isIntegral β, hpq.2⟩
|
no goals
|
baad06864f8e9f42
|
cfcHom_comp
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean
|
theorem cfcHom_comp [UniqueHom R A] (f : C(spectrum R a, R))
(f' : C(spectrum R a, spectrum R (cfcHom ha f)))
(hff' : ∀ x, f x = f' x) (g : C(spectrum R (cfcHom ha f), R)) :
cfcHom ha (g.comp f') = cfcHom (cfcHom_predicate ha f) g
|
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
a : A
ha : p a
inst✝ : UniqueHom R A
f : C(↑(spectrum R a), R)
f' : C(↑(spectrum R a), ↑(spectrum R ((cfcHom ha) f)))
hff' : ∀ (x : ↑(spectrum R a)), f x = ↑(f' x)
g : C(↑(spectrum R ((cfcHom ha) f)), R)
φ : C(↑(spectrum R ((cfcHom ha) f)), R) →⋆ₐ[R] A := (cfcHom ha).comp (compStarAlgHom' R R f')
⊢ cfcHom ⋯ = φ
|
refine cfcHom_eq_of_continuous_of_map_id (cfcHom_predicate ha f) φ ?_ ?_
|
case refine_1
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
a : A
ha : p a
inst✝ : UniqueHom R A
f : C(↑(spectrum R a), R)
f' : C(↑(spectrum R a), ↑(spectrum R ((cfcHom ha) f)))
hff' : ∀ (x : ↑(spectrum R a)), f x = ↑(f' x)
g : C(↑(spectrum R ((cfcHom ha) f)), R)
φ : C(↑(spectrum R ((cfcHom ha) f)), R) →⋆ₐ[R] A := (cfcHom ha).comp (compStarAlgHom' R R f')
⊢ Continuous ⇑φ
case refine_2
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
a : A
ha : p a
inst✝ : UniqueHom R A
f : C(↑(spectrum R a), R)
f' : C(↑(spectrum R a), ↑(spectrum R ((cfcHom ha) f)))
hff' : ∀ (x : ↑(spectrum R a)), f x = ↑(f' x)
g : C(↑(spectrum R ((cfcHom ha) f)), R)
φ : C(↑(spectrum R ((cfcHom ha) f)), R) →⋆ₐ[R] A := (cfcHom ha).comp (compStarAlgHom' R R f')
⊢ φ (restrict (spectrum R ((cfcHom ha) f)) (ContinuousMap.id R)) = (cfcHom ha) f
|
c37ba724f2010a82
|
AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv
|
Mathlib/AlgebraicGeometry/OpenImmersion.lean
|
@[reassoc (attr := simp), elementwise nosimp]
lemma appLE_appIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] {U : Y.Opens}
{V : X.Opens} (e : V ≤ f ⁻¹ᵁ U) :
f.appLE U V e ≫ (f.appIso V).inv =
Y.presheaf.map (homOfLE <| (f.image_le_image_of_le e).trans
(f.image_preimage_eq_opensRange_inter U ▸ inf_le_right)).op
|
X Y : Scheme
f : X ⟶ Y
inst✝ : IsOpenImmersion f
U : Y.Opens
V : X.Opens
e : V ≤ f ⁻¹ᵁ U
⊢ Y.presheaf.map ((homOfLE ⋯).op ≫ (homOfLE ⋯).op) = Y.presheaf.map (homOfLE ⋯).op
|
rfl
|
no goals
|
434e9026ccbdcdd4
|
Nat.le_nth_of_lt_nth_succ
|
Mathlib/Data/Nat/Nth.lean
|
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k
|
case inl.intro.intro.inr
p : ℕ → Prop
k : ℕ
hf : (setOf p).Finite
n : ℕ
hn : n < #hf.toFinset
h : nth p n < 0
ha : p (nth p n)
hk : #hf.toFinset ≤ k + 1
⊢ nth p n ≤ nth p k
|
exact absurd h (zero_le _).not_lt
|
no goals
|
dc79df7ae5e59569
|
ProbabilityTheory.Kernel.tendsto_densityProcess_fst_atTop_univ_of_monotone
|
Mathlib/Probability/Kernel/Disintegration/Density.lean
|
lemma tendsto_densityProcess_fst_atTop_univ_of_monotone (κ : Kernel α (γ × β)) (n : ℕ) (a : α)
(x : γ) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ) :
Tendsto (fun m ↦ densityProcess κ (fst κ) n a x (seq m)) atTop
(𝓝 (densityProcess κ (fst κ) n a x univ))
|
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ¬((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 ∧ (κ.fst a) (countablePartitionSet n x) = 0 ∨
(κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ ∧ (κ.fst a) (countablePartitionSet n x) ≠ ⊤)
|
push_neg
|
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → (κ.fst a) (countablePartitionSet n x) ≠ 0) ∧
((κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → (κ.fst a) (countablePartitionSet n x) = ⊤)
|
66b65cbbe911116f
|
sup_eq_of_max
|
Mathlib/Order/Filter/Extr.lean
|
theorem sup_eq_of_max [Nonempty α] {b : β} (hb : b ∈ Set.range D) (hmem : D.invFun b ∈ s)
(hmax : ∀ a ∈ s, D a ≤ b) : s.sup D = b
|
case intro
α : Type u
β : Type v
inst✝² : SemilatticeSup β
inst✝¹ : OrderBot β
D : α → β
s : Finset α
inst✝ : Nonempty α
a : α
hmem : Function.invFun D (D a) ∈ s
hmax : ∀ a_1 ∈ s, D a_1 ≤ D a
⊢ IsMaxOn D (↑s) (Function.invFun D (D a))
|
intro
|
case intro
α : Type u
β : Type v
inst✝² : SemilatticeSup β
inst✝¹ : OrderBot β
D : α → β
s : Finset α
inst✝ : Nonempty α
a : α
hmem : Function.invFun D (D a) ∈ s
hmax : ∀ a_1 ∈ s, D a_1 ≤ D a
a✝ : α
⊢ a✝ ∈ ↑s → a✝ ∈ {x | (fun x => D x ≤ D (Function.invFun D (D a))) x}
|
acd295b8a19eaff5
|
List.contains_replicate
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem contains_replicate [BEq α] {n : Nat} {a b : α} :
(replicate n b).contains a = (a == b && !n == 0)
|
α : Type u_1
inst✝ : BEq α
n : Nat
a b : α
⊢ (replicate n b).contains a = (a == b && !n == 0)
|
induction n with
| zero => simp
| succ n ih =>
simp only [replicate_succ, elem_cons]
split <;> simp_all
|
no goals
|
a50795c03787787f
|
CategoryTheory.hasInitial_of_isCoseparating
|
Mathlib/CategoryTheory/Generator/Basic.lean
|
theorem hasInitial_of_isCoseparating [LocallySmall.{w} C] [WellPowered.{w} C]
[HasLimitsOfSize.{w, w} C] {𝒢 : Set C} [Small.{w} 𝒢]
(h𝒢 : IsCoseparating 𝒢) : HasInitial C
|
case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} C
inst✝² : WellPowered.{w, v₁, u₁} C
inst✝¹ : HasLimitsOfSize.{w, w, v₁, u₁} C
𝒢 : Set C
inst✝ : Small.{w, u₁} ↑𝒢
h𝒢 : IsCoseparating 𝒢
this✝² : HasFiniteLimits C
this✝¹ : HasProductsOfShape (↑𝒢) C
this✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C
this : CompleteLattice (Subobject (∏ᶜ Subtype.val)) :=
completeLatticeOfCompleteSemilatticeInf (Subobject (∏ᶜ Subtype.val))
A : C
f : Subobject.underlying.obj ⊥ ⟶ A
⊢ ∀ (g : Subobject.underlying.obj ⊥ ⟶ A), f = g
|
intro g
|
case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} C
inst✝² : WellPowered.{w, v₁, u₁} C
inst✝¹ : HasLimitsOfSize.{w, w, v₁, u₁} C
𝒢 : Set C
inst✝ : Small.{w, u₁} ↑𝒢
h𝒢 : IsCoseparating 𝒢
this✝² : HasFiniteLimits C
this✝¹ : HasProductsOfShape (↑𝒢) C
this✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C
this : CompleteLattice (Subobject (∏ᶜ Subtype.val)) :=
completeLatticeOfCompleteSemilatticeInf (Subobject (∏ᶜ Subtype.val))
A : C
f : Subobject.underlying.obj ⊥ ⟶ A
g : Subobject.underlying.obj ⊥ ⟶ A
⊢ f = g
|
5635e0b68399a531
|
ack_strict_mono_left'
|
Mathlib/Computability/Ackermann.lean
|
theorem ack_strict_mono_left' : ∀ {m₁ m₂} (n), m₁ < m₂ → ack m₁ n < ack m₂ n
| m, 0, _ => fun h => (not_lt_zero m h).elim
| 0, m + 1, 0 => fun _h => by simpa using one_lt_ack_succ_right m 0
| 0, m + 1, n + 1 => fun h => by
rw [ack_zero, ack_succ_succ]
apply lt_of_le_of_lt (le_trans _ <| add_le_add_left (add_add_one_le_ack _ _) m) (add_lt_ack _ _)
omega
| m₁ + 1, m₂ + 1, 0 => fun h => by
simpa using ack_strict_mono_left' 1 ((add_lt_add_iff_right 1).1 h)
| m₁ + 1, m₂ + 1, n + 1 => fun h => by
rw [ack_succ_succ, ack_succ_succ]
exact
(ack_strict_mono_left' _ <| (add_lt_add_iff_right 1).1 h).trans
(ack_strictMono_right _ <| ack_strict_mono_left' n h)
|
m₁ m₂ n : ℕ
h : m₁ + 1 < m₂ + 1
⊢ ack m₁ (ack (m₁ + 1) n) < ack m₂ (ack (m₂ + 1) n)
|
exact
(ack_strict_mono_left' _ <| (add_lt_add_iff_right 1).1 h).trans
(ack_strictMono_right _ <| ack_strict_mono_left' n h)
|
no goals
|
fb15ed9d9d43e49f
|
HasStrictFDerivAt.approximates_deriv_on_open_nhds
|
Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean
|
theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
∃ s : Set E, a ∈ s ∧ IsOpen s ∧
ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2)
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
a : E
hf : HasStrictFDerivAt f (↑f') a
⊢ ∃ s, (a ∈ s ∧ IsOpen s) ∧ ApproximatesLinearOn f (↑f') s (‖↑f'.symm‖₊⁻¹ / 2)
|
refine ((nhds_basis_opens a).exists_iff fun s t => ApproximatesLinearOn.mono_set).1 ?_
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
a : E
hf : HasStrictFDerivAt f (↑f') a
⊢ ∃ s ∈ 𝓝 a, ApproximatesLinearOn f (↑f') s (‖↑f'.symm‖₊⁻¹ / 2)
|
06b9e2837029fe8b
|
SimplexCategory.σ_comp_σ
|
Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean
|
theorem σ_comp_σ {n} {i j : Fin (n + 1)} (H : i ≤ j) :
σ (Fin.castSucc i) ≫ σ j = σ j.succ ≫ σ i
|
case inr
n : ℕ
i j : Fin (n + 1)
H : i ≤ j
k : Fin (n + 1)
h : i ≤ k
hkj : j < k
⊢ i < k
|
exact H.trans_lt hkj
|
no goals
|
fc4b1a638ff94922
|
IsCompact.elim_nhds_subcover_nhdsSet'
|
Mathlib/Topology/Compactness/Compact.lean
|
lemma IsCompact.elim_nhds_subcover_nhdsSet' (hs : IsCompact s) (U : ∀ x ∈ s, Set X)
(hU : ∀ x hx, U x hx ∈ 𝓝 x) : ∃ t : Finset s, (⋃ x ∈ t, U x.1 x.2) ∈ 𝓝ˢ s
|
case intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
U : (x : X) → x ∈ s → Set X
hU : ∀ (x : X) (hx : x ∈ s), U x hx ∈ 𝓝 x
t : Finset ↑s
hst : s ⊆ ⋃ i ∈ t, interior (U ↑i ⋯)
x : X
hx : x ∈ s
⊢ ⋃ x ∈ t, U ↑x ⋯ ∈ 𝓝 x
|
rcases mem_iUnion₂.1 (hst hx) with ⟨y, hyt, hy⟩
|
case intro.intro.intro
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
U : (x : X) → x ∈ s → Set X
hU : ∀ (x : X) (hx : x ∈ s), U x hx ∈ 𝓝 x
t : Finset ↑s
hst : s ⊆ ⋃ i ∈ t, interior (U ↑i ⋯)
x : X
hx : x ∈ s
y : ↑s
hyt : y ∈ t
hy : x ∈ interior (U ↑y ⋯)
⊢ ⋃ x ∈ t, U ↑x ⋯ ∈ 𝓝 x
|
860833996d6ec9bb
|
List.zipWith_eq_cons_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean
|
theorem zipWith_eq_cons_iff {f : α → β → γ} {l₁ : List α} {l₂ : List β} :
zipWith f l₁ l₂ = g :: l ↔
∃ a l₁' b l₂', l₁ = a :: l₁' ∧ l₂ = b :: l₂' ∧ g = f a b ∧ l = zipWith f l₁' l₂'
|
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
g : γ
l : List γ
f : α → β → γ
l₁✝ : List α
l₂✝ : List β
a' : α
l₁ : List α
b' : β
l₂ : List β
⊢ (∃ a l₁' b l₂', (a' = a ∧ l₁ = l₁') ∧ (b' = b ∧ l₂ = l₂') ∧ g = f a b ∧ l = zipWith f l₁' l₂') →
zipWith f (a' :: l₁) (b' :: l₂) = g :: l
|
rintro ⟨a, l₁, b, l₂, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl, rfl⟩
|
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
l₁✝ : List α
l₂✝ : List β
a' : α
l₁ : List α
b' : β
l₂ : List β
⊢ zipWith f (a' :: l₁) (b' :: l₂) = f a' b' :: zipWith f l₁ l₂
|
02ad9c8699cdd0ae
|
cfcₙ_congr
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
|
lemma cfcₙ_congr {f g : R → R} {a : A} (hfg : (σₙ R a).EqOn f g) :
cfcₙ f a = cfcₙ g a
|
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : ¬p a ∨ ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
⊢ cfcₙ f a = cfcₙ g a
|
obtain (ha | hg | h0) := h
|
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
ha : ¬p a
⊢ cfcₙ f a = cfcₙ g a
case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
hg : ¬ContinuousOn g (σₙ R a)
⊢ cfcₙ f a = cfcₙ g a
case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
f g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h0 : ¬g 0 = 0
⊢ cfcₙ f a = cfcₙ g a
|
e9bfeb6163b24036
|
Fermat42.not_minimal
|
Mathlib/NumberTheory/FLT/Four.lean
|
theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False
|
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
ht : PythagoreanTriple (a ^ 2) (b ^ 2) c
h2 : (a ^ 2).gcd (b ^ 2) = 1
ha22 : a ^ 2 % 2 = 1
n : ℤ
h3 : a.gcd n = 1
ht1 : a ^ 2 = 0 ^ 2 - n ^ 2
ht2 : b ^ 2 = 2 * 0 * n
ht3 : c = 0 ^ 2 + n ^ 2
ht4 : Int.gcd 0 n = 1
ht5 : 0 % 2 = 0 ∧ n % 2 = 1 ∨ 0 % 2 = 1 ∧ n % 2 = 0
ht6 : 0 ≤ 0
htt : PythagoreanTriple a n 0
⊢ b ^ 2 ≠ 0 → False
|
rw [ht2]
|
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
ht : PythagoreanTriple (a ^ 2) (b ^ 2) c
h2 : (a ^ 2).gcd (b ^ 2) = 1
ha22 : a ^ 2 % 2 = 1
n : ℤ
h3 : a.gcd n = 1
ht1 : a ^ 2 = 0 ^ 2 - n ^ 2
ht2 : b ^ 2 = 2 * 0 * n
ht3 : c = 0 ^ 2 + n ^ 2
ht4 : Int.gcd 0 n = 1
ht5 : 0 % 2 = 0 ∧ n % 2 = 1 ∨ 0 % 2 = 1 ∧ n % 2 = 0
ht6 : 0 ≤ 0
htt : PythagoreanTriple a n 0
⊢ 2 * 0 * n ≠ 0 → False
|
58647b498e4824ae
|
NormedSpace.vonNBornology_eq
|
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
theorem vonNBornology_eq : Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology
|
𝕜 : Type u_1
E : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s
|
exact isVonNBounded_iff _
|
no goals
|
6b09106729310f8b
|
CategoryTheory.Functor.complete_distinguished_essImageDistTriang_morphism
|
Mathlib/CategoryTheory/Localization/Triangulated.lean
|
lemma complete_distinguished_essImageDistTriang_morphism
(H : ∀ (T₁' T₂' : Triangle C) (_ : T₁' ∈ distTriang C) (_ : T₂' ∈ distTriang C)
(a : L.obj (T₁'.obj₁) ⟶ L.obj (T₂'.obj₁)) (b : L.obj (T₁'.obj₂) ⟶ L.obj (T₂'.obj₂))
(_ : L.map T₁'.mor₁ ≫ b = a ≫ L.map T₂'.mor₁),
∃ (φ : L.mapTriangle.obj T₁' ⟶ L.mapTriangle.obj T₂'), φ.hom₁ = a ∧ φ.hom₂ = b)
(T₁ T₂ : Triangle D)
(hT₁ : T₁ ∈ Functor.essImageDistTriang L) (hT₂ : T₂ ∈ L.essImageDistTriang)
(a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃
|
case intro.intro.intro.intro.intro.intro.refine_1
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
L : C ⥤ D
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : HasShift D ℤ
inst✝ : L.CommShift ℤ
H :
∀ (T₁' T₂' : Triangle C),
T₁' ∈ distinguishedTriangles →
T₂' ∈ distinguishedTriangles →
∀ (a : L.obj T₁'.obj₁ ⟶ L.obj T₂'.obj₁) (b : L.obj T₁'.obj₂ ⟶ L.obj T₂'.obj₂),
L.map T₁'.mor₁ ≫ b = a ≫ L.map T₂'.mor₁ → ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
T₁ T₂ : Triangle D
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
T₁' : Triangle C
e₁ : T₁ ≅ L.mapTriangle.obj T₁'
hT₁' : T₁' ∈ distinguishedTriangles
T₂' : Triangle C
e₂ : T₂ ≅ L.mapTriangle.obj T₂'
hT₂' : T₂' ∈ distinguishedTriangles
comm₁ : L.map T₁'.mor₁ ≫ e₁.inv.hom₂ = e₁.inv.hom₁ ≫ T₁.mor₁
comm₁' : T₂.mor₁ ≫ e₂.hom.hom₂ = e₂.hom.hom₁ ≫ L.map T₂'.mor₁
comm₂ : T₁.mor₂ ≫ e₁.hom.hom₃ = e₁.hom.hom₂ ≫ L.map T₁'.mor₂
comm₂' : T₂.mor₂ ≫ e₂.hom.hom₃ = e₂.hom.hom₂ ≫ L.map T₂'.mor₂
comm₃' :
T₂.mor₃ ≫ (shiftFunctor D 1).map e₂.hom.hom₁ = e₂.hom.hom₃ ≫ L.map T₂'.mor₃ ≫ (L.commShiftIso 1).hom.app T₂'.obj₁
comm₃ :
L.map T₁'.mor₃ ≫ (L.commShiftIso 1).hom.app T₁'.obj₁ ≫ (shiftFunctor D 1).map e₁.inv.hom₁ = e₁.inv.hom₃ ≫ T₁.mor₃
φ : L.mapTriangle.obj T₁' ⟶ L.mapTriangle.obj T₂'
hφ₁ : φ.hom₁ = e₁.inv.hom₁ ≫ a ≫ e₂.hom.hom₁
hφ₂ : φ.hom₂ = e₁.inv.hom₂ ≫ b ≫ e₂.hom.hom₂
h₂ : L.map T₁'.mor₂ ≫ φ.hom₃ = φ.hom₂ ≫ L.map T₂'.mor₂
h₃ :
L.map T₁'.mor₃ ≫ (L.commShiftIso 1).hom.app T₁'.obj₁ ≫ (shiftFunctor D 1).map φ.hom₁ =
φ.hom₃ ≫ L.map T₂'.mor₃ ≫ (L.commShiftIso 1).hom.app T₂'.obj₁
⊢ T₁.mor₂ ≫ e₁.hom.hom₃ ≫ φ.hom₃ ≫ e₂.inv.hom₃ = b ≫ T₂.mor₂
|
rw [reassoc_of% comm₂, reassoc_of% h₂, hφ₂, assoc, assoc,
Iso.hom_inv_id_triangle_hom₂_assoc, ← reassoc_of% comm₂',
Iso.hom_inv_id_triangle_hom₃, comp_id]
|
no goals
|
f4bc8e1c8f902228
|
Finset.Nonempty.of_disjSups_left
|
Mathlib/Data/Finset/Sups.lean
|
theorem Nonempty.of_disjSups_left : (s ○ t).Nonempty → s.Nonempty
|
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
⊢ (∃ x, ∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x) → ∃ x, x ∈ s
|
exact fun ⟨_, a, ha, _⟩ => ⟨a, ha⟩
|
no goals
|
63779e3956b0ff3d
|
integral_mul_cpow_one_add_sq
|
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
theorem integral_mul_cpow_one_add_sq {t : ℂ} (ht : t ≠ -1) :
(∫ x : ℝ in a..b, (x : ℂ) * ((1 : ℂ) + ↑x ^ 2) ^ t) =
((1 : ℂ) + (b : ℂ) ^ 2) ^ (t + 1) / (2 * (t + ↑1)) -
((1 : ℂ) + (a : ℂ) ^ 2) ^ (t + 1) / (2 * (t + ↑1))
|
a b : ℝ
t : ℂ
ht : t ≠ -1
⊢ t + 1 ≠ 0
|
contrapose! ht
|
a b : ℝ
t : ℂ
ht : t + 1 = 0
⊢ t = -1
|
5e0a067782411706
|
Finset.centerMass_le_sup
|
Mathlib/Analysis/Convex/Combination.lean
|
theorem centerMass_le_sup {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i)
(hw₁ : 0 < ∑ i ∈ s, w i) :
s.centerMass w f ≤ s.sup' (nonempty_of_ne_empty <| by rintro rfl; simp at hw₁) f
|
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s✝ : Set E
i j : ι
c : R
t : Finset ι
w✝ : ι → R
z : ι → E
s : Finset ι
f : ι → α
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ s, w i
⊢ s ≠ ∅
|
rintro rfl
|
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w✝ : ι → R
z : ι → E
f : ι → α
w : ι → R
hw₀ : ∀ i ∈ ∅, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ ∅, w i
⊢ False
|
da5b7b31c2eefa72
|
Set.ncard_eq_of_bijective
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_of_bijective {n : ℕ} (f : ∀ i, i < n → α)
(hf : ∀ a ∈ s, ∃ i, ∃ h : i < n, f i h = a) (hf' : ∀ (i) (h : i < n), f i h ∈ s)
(f_inj : ∀ (i j) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j) : s.ncard = n
|
α : Type u_1
s : Set α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ i, ∃ (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
f' : Fin n → α := fun i => f ↑i ⋯
himage : s = f' '' univ
⊢ s.ncard = n
|
rw [← Fintype.card_fin n, ← Nat.card_eq_fintype_card, ← Set.ncard_univ, himage]
|
α : Type u_1
s : Set α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ i, ∃ (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
f' : Fin n → α := fun i => f ↑i ⋯
himage : s = f' '' univ
⊢ (f' '' univ).ncard = univ.ncard
|
e5caade2630c707b
|
Finset.noncommProd_erase_mul
|
Mathlib/Data/Finset/NoncommProd.lean
|
theorem noncommProd_erase_mul [DecidableEq α] (s : Finset α) {a : α} (h : a ∈ s) (f : α → β) (comm)
(comm' := fun _ hx _ hy hxy ↦ comm (s.mem_of_mem_erase hx) (s.mem_of_mem_erase hy) hxy) :
(s.erase a).noncommProd f comm' * f a = s.noncommProd f comm
|
α : Type u_3
β : Type u_4
inst✝¹ : Monoid β
inst✝ : DecidableEq α
s : Finset α
a : α
h : a ∈ s
f : α → β
comm : (↑s).Pairwise (Commute on f)
comm' : optParam (∀ x ∈ ↑(s.erase a), ∀ x_1 ∈ ↑(s.erase a), x ≠ x_1 → (Commute on f) x x_1) ⋯
⊢ (s.erase a).noncommProd f comm' * f a = s.noncommProd f comm
|
simpa only [← Multiset.map_erase_of_mem _ _ h] using
Multiset.noncommProd_erase_mul (s.1.map f) (Multiset.mem_map_of_mem f h) _
|
no goals
|
005bcbf5b5e32eec
|
Hopf_.mul_antipode
|
Mathlib/CategoryTheory/Monoidal/Hopf_.lean
|
theorem mul_antipode (A : Hopf_ C) :
A.X.X.mul ≫ A.antipode = (A.antipode ⊗ A.antipode) ≫ (β_ _ _).hom ≫ A.X.X.mul
|
case hba
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
⊢ (A.X.comul.hom ⊗ A.X.comul.hom) ≫
(tensorμ (Opposite.op A.X.X.X) (Opposite.op A.X.X.X) (Opposite.op A.X.X.X) (Opposite.op A.X.X.X)).unop ≫
(α_ (A.X.X.X ⊗ A.X.X.X) A.X.X.X A.X.X.X).inv ≫
A.X.X.mul ▷ A.X.X.X ▷ A.X.X.X ≫
A.antipode ▷ A.X.X.X ▷ A.X.X.X ≫ (α_ A.X.X.X A.X.X.X A.X.X.X).hom ≫ A.X.X.X ◁ A.X.X.mul ≫ A.X.X.mul =
(A.X.counit.hom ⊗ A.X.counit.hom) ≫ (λ_ (𝟙_ C)).hom ≫ A.X.X.one
|
simp only [tensorμ]
|
case hba
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
⊢ (A.X.comul.hom ⊗ A.X.comul.hom) ≫
((α_ (Opposite.op A.X.X.X) (Opposite.op A.X.X.X) (Opposite.op A.X.X.X ⊗ Opposite.op A.X.X.X)).hom ≫
Opposite.op A.X.X.X ◁ (α_ (Opposite.op A.X.X.X) (Opposite.op A.X.X.X) (Opposite.op A.X.X.X)).inv ≫
Opposite.op A.X.X.X ◁ (β_ (Opposite.op A.X.X.X) (Opposite.op A.X.X.X)).hom ▷ Opposite.op A.X.X.X ≫
Opposite.op A.X.X.X ◁ (α_ (Opposite.op A.X.X.X) (Opposite.op A.X.X.X) (Opposite.op A.X.X.X)).hom ≫
(α_ (Opposite.op A.X.X.X) (Opposite.op A.X.X.X)
(Opposite.op A.X.X.X ⊗ Opposite.op A.X.X.X)).inv).unop ≫
(α_ (A.X.X.X ⊗ A.X.X.X) A.X.X.X A.X.X.X).inv ≫
A.X.X.mul ▷ A.X.X.X ▷ A.X.X.X ≫
A.antipode ▷ A.X.X.X ▷ A.X.X.X ≫ (α_ A.X.X.X A.X.X.X A.X.X.X).hom ≫ A.X.X.X ◁ A.X.X.mul ≫ A.X.X.mul =
(A.X.counit.hom ⊗ A.X.counit.hom) ≫ (λ_ (𝟙_ C)).hom ≫ A.X.X.one
|
fc94aad8d9cc8423
|
Preperfect.open_inter
|
Mathlib/Topology/Perfect.lean
|
theorem Preperfect.open_inter {U : Set α} (hC : Preperfect C) (hU : IsOpen U) :
Preperfect (U ∩ C)
|
case intro
α : Type u_1
inst✝ : TopologicalSpace α
C U : Set α
hC : Preperfect C
hU : IsOpen U
x : α
xU : x ∈ U
xC : x ∈ C
⊢ U ∈ 𝓝 x
|
exact hU.mem_nhds xU
|
no goals
|
9bea63263d482796
|
CategoryTheory.Limits.colimitLimitToLimitColimit_injective
|
Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean
|
theorem colimitLimitToLimitColimit_injective :
Function.Injective (colimitLimitToLimitColimit F)
|
J : Type u₁
K : Type u₂
inst✝⁴ : Category.{v₁, u₁} J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
F : J × K ⥤ Type v
inst✝¹ : IsFiltered K
inst✝ : Finite J
val✝ : Fintype J
kx : K
x : limit ((curry.obj (swap K J ⋙ F)).obj kx)
ky : K
y : limit ((curry.obj (swap K J ⋙ F)).obj ky)
h :
∀ (j : J),
∃ k f g,
F.map (𝟙 j, f) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =
F.map (𝟙 j, g) (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y)
k : J → K := fun j => ⋯.choose
f : (j : J) → kx ⟶ k j := fun j => ⋯.choose
g : (j : J) → ky ⟶ k j := fun j => ⋯.choose
w :
∀ (j : J),
F.map (𝟙 j, f j) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =
F.map (𝟙 j, g j) (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y)
O : Finset K := Finset.image k Finset.univ ∪ {kx, ky}
kxO : kx ∈ O
kyO : ky ∈ O
kjO : ∀ (j : J), k j ∈ O
H : Finset ((X : K) ×' (Y : K) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)) :=
Finset.image (fun j => ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ ∪
Finset.image (fun j => ⟨ky, ⟨k j, ⟨kyO, ⟨⋯, g j⟩⟩⟩⟩) Finset.univ
S : K
T : {X : K} → X ∈ O → (X ⟶ S)
W : ∀ {X Y : K} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}, ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈ H → f ≫ T mY = T mX
j : J
⊢ ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩ = ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩
|
simp only [heq_iff_eq]
|
no goals
|
dd8ec51392784757
|
collapse_modular
|
Mathlib/Combinatorics/SetFamily/FourFunctions.lean
|
lemma collapse_modular [ExistsAddOfLE β]
(hu : a ∉ u) (h₁ : 0 ≤ f₁) (h₂ : 0 ≤ f₂) (h₃ : 0 ≤ f₃) (h₄ : 0 ≤ f₄)
(h : ∀ ⦃s⦄, s ⊆ insert a u → ∀ ⦃t⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t))
(𝒜 ℬ : Finset (Finset α)) :
∀ ⦃s⦄, s ⊆ u → ∀ ⦃t⦄, t ⊆ u → collapse 𝒜 a f₁ s * collapse ℬ a f₂ t ≤
collapse (𝒜 ⊼ ℬ) a f₃ (s ∩ t) * collapse (𝒜 ⊻ ℬ) a f₄ (s ∪ t)
|
α : Type u_1
β : Type u_2
inst✝² : DecidableEq α
inst✝¹ : LinearOrderedCommSemiring β
a : α
f₁ f₂ f₃ f₄ : Finset α → β
u : Finset α
inst✝ : ExistsAddOfLE β
hu : a ∉ u
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
s : Finset α
hsu : s ⊆ u
t : Finset α
htu : t ⊆ u
this✝¹ : s ⊆ insert a u
this✝ : t ⊆ insert a u
this : insert a s ⊆ insert a u
⊢ collapse 𝒜 a f₁ s * collapse ℬ a f₂ t ≤ collapse (𝒜 ⊼ ℬ) a f₃ (s ∩ t) * collapse (𝒜 ⊻ ℬ) a f₄ (s ∪ t)
|
have := insert_subset_insert a htu
|
α : Type u_1
β : Type u_2
inst✝² : DecidableEq α
inst✝¹ : LinearOrderedCommSemiring β
a : α
f₁ f₂ f₃ f₄ : Finset α → β
u : Finset α
inst✝ : ExistsAddOfLE β
hu : a ∉ u
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
s : Finset α
hsu : s ⊆ u
t : Finset α
htu : t ⊆ u
this✝² : s ⊆ insert a u
this✝¹ : t ⊆ insert a u
this✝ : insert a s ⊆ insert a u
this : insert a t ⊆ insert a u
⊢ collapse 𝒜 a f₁ s * collapse ℬ a f₂ t ≤ collapse (𝒜 ⊼ ℬ) a f₃ (s ∩ t) * collapse (𝒜 ⊻ ℬ) a f₄ (s ∪ t)
|
4b3d90b1fc92ae23
|
LieModuleHom.coe_injective
|
Mathlib/Algebra/Lie/Basic.lean
|
theorem coe_injective : @Function.Injective (M →ₗ⁅R,L⁆ N) (M → N) (↑)
|
R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : LieRingModule L M
inst✝ : LieRingModule L N
⊢ Injective DFunLike.coe
|
rintro ⟨⟨⟨f, _⟩⟩⟩ ⟨⟨⟨g, _⟩⟩⟩ h
|
case mk.mk.mk.mk.mk.mk
R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : LieRingModule L M
inst✝ : LieRingModule L N
f : M → N
map_add'✝¹ : ∀ (x y : M), f (x + y) = f x + f y
map_smul'✝¹ :
∀ (m : R) (x : M),
{ toFun := f, map_add' := map_add'✝¹ }.toFun (m • x) =
(RingHom.id R) m • { toFun := f, map_add' := map_add'✝¹ }.toFun x
map_lie'✝¹ :
∀ {x : L} {m : M},
{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun ⁅x, m⁆ =
⁅x, { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun m⁆
g : M → N
map_add'✝ : ∀ (x y : M), g (x + y) = g x + g y
map_smul'✝ :
∀ (m : R) (x : M),
{ toFun := g, map_add' := map_add'✝ }.toFun (m • x) =
(RingHom.id R) m • { toFun := g, map_add' := map_add'✝ }.toFun x
map_lie'✝ :
∀ {x : L} {m : M},
{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun ⁅x, m⁆ =
⁅x, { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun m⁆
h :
⇑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ } =
⇑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }
⊢ { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ } =
{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }
|
b0f29a1729ee1483
|
ContinuousLinearMap.adjoint_comp
|
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
theorem adjoint_comp (A : F →L[𝕜] G) (B : E →L[𝕜] F) : (A ∘L B)† = B† ∘L A†
|
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : F →L[𝕜] G
B : E →L[𝕜] F
v : G
⊢ (adjoint (A.comp B)) v = ((adjoint B).comp (adjoint A)) v
|
refine ext_inner_left 𝕜 fun w => ?_
|
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : F →L[𝕜] G
B : E →L[𝕜] F
v : G
w : E
⊢ ⟪w, (adjoint (A.comp B)) v⟫_𝕜 = ⟪w, ((adjoint B).comp (adjoint A)) v⟫_𝕜
|
f64eda0712198575
|
IsPrincipalIdealRing.of_prime
|
Mathlib/RingTheory/PrincipalIdealDomainOfPrime.lean
|
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
IsPrincipalIdealRing R
|
R : Type u_1
inst✝ : CommRing R
H : ∀ (P : Ideal R), P.IsPrime → Submodule.IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
hJI : J ≤ I
hI : Maximal (fun x => x ∈ nonPrincipals R) I
Imax' : ∀ {J : Ideal R}, I < J → Submodule.IsPrincipal J
hI1 : ¬I = ⊤
x a b v : R
hxy : x * (v * a) ∈ I
hy : v * a ∉ I
ha : I ⊔ span {v * a} = Submodule.span R {a}
hb : Submodule.colon I (span {v * a}) = Submodule.span R {b}
u : R
hi : u * a ∈ I
⊢ u * a * v ∈ I
|
exact mul_mem_right _ _ hi
|
no goals
|
a2a0a20bfddf7c1c
|
LaurentSeries.Cauchy.exists_lb_eventual_support
|
Mathlib/RingTheory/LaurentSeries.lean
|
lemma Cauchy.exists_lb_eventual_support {ℱ : Filter K⸨X⸩} (hℱ : Cauchy ℱ) :
∃ N, ∀ᶠ f : K⸨X⸩ in ℱ, ∀ n < N, f.coeff n = (0 : K)
|
case pos
K : Type u_2
inst✝ : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) := {P | Valued.v (P.2 - P.1) < ↑(ofAdd 0)}
ζ : ℤₘ₀ˣ := Units.mk0 ↑(ofAdd 0) ⋯
S : Set K⸨X⸩
hS : S ∈ ℱ
T : Set K⸨X⸩
hT : T ∈ ℱ
H : S ×ˢ T ⊆ entourage
f : K⸨X⸩
hf✝ : f ∈ S ∩ T
hf : f = 0
x : K⸨X⸩
hg : Valued.v x ≤ ↑(ofAdd 0)
⊢ ∀ n < 0, x.coeff n = 0
|
exact (valuation_le_iff_coeff_lt_eq_zero K).mp hg
|
no goals
|
3e2bae03010f2af6
|
spectrum.subset_polynomial_aeval
|
Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean
|
theorem subset_polynomial_aeval (a : A) (p : 𝕜[X]) : (eval · p) '' σ a ⊆ σ (aeval a p)
|
𝕜 : Type u
A : Type v
inst✝² : Field 𝕜
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
a : A
p : 𝕜[X]
⊢ (fun x => eval x p) '' σ a ⊆ σ ((aeval a) p)
|
rintro _ ⟨k, hk, rfl⟩
|
case intro.intro
𝕜 : Type u
A : Type v
inst✝² : Field 𝕜
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
a : A
p : 𝕜[X]
k : 𝕜
hk : k ∈ σ a
⊢ (fun x => eval x p) k ∈ σ ((aeval a) p)
|
0d43137f73938891
|
MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
|
Mathlib/Probability/Martingale/Upcrossing.lean
|
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
M : ℕ
hNM : N ≤ M
h : lowerCrossingTime a b f N n ω < N
⊢ upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
|
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
M : ℕ
hNM : N ≤ M
h : lowerCrossingTime a b f N n ω < N
h' : upperCrossingTime a b f N n ω < N
⊢ upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
defdbed08859d272
|
Continuous.integrableAt_nhds
|
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
theorem Continuous.integrableAt_nhds [TopologicalSpace α] [SecondCountableTopologyEither α E]
[OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ] {f : α → E}
(hf : Continuous f) (a : α) : IntegrableAtFilter f (𝓝 a) μ
|
α : Type u_1
E : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous f
a : α
⊢ IntegrableAtFilter f (𝓝 a) μ
|
rw [← nhdsWithin_univ]
|
α : Type u_1
E : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous f
a : α
⊢ IntegrableAtFilter f (𝓝[univ] a) μ
|
61ea7baddfe6310e
|
ComplexShape.Embedding.boundaryLE
|
Mathlib/Algebra/Homology/Embedding/Boundary.lean
|
lemma boundaryLE {k' : ι'} {j : ι} (hj : c'.Rel (e.f j) k') (hk' : ∀ i, e.f i ≠ k') :
e.BoundaryLE j
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
⊢ e.BoundaryLE j
|
constructor
|
case left
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
⊢ c'.Rel (e.f j) (c'.next (e.f j))
case right
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
⊢ ∀ (k : ι), ¬c'.Rel (e.f j) (e.f k)
|
7500d492e2c04792
|
Con.map_of_mul_left_rel_one
|
Mathlib/GroupTheory/Congruence/Defs.lean
|
theorem map_of_mul_left_rel_one [Monoid M] (c : Con M)
(f : M → M) (hf : ∀ x, c (f x * x) 1) {x y} (h : c x y) : c (f x) (f y)
|
M : Type u_1
inst✝ : Monoid M
c : Con M
f : M → M
hf : ∀ (x : M), c (f x * x) 1
x y : M
h : c x y
⊢ c (f x) (f y)
|
simp only [← Con.eq, coe_one, coe_mul] at *
|
M : Type u_1
inst✝ : Monoid M
c : Con M
f : M → M
x y : M
hf : ∀ (x : M), ↑(f x) * ↑x = 1
h : ↑x = ↑y
⊢ ↑(f x) = ↑(f y)
|
d7e45b31b3319c7c
|
Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
|
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
theorem two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable {R : ℝ}
{c w : ℂ} {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ ball c R)
(hc : ContinuousOn f (closedBall c R)) (hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x) :
((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) = f w
|
case intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
R : ℝ≥0
hw : w ∈ ball c ↑R
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ x ∈ ball c ↑R \ s, DifferentiableAt ℂ f x
hR : 0 < ↑R
this : w ∈ closure (ball c ↑R \ s)
A : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w
B : ContinuousAt f w
⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w
|
refine tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono ?_)
|
case intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
R : ℝ≥0
hw : w ∈ ball c ↑R
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ x ∈ ball c ↑R \ s, DifferentiableAt ℂ f x
hR : 0 < ↑R
this : w ∈ closure (ball c ↑R \ s)
A : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w
B : ContinuousAt f w
⊢ ∀ x ∈ ball c ↑R \ s, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - x)⁻¹ • f z) = f x
|
6fb0641025ed7482
|
SimpleGraph.Walk.map_append
|
Mathlib/Combinatorics/SimpleGraph/Walk.lean
|
theorem map_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :
(p.append q).map f = (p.map f).append (q.map f)
|
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u v w : V
p : G.Walk u v
q : G.Walk v w
⊢ Walk.map f (p.append q) = (Walk.map f p).append (Walk.map f q)
|
induction p <;> simp [*]
|
no goals
|
aed9d2f973b758e7
|
AlgebraicGeometry.Scheme.mem_toGrothendieck_smallPretopology
|
Mathlib/AlgebraicGeometry/Sites/Small.lean
|
lemma mem_toGrothendieck_smallPretopology (X : Q.Over ⊤ S) (R : Sieve X) :
R ∈ (S.smallPretopology P Q).toGrothendieck _ X ↔
∀ x : X.left, ∃ (Y : Q.Over ⊤ S) (f : Y ⟶ X) (y : Y.left),
R f ∧ P f.left ∧ f.left.base y = x
|
case refine_1.intro.intro.intro.intro.intro.intro
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁶ : P.IsMultiplicative
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.IsStableUnderBaseChange
inst✝³ : IsJointlySurjectivePreserving P
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover P X.left
h : Cover.Over S 𝒰
p : ∀ (j : 𝒰.J), Q (𝒰.obj j ↘ S)
hle : 𝒰.toPresieveOverProp p ≤ R.arrows
x : ↑↑X.left.toPresheafedSpace
y : ↑↑(𝒰.obj (𝒰.f x)).toPresheafedSpace
hy : (ConcreteCategory.hom (𝒰.map (𝒰.f x)).base) y = x
⊢ ∃ Y f y, R.arrows f ∧ P f.left ∧ (ConcreteCategory.hom f.left.base) y = x
|
refine ⟨(𝒰.obj (𝒰.f x)).asOverProp S (p _), (𝒰.map (𝒰.f x)).asOverProp S, y, hle _ ?_,
𝒰.map_prop _, hy⟩
|
case refine_1.intro.intro.intro.intro.intro.intro
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁶ : P.IsMultiplicative
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.IsStableUnderBaseChange
inst✝³ : IsJointlySurjectivePreserving P
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
X : Q.Over ⊤ S
R : Sieve X
𝒰 : Cover P X.left
h : Cover.Over S 𝒰
p : ∀ (j : 𝒰.J), Q (𝒰.obj j ↘ S)
hle : 𝒰.toPresieveOverProp p ≤ R.arrows
x : ↑↑X.left.toPresheafedSpace
y : ↑↑(𝒰.obj (𝒰.f x)).toPresheafedSpace
hy : (ConcreteCategory.hom (𝒰.map (𝒰.f x)).base) y = x
⊢ Hom.asOverProp (𝒰.map (𝒰.f x)) S ∈ 𝒰.toPresieveOverProp p
|
f1138a154afc8db2
|
monadLift_seq
|
Mathlib/.lake/packages/batteries/Batteries/Control/Lawful/MonadLift.lean
|
theorem monadLift_seq [LawfulMonad m] [LawfulMonad n] (mf : m (α → β)) (ma : m α) :
monadLift (mf <*> ma) = monadLift mf <*> (monadLift ma : n α)
|
m : Type u_1 → Type u_2
n : Type u_1 → Type u_3
inst✝⁵ : Monad m
inst✝⁴ : Monad n
inst✝³ : MonadLiftT m n
inst✝² : LawfulMonadLiftT m n
α β : Type u_1
inst✝¹ : LawfulMonad m
inst✝ : LawfulMonad n
mf : m (α → β)
ma : m α
⊢ monadLift (mf <*> ma) = monadLift mf <*> monadLift ma
|
simp only [seq_eq_bind, monadLift_map, monadLift_bind]
|
no goals
|
e28632888a82ae78
|
Array.countP_eq_size
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Count.lean
|
theorem countP_eq_size {p} : countP p l = l.size ↔ ∀ a ∈ l, p a
|
α✝ : Type u_1
l : Array α✝
p : α✝ → Bool
⊢ countP p l = l.size ↔ ∀ (a : α✝), a ∈ l → p a = true
|
cases l
|
case mk
α✝ : Type u_1
p : α✝ → Bool
toList✝ : List α✝
⊢ countP p { toList := toList✝ } = { toList := toList✝ }.size ↔ ∀ (a : α✝), a ∈ { toList := toList✝ } → p a = true
|
126e68e7ae0b7625
|
HasFPowerSeriesWithinOnBall.fderivWithin_of_mem
|
Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
|
theorem HasFPowerSeriesWithinOnBall.fderivWithin_of_mem [CompleteSpace F]
(h : HasFPowerSeriesWithinOnBall f p s x r) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r
|
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r
|
have : insert x s = s := insert_eq_of_mem hx
|
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
this : insert x s = s
⊢ HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r
|
3b099c76b2a60ea3
|
Filter.filter_injOn_Iic_iff_injOn
|
Mathlib/Order/Filter/Map.lean
|
theorem Filter.filter_injOn_Iic_iff_injOn {s : Set α} {m : α → β} :
InjOn (map m) (Iic <| 𝓟 s) ↔ InjOn m s
|
case refine_2
α : Type u_1
β : Type u_2
s : Set α
m : α → β
hm : InjOn m s
F : Filter α
hF : F ∈ Iic (𝓟 s)
G : Filter α
hG : G ∈ Iic (𝓟 s)
⊢ map m F = map m G → F = G
|
simp [map_eq_map_iff_of_injOn (le_principal_iff.mp hF) (le_principal_iff.mp hG) hm]
|
no goals
|
d6e29efebf50b69f
|
MeasureTheory.mul_le_integral_rnDeriv_of_ac
|
Mathlib/MeasureTheory/Decomposition/IntegralRNDeriv.lean
|
/-- For a convex continuous function `f` on `[0, ∞)`, if `μ` is absolutely continuous
with respect to `ν`, then
`(ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal) ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν`. -/
lemma mul_le_integral_rnDeriv_of_ac [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hf_cvx : ConvexOn ℝ (Ici 0) f) (hf_cont : ContinuousWithinAt f (Ici 0) 0)
(hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν) (hμν : μ ≪ ν) :
(ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal)
≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν
|
case neg.refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x => f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure α := (ν univ)⁻¹ • μ
ν' : Measure α := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᶠ[ae ν] μ.rnDeriv ν
h_eq : ∫ (x : α), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν univ).toReal⁻¹ * ∫ (x : α), f (μ.rnDeriv ν x).toReal ∂ν
⊢ Integrable (fun x => f (μ'.rnDeriv ν' x).toReal) ν
|
refine (integrable_congr ?_).mpr hf_int
|
case neg.refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x => f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure α := (ν univ)⁻¹ • μ
ν' : Measure α := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᶠ[ae ν] μ.rnDeriv ν
h_eq : ∫ (x : α), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν univ).toReal⁻¹ * ∫ (x : α), f (μ.rnDeriv ν x).toReal ∂ν
⊢ (fun x => f (μ'.rnDeriv ν' x).toReal) =ᶠ[ae ν] fun x => f (μ.rnDeriv ν x).toReal
|
0004fdd48300255a
|
DualNumber.lift_apply_inl
|
Mathlib/Algebra/DualNumber.lean
|
theorem lift_apply_inl (fe : {fe : (A →ₐ[R] B) × B // _}) (a : A) :
lift fe (inl a : A[ε]) = fe.val.1 a
|
R : Type u_1
B : Type u_3
A : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
fe : { fe // fe.2 * fe.2 = 0 ∧ ∀ (a : A), Commute fe.2 (fe.1 a) }
a : A
⊢ (lift fe) (inl a) = (↑fe).1 a
|
rw [lift_apply_apply, fst_inl, snd_inl, map_zero, zero_mul, add_zero]
|
no goals
|
77744d62c26c2692
|
AddMonoidHom.toNatLinearMap_injective
|
Mathlib/Algebra/Module/LinearMap/Defs.lean
|
theorem AddMonoidHom.toNatLinearMap_injective [AddCommMonoid M] [AddCommMonoid M₂] :
Function.Injective (@AddMonoidHom.toNatLinearMap M M₂ _ _)
|
case h
M : Type u_8
M₂ : Type u_10
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid M₂
f g : M →+ M₂
h : f.toNatLinearMap = g.toNatLinearMap
x : M
⊢ f x = g x
|
exact LinearMap.congr_fun h x
|
no goals
|
1670b4edf3bacc33
|
Set.Finite.toFinset_compl
|
Mathlib/Data/Set/Finite/Basic.lean
|
theorem toFinset_compl [DecidableEq α] [Fintype α] (hs : s.Finite) (h : sᶜ.Finite) :
h.toFinset = hs.toFinsetᶜ
|
α : Type u
s : Set α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hs : s.Finite
h : sᶜ.Finite
⊢ h.toFinset = hs.toFinsetᶜ
|
ext
|
case h
α : Type u
s : Set α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hs : s.Finite
h : sᶜ.Finite
a✝ : α
⊢ a✝ ∈ h.toFinset ↔ a✝ ∈ hs.toFinsetᶜ
|
0419e5c1954818e8
|
Fin.snoc_update
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem snoc_update : snoc (update p i y) x = update (snoc p x) i.castSucc y
|
case h.e'_2.h.e'_6
n : ℕ
α : Fin (n + 1) → Sort u_1
x : α (last n)
p : (i : Fin n) → α i.castSucc
i : Fin n
y : α i.castSucc
j : Fin (n + 1)
h : ↑j < n
h' : j = i.castSucc
C1 : α i.castSucc = α j
this : update (snoc p x) j (cast C1 y) j = cast C1 y
e_5✝ : i.castSucc = j
⊢ HEq y (cast C1 y)
|
exact heq_of_cast_eq (congr_arg α (Eq.symm h')) rfl
|
no goals
|
4eef9dec140e05c3
|
Submodule.spanRank_finite_iff_fg
|
Mathlib/Algebra/Module/SpanRank.lean
|
/-- A submodule's `spanRank` is finite if and only if it is finitely generated. -/
@[simp]
lemma spanRank_finite_iff_fg {p : Submodule R M} : p.spanRank < aleph0 ↔ p.FG
|
case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, #↑↑s < ℵ₀ → ∃ S, S.Finite ∧ span R S = p
|
rintro h
|
case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : ⨅ s, #↑↑s < ℵ₀
⊢ ∃ S, S.Finite ∧ span R S = p
|
57fd9ea29407f8a5
|
MvPolynomial.aeval_sumElim
|
Mathlib/Algebra/MvPolynomial/Eval.lean
|
lemma aeval_sumElim {σ τ : Type*} (p : MvPolynomial (σ ⊕ τ) R) (f : τ → S) (g : σ → T) :
(aeval (Sum.elim g (algebraMap S T ∘ f))) p =
(aeval g) ((aeval (Sum.elim X (C ∘ f))) p)
|
case h_add
R : Type u
inst✝⁶ : CommSemiring R
S : Type u_2
T : Type u_3
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
inst✝³ : CommSemiring T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
σ : Type u_4
τ : Type u_5
f : τ → S
g : σ → T
p q : MvPolynomial (σ ⊕ τ) R
hp : (aeval (Sum.elim g (⇑(algebraMap S T) ∘ f))) p = (aeval g) ((aeval (Sum.elim X (⇑C ∘ f))) p)
hq : (aeval (Sum.elim g (⇑(algebraMap S T) ∘ f))) q = (aeval g) ((aeval (Sum.elim X (⇑C ∘ f))) q)
⊢ (aeval (Sum.elim g (⇑(algebraMap S T) ∘ f))) (p + q) = (aeval g) ((aeval (Sum.elim X (⇑C ∘ f))) (p + q))
|
simp [hp, hq]
|
no goals
|
d1f6b73e513987b6
|
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
i : ℕ
x✝ : i ∈ range p
⊢ bernoulli (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2) =
bernoulli' (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2)
|
rw [bernoulli_eq_bernoulli'_of_ne_one (succ_succ_ne_one i)]
|
no goals
|
0112bf36c164f141
|
Finsupp.range_mapRange
|
Mathlib/Data/Finsupp/Defs.lean
|
lemma range_mapRange (e : M → N) (he₀ : e 0 = 0) :
Set.range (Finsupp.mapRange (α := α) e he₀) = {g | ∀ i, g i ∈ Set.range e}
|
case h.mpr
α : Type u_1
M : Type u_5
N : Type u_7
inst✝¹ : Zero M
inst✝ : Zero N
e : M → N
he₀ : e 0 = 0
g : α →₀ N
h : ∀ (i : α), ∃ y, e y = g i
⊢ ∃ y, mapRange e he₀ y = g
|
classical
choose f h using h
use onFinset g.support (Set.indicator g.support f) (by aesop)
ext i
simp only [mapRange_apply, onFinset_apply, Set.indicator_apply]
split_ifs <;> simp_all
|
no goals
|
8e042fcc877c4ad7
|
List.erase_toArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean
|
theorem erase_toArray [BEq α] {as : List α} {a : α} :
as.toArray.erase a = (as.erase a).toArray
|
α : Type u_1
inst✝ : BEq α
as : List α
a : α
⊢ (match finIdxOf? a as with
| none => as.toArray
| some i => as.toArray.eraseIdx ↑i ⋯) =
(match idxOf? a as with
| none => as
| some i => as.eraseIdx i).toArray
|
rw [idxOf?_eq_map_finIdxOf?_val]
|
α : Type u_1
inst✝ : BEq α
as : List α
a : α
⊢ (match finIdxOf? a as with
| none => as.toArray
| some i => as.toArray.eraseIdx ↑i ⋯) =
(match Option.map (fun x => ↑x) (finIdxOf? a as) with
| none => as
| some i => as.eraseIdx i).toArray
|
a257ecf7833886f3
|
VectorField.pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt_of_eventuallyEq
|
Mathlib/Analysis/Calculus/VectorField.lean
|
/-- The Lie bracket commutes with taking pullbacks. This requires the function to have symmetric
second derivative. Version in a complete space. One could also give a version avoiding
completeness but requiring that `f` is a local diffeo. Variant where unique differentiability and
the invariance property are only required in a smaller set `u`. -/
lemma pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt_of_eventuallyEq
{f : E → F} {V W : F → F} {x : E} {t : Set F} {u : Set E}
(hf : IsSymmSndFDerivWithinAt 𝕜 f s x) (h'f : ContDiffWithinAt 𝕜 2 f s x)
(hV : DifferentiableWithinAt 𝕜 V t (f x)) (hW : DifferentiableWithinAt 𝕜 W t (f x))
(hu : UniqueDiffOn 𝕜 u) (hx : x ∈ u) (hst : MapsTo f u t) (hus : u =ᶠ[𝓝 x] s) :
pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) s x
= lieBracketWithin 𝕜 (pullbackWithin 𝕜 f V s) (pullbackWithin 𝕜 f W s) s x := calc
pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) s x
_ = pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) u x
|
case a
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
inst✝ : CompleteSpace E
f : E → F
V W : F → F
x : E
t : Set F
u : Set E
hf : IsSymmSndFDerivWithinAt 𝕜 f s x
h'f : ContDiffWithinAt 𝕜 2 f s x
hV : DifferentiableWithinAt 𝕜 V t (f x)
hW : DifferentiableWithinAt 𝕜 W t (f x)
hu : UniqueDiffOn 𝕜 u
hx : x ∈ u
hst : MapsTo f u t
hus : u =ᶠ[𝓝 x] s
⊢ {x | (fun x => pullbackWithin 𝕜 f W u x = pullbackWithin 𝕜 f W s x) x} ∈ 𝓝 x
|
filter_upwards [fderivWithin_eventually_congr_set (𝕜 := 𝕜) (f := f) hus] with y hy
|
case h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
inst✝ : CompleteSpace E
f : E → F
V W : F → F
x : E
t : Set F
u : Set E
hf : IsSymmSndFDerivWithinAt 𝕜 f s x
h'f : ContDiffWithinAt 𝕜 2 f s x
hV : DifferentiableWithinAt 𝕜 V t (f x)
hW : DifferentiableWithinAt 𝕜 W t (f x)
hu : UniqueDiffOn 𝕜 u
hx : x ∈ u
hst : MapsTo f u t
hus : u =ᶠ[𝓝 x] s
y : E
hy : fderivWithin 𝕜 f u y = fderivWithin 𝕜 f s y
⊢ pullbackWithin 𝕜 f W u y = pullbackWithin 𝕜 f W s y
|
157b25f890fe6b72
|
CategoryTheory.ReflQuiver.homOfEq_id
|
Mathlib/Combinatorics/Quiver/ReflQuiver.lean
|
theorem ReflQuiver.homOfEq_id {V : Type*} [ReflQuiver V] {X X' : V} (hX : X = X') :
Quiver.homOfEq (𝟙rq X) hX hX = 𝟙rq X'
|
V : Type u_1
inst✝ : ReflQuiver V
X : V
⊢ Quiver.homOfEq (𝟙rq X) ⋯ ⋯ = 𝟙rq X
|
rfl
|
no goals
|
7ef3a697356fdcdf
|
MeasureTheory.MemLp.locallyIntegrable
|
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
theorem MemLp.locallyIntegrable [IsLocallyFiniteMeasure μ] {f : X → E} {p : ℝ≥0∞}
(hf : MemLp f p μ) (hp : 1 ≤ p) : LocallyIntegrable f μ
|
case intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
μ : Measure X
inst✝ : IsLocallyFiniteMeasure μ
f : X → E
p : ℝ≥0∞
hf : MemLp f p μ
hp : 1 ≤ p
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : μ U < ⊤
this : Fact (μ U < ⊤)
⊢ MemLp f 1 (μ.restrict U)
|
apply (hf.restrict U).mono_exponent hp
|
no goals
|
52b8c32a3c7b14a7
|
Nat.Subtype.toFunAux_eq
|
Mathlib/Logic/Denumerable.lean
|
theorem toFunAux_eq {s : Set ℕ} [DecidablePred (· ∈ s)] (x : s) :
toFunAux x = #{y ∈ Finset.range x | y ∈ s}
|
s : Set ℕ
inst✝ : DecidablePred fun x => x ∈ s
x : ↑s
⊢ (List.filter (fun x => decide (x ∈ s)) (List.range ↑x)).length = #(filter (fun y => y ∈ s) (range ↑x))
|
rfl
|
no goals
|
5627fbb4462b23ef
|
Finset.mem_insertNone
|
Mathlib/Data/Finset/Option.lean
|
theorem mem_insertNone {s : Finset α} : ∀ {o : Option α}, o ∈ insertNone s ↔ ∀ a ∈ o, a ∈ s
| none => iff_of_true (Multiset.mem_cons_self _ _) fun a h => by cases h
| some a => Multiset.mem_cons.trans <| by simp
|
α : Type u_1
s : Finset α
a : α
h : a ∈ none
⊢ a ∈ s
|
cases h
|
no goals
|
b90e86c3f5729ded
|
HolderWith.restrict_iff
|
Mathlib/Topology/MetricSpace/Holder.lean
|
theorem restrict_iff {s : Set X} : HolderWith C r (s.restrict f) ↔ HolderOnWith C r f s
|
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
⊢ HolderWith C r (s.restrict f) ↔ HolderOnWith C r f s
|
simp [HolderWith, HolderOnWith]
|
no goals
|
493be6dff4935bba
|
MeasureTheory.Measure.measure_preimage_of_map_eq_self
|
Mathlib/MeasureTheory/Measure/Map.lean
|
/-- If `map f μ = μ`, then the measure of the preimage of any null measurable set `s`
is equal to the measure of `s`.
Note that this lemma does not assume (a.e.) measurability of `f`. -/
lemma measure_preimage_of_map_eq_self {f : α → α} (hf : map f μ = μ)
{s : Set α} (hs : NullMeasurableSet s μ) : μ (f ⁻¹' s) = μ s
|
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → α
hf : map f μ = μ
s : Set α
hs : NullMeasurableSet s μ
hfm : AEMeasurable f μ
⊢ NullMeasurableSet s (map f μ)
|
rwa [hf]
|
no goals
|
1342c2772e8df2cf
|
HasCompactSupport.exists_simpleFunc_approx_of_prod
|
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
|
/-- A continuous function with compact support on a product space can be uniformly approximated by
simple functions. The subtlety is that we do not assume that the spaces are separable, so the
product of the Borel sigma algebras might not contain all open sets, but still it contains enough
of them to approximate compactly supported continuous functions. -/
lemma HasCompactSupport.exists_simpleFunc_approx_of_prod [PseudoMetricSpace α]
{f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f)
{ε : ℝ} (hε : 0 < ε) :
∃ (g : SimpleFunc (X × Y) α), ∀ x, dist (f x) (g x) < ε
|
case hunion
X : Type u_7
Y : Type u_8
α : Type u_9
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
⊢ ∀ ⦃s t : Set (X × Y)⦄,
(∃ g s_1, MeasurableSet s_1 ∧ s ⊆ s_1 ∧ ∀ x ∈ s_1, dist (f x) (g x) < ε) →
(∃ g s, MeasurableSet s ∧ t ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε) →
∃ g s_1, MeasurableSet s_1 ∧ s ∪ t ⊆ s_1 ∧ ∀ x ∈ s_1, dist (f x) (g x) < ε
|
intro t t' ⟨g, s, s_meas, ts, hg⟩ ⟨g', s', s'_meas, t's', hg'⟩
|
case hunion
X : Type u_7
Y : Type u_8
α : Type u_9
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
t t' : Set (X × Y)
g : SimpleFunc (X × Y) α
s : Set (X × Y)
s_meas : MeasurableSet s
ts : t ⊆ s
hg : ∀ x ∈ s, dist (f x) (g x) < ε
g' : SimpleFunc (X × Y) α
s' : Set (X × Y)
s'_meas : MeasurableSet s'
t's' : t' ⊆ s'
hg' : ∀ x ∈ s', dist (f x) (g' x) < ε
⊢ ∃ g s, MeasurableSet s ∧ t ∪ t' ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
|
45212510ef563462
|
DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta
|
Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
|
/-- The L function of the trivial Dirichlet character mod `N` is obtained from the Riemann
zeta function by multiplying with `∏ p ∈ N.primeFactors, (1 - (p : ℂ) ^ (-s))`. -/
lemma LFunctionTrivChar_eq_mul_riemannZeta {s : ℂ} (hs : s ≠ 1) :
LFunctionTrivChar N s = (∏ p ∈ N.primeFactors, (1 - (p : ℂ) ^ (-s))) * riemannZeta s
|
case h.e'_3.h.e'_6.a.h.e'_6
N : ℕ
inst✝ : NeZero N
s : ℂ
hs : s ≠ 1
p : ℕ
a✝ : p ∈ N.primeFactors
⊢ ↑p ^ (-s) = 1 ↑p * ↑p ^ (-s)
|
rw [MulChar.one_apply <| isUnit_of_subsingleton _, one_mul]
|
no goals
|
2ddfcc7305b76a9d
|
Bornology.isVonNBounded_of_smul_tendsto_zero
|
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
theorem isVonNBounded_of_smul_tendsto_zero {ε : ι → 𝕜} {l : Filter ι} [l.NeBot]
(hε : ∀ᶠ n in l, ε n ≠ 0) {S : Set E}
(H : ∀ x : ι → E, (∀ n, x n ∈ S) → Tendsto (ε • x) l (𝓝 0)) : IsVonNBounded 𝕜 S
|
case h.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
ε : ι → 𝕜
l : Filter ι
inst✝ : l.NeBot
hε : ∀ᶠ (n : ι) in l, ε n ≠ 0
S : Set E
H : ∀ (x : ι → E), (∀ (n : ι), x n ∈ S) → Tendsto (ε • x) l (𝓝 0)
V : Set E
hV : V ∈ 𝓝 0
hVb : Balanced 𝕜 V
n : ι
hn : ε n ≠ 0
hVS : ∀ (r : ℝ), ∃ c, r ≤ ‖c‖ ∧ ¬S ⊆ c • id V
a : 𝕜
haε : ‖(ε n)⁻¹‖ ≤ ‖a‖
haS : ¬S ⊆ a • id V
x : E
hxS : x ∈ S
hx : x ∉ a • id V
hnx : ε n • ↑⟨x, hxS⟩ ∈ V
⊢ False
|
rw [← Set.mem_inv_smul_set_iff₀ hn] at hnx
|
case h.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
ε : ι → 𝕜
l : Filter ι
inst✝ : l.NeBot
hε : ∀ᶠ (n : ι) in l, ε n ≠ 0
S : Set E
H : ∀ (x : ι → E), (∀ (n : ι), x n ∈ S) → Tendsto (ε • x) l (𝓝 0)
V : Set E
hV : V ∈ 𝓝 0
hVb : Balanced 𝕜 V
n : ι
hn : ε n ≠ 0
hVS : ∀ (r : ℝ), ∃ c, r ≤ ‖c‖ ∧ ¬S ⊆ c • id V
a : 𝕜
haε : ‖(ε n)⁻¹‖ ≤ ‖a‖
haS : ¬S ⊆ a • id V
x : E
hxS : x ∈ S
hx : x ∉ a • id V
hnx : ↑⟨x, hxS⟩ ∈ (ε n)⁻¹ • V
⊢ False
|
d3a90bc513e13615
|
MvPolynomial.constantCoeff_rename
|
Mathlib/Algebra/MvPolynomial/Rename.lean
|
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ
|
case h_X
σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
constantCoeff ((rename f) p) = constantCoeff p → constantCoeff ((rename f) (p * X n)) = constantCoeff (p * X n)
|
intro p n hp
|
case h_X
σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
τ : Type u_6
f : σ → τ
φ p : MvPolynomial σ R
n : σ
hp : constantCoeff ((rename f) p) = constantCoeff p
⊢ constantCoeff ((rename f) (p * X n)) = constantCoeff (p * X n)
|
99e6581b3c86be32
|
IsUltrametricDist.isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm
|
Mathlib/Analysis/Normed/Field/Ultra.lean
|
/-- This technical lemma is used in the proof of
`isUltrametricDist_of_forall_norm_natCast_le_one`. -/
lemma isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm
(h : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)) :
IsUltrametricDist R
|
case intro
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • (1 ⊔ ‖x‖ ^ m)
x : R
a : ℝ
ha : 1 ⊔ ‖x‖ < a
ha' : 1 < a
m : ℕ
hm : (m + 1) • (1 ⊔ ‖x‖) ^ m < a ^ m
hp : (1 ⊔ ‖x‖) ^ m = 1 ⊔ ‖x‖ ^ m
⊢ ‖x + 1‖ ≤ a
|
rw [hp] at hm
|
case intro
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • (1 ⊔ ‖x‖ ^ m)
x : R
a : ℝ
ha : 1 ⊔ ‖x‖ < a
ha' : 1 < a
m : ℕ
hm : (m + 1) • (1 ⊔ ‖x‖ ^ m) < a ^ m
hp : (1 ⊔ ‖x‖) ^ m = 1 ⊔ ‖x‖ ^ m
⊢ ‖x + 1‖ ≤ a
|
226284b7952ecdae
|
neg_one_geom_sum
|
Mathlib/Algebra/GeomSum.lean
|
theorem neg_one_geom_sum [Ring α] {n : ℕ} :
∑ i ∈ range n, (-1 : α) ^ i = if Even n then 0 else 1
|
case pos
α : Type u
inst✝ : Ring α
k : ℕ
hk : ∑ i ∈ range k, (-1) ^ i = if Even k then 0 else 1
h : Even k
⊢ (-1) ^ k + 0 = 1
|
rw [h.neg_one_pow, add_zero]
|
no goals
|
718ff43bff65c1c1
|
Module.finitePresentation_of_surjective
|
Mathlib/Algebra/Module/FinitePresentation.lean
|
lemma Module.finitePresentation_of_surjective [h : Module.FinitePresentation R M] (l : M →ₗ[R] N)
(hl : Function.Surjective l) (hl' : (LinearMap.ker l).FG) :
Module.FinitePresentation R N
|
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : FinitePresentation R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : (LinearMap.ker l).FG
⊢ FinitePresentation R N
|
classical
obtain ⟨s, hs, hs'⟩ := h
obtain ⟨t, ht⟩ := hl'
have H : Function.Surjective (Finsupp.linearCombination R ((↑) : s → M)) :=
LinearMap.range_eq_top.mp
(by rw [range_linearCombination, Subtype.range_val, ← hs]; rfl)
apply Module.finitePresentation_of_free_of_surjective (l ∘ₗ linearCombination R Subtype.val)
(hl.comp H)
choose σ hσ using (show _ from H)
have : Finsupp.linearCombination R Subtype.val '' (σ '' t) = t := by
simp only [Set.image_image, hσ, Set.image_id']
rw [LinearMap.ker_comp, ← ht, ← this, ← Submodule.map_span, Submodule.comap_map_eq,
← Finset.coe_image]
exact Submodule.FG.sup ⟨_, rfl⟩ hs'
|
no goals
|
91cced66bee4e79e
|
Finset.filter_snd_eq_antidiagonal
|
Mathlib/Algebra/Order/Antidiag/Prod.lean
|
theorem filter_snd_eq_antidiagonal (n m : A) [DecidablePred (· = m)] [Decidable (m ≤ n)] :
filter (fun x : A × A ↦ x.snd = m) (antidiagonal n) = if m ≤ n then {(n - m, m)} else ∅
|
A : Type u_1
inst✝⁷ : OrderedAddCommMonoid A
inst✝⁶ : CanonicallyOrderedAdd A
inst✝⁵ : Sub A
inst✝⁴ : OrderedSub A
inst✝³ : AddLeftReflectLE A
inst✝² : HasAntidiagonal A
n m : A
inst✝¹ : DecidablePred fun x => x = m
inst✝ : Decidable (m ≤ n)
⊢ (fun x => x.2 = m) ∘ Prod.swap = fun x => x.1 = m
|
ext
|
case h.a
A : Type u_1
inst✝⁷ : OrderedAddCommMonoid A
inst✝⁶ : CanonicallyOrderedAdd A
inst✝⁵ : Sub A
inst✝⁴ : OrderedSub A
inst✝³ : AddLeftReflectLE A
inst✝² : HasAntidiagonal A
n m : A
inst✝¹ : DecidablePred fun x => x = m
inst✝ : Decidable (m ≤ n)
x✝ : A × A
⊢ ((fun x => x.2 = m) ∘ Prod.swap) x✝ ↔ x✝.1 = m
|
d99ee23af9b14623
|
FiniteField.Matrix.charpoly_pow_card
|
Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean
|
theorem FiniteField.Matrix.charpoly_pow_card {K : Type*} [Field K] [Fintype K] (M : Matrix n n K) :
(M ^ Fintype.card K).charpoly = M.charpoly
|
case inl.intro
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
h✝ : Nonempty n
p : ℕ
hp : CharP K p
this : CharP K p := hp
⊢ (M ^ Fintype.card K).charpoly = M.charpoly
|
rcases FiniteField.card K p with ⟨⟨k, kpos⟩, ⟨hp, hk⟩⟩
|
case inl.intro.intro.mk.intro
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
h✝ : Nonempty n
p : ℕ
hp✝ : CharP K p
this : CharP K p := hp✝
k : ℕ
kpos : 0 < k
hp : Nat.Prime p
hk : Fintype.card K = p ^ ↑⟨k, kpos⟩
⊢ (M ^ Fintype.card K).charpoly = M.charpoly
|
3e36a605f896f181
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.blastDivSubtractShift_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Operations/Udiv.lean
|
theorem blastDivSubtractShift_decl_eq (aig : AIG α) (falseRef trueRef : AIG.Ref aig)
(n d : AIG.RefVec aig w) (wn wr : Nat) (q r : AIG.RefVec aig w) :
∀ (idx : Nat) (h1) (h2),
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig.decls[idx]'h2 = aig.decls[idx]'h1
|
case h2.h.h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
res : BlastDivSubtractShiftOutput aig w
hres :
{
aig :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig,
wn := wn - 1, wr := wr + 1,
q :=
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec.cast
⋯,
r :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec,
hle := ⋯ } =
res
idx✝ : Nat
h1✝ : idx✝ < aig.decls.size
h2✝ :
idx✝ <
{
aig :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig,
wn := wn - 1, wr := wr + 1,
q :=
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec.cast
⋯,
r :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec,
hle := ⋯ }.aig.decls.size
⊢ idx✝ < (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig.decls.size
|
apply AIG.LawfulVecOperator.lt_size_of_lt_aig_size (f := blastUdiv.blastShiftConcat)
|
case h2.h.h.h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
res : BlastDivSubtractShiftOutput aig w
hres :
{
aig :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig,
wn := wn - 1, wr := wr + 1,
q :=
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec.cast
⋯,
r :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec,
hle := ⋯ } =
res
idx✝ : Nat
h1✝ : idx✝ < aig.decls.size
h2✝ :
idx✝ <
{
aig :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig,
wn := wn - 1, wr := wr + 1,
q :=
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec.cast
⋯,
r :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref,
lhs :=
(((blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).aig
{
discr :=
(BVPred.mkUlt
(blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).aig
{
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }).ref.cast
⋯,
lhs :=
(((((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast
⋯).cast
⋯).cast
⋯,
rhs :=
((blastSub
(blastShiftConcat
(blastShiftConcat
(blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig
{
lhs :=
((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast
⋯).cast
⋯,
rhs := ((d.cast ⋯).cast ⋯).cast ⋯ }).vec.cast
⋯).cast
⋯ }).vec,
hle := ⋯ }.aig.decls.size
⊢ idx✝ < aig.decls.size
|
76230fc57d8c20e2
|
Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne
|
Mathlib/RingTheory/DedekindDomain/PID.lean
|
theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {P : Ideal R}
(hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P ^ 2)
(hxQ : ∀ Q : Ideal R, IsPrime Q → Q ≠ P → x ∉ Q) : P = Ideal.span {x}
|
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
hP : P.IsPrime
inst✝ : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ P ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
this : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hx0 : x ≠ 0
⊢ P = span {x}
|
by_cases hP0 : P = ⊥
|
case pos
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
hP : P.IsPrime
inst✝ : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ P ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
this : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hx0 : x ≠ 0
hP0 : P = ⊥
⊢ P = span {x}
case neg
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
hP : P.IsPrime
inst✝ : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ P ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
this : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
hx0 : x ≠ 0
hP0 : ¬P = ⊥
⊢ P = span {x}
|
2ba74b55b2935b6c
|
AlgebraicGeometry.Scheme.zeroLocus_biInf
|
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
lemma Scheme.zeroLocus_biInf
{X : Scheme.{u}} {U : X.Opens} {ι : Type*}
(I : ι → Ideal Γ(X, U)) {t : Set ι} (ht : t.Finite) :
X.zeroLocus (U := U) ↑(⨅ i ∈ t, I i) = (⋃ i ∈ t, X.zeroLocus (U := U) (I i)) ∪ (↑U)ᶜ
|
X : Scheme
U : X.Opens
ι : Type u_1
I : ι → Ideal ↑Γ(X, U)
t✝ : Set ι
ht✝ : t✝.Finite
i : ι
t : Set ι
hit : i ∉ t
ht : t.Finite
IH : X.zeroLocus ↑(⨅ i ∈ t, I i) = (⋃ i ∈ t, X.zeroLocus ↑(I i)) ∪ (↑U)ᶜ
⊢ X.zeroLocus ↑(⨅ i_1 ∈ insert i t, I i_1) = (⋃ i_1 ∈ insert i t, X.zeroLocus ↑(I i_1)) ∪ (↑U)ᶜ
|
simp only [Set.mem_insert_iff, Set.iUnion_iUnion_eq_or_left, ← IH, ← zeroLocus_inf,
Submodule.inf_coe, Set.union_assoc]
|
X : Scheme
U : X.Opens
ι : Type u_1
I : ι → Ideal ↑Γ(X, U)
t✝ : Set ι
ht✝ : t✝.Finite
i : ι
t : Set ι
hit : i ∉ t
ht : t.Finite
IH : X.zeroLocus ↑(⨅ i ∈ t, I i) = (⋃ i ∈ t, X.zeroLocus ↑(I i)) ∪ (↑U)ᶜ
⊢ X.zeroLocus ↑(⨅ i_1, ⨅ (_ : i_1 = i ∨ i_1 ∈ t), I i_1) = X.zeroLocus (↑(I i) ∩ ↑(⨅ i ∈ t, I i))
|
f9c9044705bdeb37
|
Submodule.IsLattice.rank_of_pi
|
Mathlib/Algebra/Module/Lattice.lean
|
/-- Any `R`-lattice in `ι → K` has `#ι` as `R`-rank. -/
lemma rank_of_pi {ι : Type*} [Fintype ι] [IsFractionRing R K] (M : Submodule R (ι → K))
[IsLattice K M] : Module.rank R M = Fintype.card ι
|
R : Type u_1
inst✝⁷ : CommRing R
K : Type u_2
inst✝⁶ : Field K
inst✝⁵ : Algebra R K
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : IsFractionRing R K
M : Submodule R (ι → K)
inst✝ : IsLattice K M
⊢ Module.rank K (ι → K) = ↑(Fintype.card ι)
|
simp
|
no goals
|
242153634d45150d
|
balancedCoreAux_empty
|
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
theorem balancedCoreAux_empty : balancedCoreAux 𝕜 (∅ : Set E) = ∅
|
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
⊢ ∀ (a : E), ∃ i, ∃ (_ : 1 ≤ ‖i‖), a ∉ ∅
|
exact fun _ => ⟨1, norm_one.ge, not_mem_empty _⟩
|
no goals
|
888cdbecd2e5a5ba
|
MeasureTheory.SimpleFunc.induction
|
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
theorem induction {α γ} [MeasurableSpace α] [AddMonoid γ] {P : SimpleFunc α γ → Prop}
(h_ind :
∀ (c) {s} (hs : MeasurableSet s),
P (SimpleFunc.piecewise s hs (SimpleFunc.const _ c) (SimpleFunc.const _ 0)))
(h_add : ∀ ⦃f g : SimpleFunc α γ⦄, Disjoint (support f) (support g) → P f → P g → P (f + g))
(f : SimpleFunc α γ) : P f
|
α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → P f → P g → P (f + g)
f : α →ₛ γ
s : Finset γ
h : f.range \ {0} = s
⊢ P f
|
rw [← Finset.coe_inj, Finset.coe_sdiff, Finset.coe_singleton, SimpleFunc.coe_range] at h
|
α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → P f → P g → P (f + g)
f : α →ₛ γ
s : Finset γ
h : range ⇑f \ {0} = ↑s
⊢ P f
|
01b1b313375647ab
|
MeasureTheory.IntegrableOn.hasBoxIntegral
|
Mathlib/Analysis/BoxIntegral/Integrability.lean
|
theorem IntegrableOn.hasBoxIntegral [CompleteSpace E] {f : (ι → ℝ) → E} {μ : Measure (ι → ℝ)}
[IsLocallyFiniteMeasure μ] {I : Box ι} (hf : IntegrableOn f I μ) (l : IntegrationParams)
(hl : l.bRiemann = false) :
HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul (∫ x in I, f x ∂μ)
|
case intro.intro.intro.intro.intro.intro.intro.intro.refine_3
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi✝ : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
hfi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (⇑(f n)) (↑J) μ
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ
|
have hgi : ∀ J ∈ π, IntegrableOn g (↑J) μ := fun J hJ => hgi.mono_set (π.le_of_mem' J hJ)
|
case intro.intro.intro.intro.intro.intro.intro.intro.refine_3
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi✝ : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi✝ : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
hfi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (⇑(f n)) (↑J) μ
hgi : ∀ J ∈ π, IntegrableOn g (↑J) μ
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ
|
1cd867d174f2adc3
|
List.eraseIdx_modify_of_gt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean
|
theorem eraseIdx_modify_of_gt (f : α → α) (i j) (l : List α) (h : j > i) :
(modify f i l).eraseIdx j = (l.eraseIdx j).modify f i
|
case h
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j > i
k : Nat
h₁ : k < ((modify f i l).eraseIdx j).length
h₂ : k < (modify f i (l.eraseIdx j)).length
⊢ ((modify f i l).eraseIdx j)[k] = (modify f i (l.eraseIdx j))[k]
|
simp only [getElem_eraseIdx, getElem_modify]
|
case h
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j > i
k : Nat
h₁ : k < ((modify f i l).eraseIdx j).length
h₂ : k < (modify f i (l.eraseIdx j)).length
⊢ (if h : k < j then if i = k then f l[k] else l[k] else if i = k + 1 then f l[k + 1] else l[k + 1]) =
if i = k then f (if h' : k < j then l[k] else l[k + 1]) else if h' : k < j then l[k] else l[k + 1]
|
26512ea5cfe6dba0
|
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod
|
Mathlib/LinearAlgebra/FreeModule/Int.lean
|
/-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
(snf : Basis.SmithNormalForm N ι n) :
N.toAddSubgroup.index = Nat.card R ^ (Fintype.card ι - n) *
∏ i : Fin n, (Ideal.span {snf.a i}).toAddSubgroup.index
|
case mk
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN'✝ : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
hN' :
N'.toAddSubgroup =
AddSubgroup.pi Set.univ fun i => Submodule.toAddSubgroup (Ideal.span {if h : ∃ j, f j = i then a h.choose else 0})
⊢ Nat.card R ^ (Finset.filter (fun x => ¬∃ j, f j = x) Finset.univ).card *
∏ x ∈ (Finset.filter (fun x => ∃ j, f j = x) Finset.univ).attach,
(Submodule.toAddSubgroup (Ideal.span {a ⋯.choose})).index =
Nat.card R ^ (Fintype.card ι - n) * ∏ i : Fin n, (Submodule.toAddSubgroup (Ideal.span {a i})).index
|
congr
|
case mk.e_a.e_a
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN'✝ : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
hN' :
N'.toAddSubgroup =
AddSubgroup.pi Set.univ fun i => Submodule.toAddSubgroup (Ideal.span {if h : ∃ j, f j = i then a h.choose else 0})
⊢ (Finset.filter (fun x => ¬∃ j, f j = x) Finset.univ).card = Fintype.card ι - n
case mk.e_a
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN'✝ : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
hN' :
N'.toAddSubgroup =
AddSubgroup.pi Set.univ fun i => Submodule.toAddSubgroup (Ideal.span {if h : ∃ j, f j = i then a h.choose else 0})
⊢ ∏ x ∈ (Finset.filter (fun x => ∃ j, f j = x) Finset.univ).attach,
(Submodule.toAddSubgroup (Ideal.span {a ⋯.choose})).index =
∏ i : Fin n, (Submodule.toAddSubgroup (Ideal.span {a i})).index
|
3d7a9cb7455a5c0f
|
Complex.partialGamma_add_one
|
Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
|
theorem partialGamma_add_one {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X) :
partialGamma (s + 1) X = s * partialGamma s X - (-X).exp * X ^ s
|
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
x : ℝ
hx : x ∈ Ioo 0 X
d1 : HasDerivAt (fun y => rexp (-y)) (-rexp (-x)) x
⊢ HasDerivAt (fun y => ↑y ^ s) (s * ↑x ^ (s - 1)) x
|
have t := @HasDerivAt.cpow_const _ _ _ s (hasDerivAt_id ↑x) ?_
|
case refine_2
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
x : ℝ
hx : x ∈ Ioo 0 X
d1 : HasDerivAt (fun y => rexp (-y)) (-rexp (-x)) x
t : HasDerivAt (fun x => id x ^ s) (s * id ↑x ^ (s - 1) * 1) ↑x
⊢ HasDerivAt (fun y => ↑y ^ s) (s * ↑x ^ (s - 1)) x
case refine_1
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
x : ℝ
hx : x ∈ Ioo 0 X
d1 : HasDerivAt (fun y => rexp (-y)) (-rexp (-x)) x
⊢ id ↑x ∈ slitPlane
|
64c3078f05c8af7b
|
CauSeq.lim_inv
|
Mathlib/Algebra/Order/CauSeq/Completion.lean
|
theorem lim_inv {f : CauSeq β abv} (hf : ¬LimZero f) : lim (inv f hf) = (lim f)⁻¹ :=
have hl : lim f ≠ 0
|
α : Type u_1
inst✝³ : LinearOrderedField α
β : Type u_2
inst✝² : Field β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : IsComplete β abv
f✝ : CauSeq β abv
hf✝ : ¬f✝.LimZero
hl : f✝.lim ≠ 0
g f : CauSeq β abv
hf : ¬f.LimZero
h₂ : g - f * f.inv hf * g = 1 * g - f * f.inv hf * g
h₃ : f * f.inv hf * g = f * f.inv hf * g
⊢ (g - f * f.inv hf * g).LimZero
|
have h₄ : g - f * inv f hf * g = (1 - f * inv f hf) * g := by rw [h₂, h₃, ← sub_mul]
|
α : Type u_1
inst✝³ : LinearOrderedField α
β : Type u_2
inst✝² : Field β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : IsComplete β abv
f✝ : CauSeq β abv
hf✝ : ¬f✝.LimZero
hl : f✝.lim ≠ 0
g f : CauSeq β abv
hf : ¬f.LimZero
h₂ : g - f * f.inv hf * g = 1 * g - f * f.inv hf * g
h₃ : f * f.inv hf * g = f * f.inv hf * g
h₄ : g - f * f.inv hf * g = (1 - f * f.inv hf) * g
⊢ (g - f * f.inv hf * g).LimZero
|
24d312013bd260a5
|
List.Nodup.sym2
|
Mathlib/Data/List/Sym.lean
|
theorem Nodup.sym2 {xs : List α} (h : xs.Nodup) : xs.sym2.Nodup
|
case cons
α : Type u_1
x : α
xs : List α
h : x ∉ xs ∧ xs.Nodup
ih : xs.sym2.Nodup
⊢ (map (fun y => s(x, y)) (x :: xs) ++ xs.sym2).Nodup
|
refine Nodup.append (Nodup.cons ?notmem (h.2.map ?inj)) ih ?disj
|
case notmem
α : Type u_1
x : α
xs : List α
h : x ∉ xs ∧ xs.Nodup
ih : xs.sym2.Nodup
⊢ (fun y => s(x, y)) x ∉ map (fun y => s(x, y)) xs
case inj
α : Type u_1
x : α
xs : List α
h : x ∉ xs ∧ xs.Nodup
ih : xs.sym2.Nodup
⊢ Function.Injective fun y => s(x, y)
case disj
α : Type u_1
x : α
xs : List α
h : x ∉ xs ∧ xs.Nodup
ih : xs.sym2.Nodup
⊢ (map (fun y => s(x, y)) (x :: xs)).Disjoint xs.sym2
|
85e54f9f7415b3ef
|
Pell.eq_pell_lem
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
theorem eq_pell_lem : ∀ (n) (b : ℤ√(d a1)), 1 ≤ b → IsPell b →
b ≤ pellZd a1 n → ∃ n, b = pellZd a1 n
| 0, _ => fun h1 _ hl => ⟨0, @Zsqrtd.le_antisymm _ (dnsq a1) _ _ hl h1⟩
| n + 1, b => fun h1 hp h =>
have a1p : (0 : ℤ√(d a1)) ≤ ⟨a, 1⟩ := trivial
have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp; exact Nat.pred_le _
have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)
if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then
let ⟨m, e⟩ :=
eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)
(isPell_mul hp (isPell_star.1 (isPell_one a1)))
(by
have t := mul_le_mul_of_nonneg_right h am1p
rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t)
⟨m + 1, by
rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,
pellZd_succ, e]⟩
else
suffices ¬1 < b from ⟨0, show b = 1 from (Or.resolve_left (lt_or_eq_of_le h1) this).symm⟩
fun h1l => by
obtain ⟨x, y⟩ := b
exact by
have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp
have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=
sub_lt_sub h1l fun hn : (1 : ℤ√d a1) ≤ ⟨x, -y⟩ => by
have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)
rw [bm, mul_one] at t
exact h1l t
have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=
show (⟨x, y⟩ - ⟨x, -y⟩ : ℤ√d a1) < ⟨a, 1⟩ - ⟨a, -1⟩ from
sub_lt_sub ha fun hn : (⟨x, -y⟩ : ℤ√d a1) ≤ ⟨a, -1⟩ => by
have t := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p
rw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t
exact ha t
simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.neg_re, add_neg_cancel,
Zsqrtd.neg_im, neg_neg] at yl2
exact
match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with
| 0, y0l, _ => y0l (le_refl 0)
| (y + 1 : ℕ), _, yl2 =>
yl2
(Zsqrtd.le_of_le_le (by simp [sub_eq_add_neg])
(let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)
add_le_add t t))
| Int.negSucc _, y0l, _ => y0l trivial
|
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
⊢ { re := ↑a, im := 1 } * { re := ↑a, im := -1 } ≤ b * { re := ↑a, im := -1 }
|
exact mul_le_mul_of_nonneg_right ha am1p
|
no goals
|
f5b138704daee74a
|
SetTheory.PGame.le_neg_iff
|
Mathlib/SetTheory/Game/PGame.lean
|
theorem le_neg_iff {x y : PGame} : y ≤ -x ↔ x ≤ -y
|
x y : PGame
⊢ y ≤ -x ↔ x ≤ -y
|
rw [← neg_neg x, neg_le_neg_iff, neg_neg]
|
no goals
|
6699b30833875119
|
isIntegral_discr_mul_of_mem_traceDual
|
Mathlib/RingTheory/DedekindDomain/Different.lean
|
/-- If `b` is an `A`-integral basis of `L` with discriminant `b`, then `d • a * x` is integral over
`A` for all `a ∈ I` and `x ∈ Iᵛ`. -/
lemma isIntegral_discr_mul_of_mem_traceDual
(I : Submodule B L) {ι} [DecidableEq ι] [Fintype ι]
{b : Basis ι K L} (hb : ∀ i, IsIntegral A (b i))
{a x : L} (ha : a ∈ I) (hx : x ∈ Iᵛ) :
IsIntegral A ((discr K b) • a * x)
|
case h
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H : ((traceMatrix K ⇑b).cramer fun i => (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
⊢ IsIntegral A (b.equivFun (discr K ⇑b • a * x) i • b i)
|
rw [smul_mul_assoc, b.equivFun.map_smul, discr_def, mul_comm, ← H, Algebra.smul_def]
|
case h
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H : ((traceMatrix K ⇑b).cramer fun i => (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
⊢ IsIntegral A ((algebraMap K L) ((traceMatrix K ⇑b).cramer (fun i => (Algebra.trace K L) (x * a * b i)) i) * b i)
|
14fab2851c4d1928
|
UpperHalfPlane.c_mul_im_sq_le_normSq_denom
|
Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
|
theorem c_mul_im_sq_le_normSq_denom : (g 1 0 * z.im) ^ 2 ≤ Complex.normSq (denom g z)
|
g : ↥GL(2, ℝ)⁺
z : ℍ
c : ℝ := ↑↑g 1 0
d : ℝ := ↑↑g 1 1
⊢ (c * z.im) ^ 2 ≤ Complex.normSq (denom g z)
|
calc
(c * z.im) ^ 2 ≤ (c * z.im) ^ 2 + (c * z.re + d) ^ 2 := by nlinarith
_ = Complex.normSq (denom g z) := by dsimp [c, d, denom, Complex.normSq]; ring
|
no goals
|
8ae3c8cd85e97f70
|
FormalMultilinearSeries.compContinuousLinearMap_applyComposition
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem compContinuousLinearMap_applyComposition {n : ℕ} (p : FormalMultilinearSeries 𝕜 F G)
(f : E →L[𝕜] F) (c : Composition n) (v : Fin n → E) :
(p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (f ∘ v)
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 F G
f : E →L[𝕜] F
c : Composition n
v : Fin n → E
⊢ (p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (⇑f ∘ v)
|
simp (config := {unfoldPartialApp := true}) [applyComposition]
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 F G
f : E →L[𝕜] F
c : Composition n
v : Fin n → E
⊢ (fun i => (p (c.blocksFun i)) (⇑f ∘ v ∘ ⇑(c.embedding i))) = fun i =>
(p (c.blocksFun i)) ((⇑f ∘ v) ∘ ⇑(c.embedding i))
|
dc21c96095630cbb
|
Submodule.fg_iff_compact
|
Mathlib/RingTheory/Finiteness/Basic.lean
|
theorem fg_iff_compact (s : Submodule R M) : s.FG ↔ CompleteLattice.IsCompactElement s
|
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
⊢ s.FG ↔ CompleteLattice.IsCompactElement s
|
let sp : M → Submodule R M := fun a => span R {a}
|
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
sp : M → Submodule R M := fun a => span R {a}
⊢ s.FG ↔ CompleteLattice.IsCompactElement s
|
caae8d7d512db78b
|
List.modify_modify_ne
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean
|
theorem modify_modify_ne (f g : α → α) {m n} (l : List α) (h : m ≠ n) :
(modify f m l).modify g n = (l.modify g n).modify f m
|
case h
α : Type u_1
f g : α → α
m n : Nat
l : List α
h : m ≠ n
⊢ ∀ (i : Nat) (h₁ : i < (modify g n (modify f m l)).length) (h₂ : i < (modify f m (modify g n l)).length),
(modify g n (modify f m l))[i] = (modify f m (modify g n l))[i]
|
intro m' h₁ h₂
|
case h
α : Type u_1
f g : α → α
m n : Nat
l : List α
h : m ≠ n
m' : Nat
h₁ : m' < (modify g n (modify f m l)).length
h₂ : m' < (modify f m (modify g n l)).length
⊢ (modify g n (modify f m l))[m'] = (modify f m (modify g n l))[m']
|
0e2d5adc4b70baf2
|
TopologicalSpace.exists_isInducing_l_infty
|
Mathlib/Topology/Metrizable/Urysohn.lean
|
theorem exists_isInducing_l_infty : ∃ f : X → ℕ →ᵇ ℝ, IsInducing f
|
case intro.intro.refine_1.inl
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.1 ⊆ UV.2}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x => ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x => { toFun := fun UV => (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : {UV | δ ≤ ε UV}.Finite
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist ((F y) UV) ((F x) UV) ≤ δ
y : X
hy : ∀ (UV : ↑s), δ ≤ ε UV → dist ((F y) UV) ((F x) UV) ≤ δ
UV : ↑s
hle : δ ≤ ε UV
⊢ dist ((F y) UV) ((F x) UV) ≤ δ
case intro.intro.refine_1.inr
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.1 ⊆ UV.2}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x => ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x => { toFun := fun UV => (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : {UV | δ ≤ ε UV}.Finite
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist ((F y) UV) ((F x) UV) ≤ δ
y : X
hy : ∀ (UV : ↑s), δ ≤ ε UV → dist ((F y) UV) ((F x) UV) ≤ δ
UV : ↑s
hle : ε UV ≤ δ
⊢ dist ((F y) UV) ((F x) UV) ≤ δ
|
exacts [hy _ hle, (Real.dist_le_of_mem_Icc (hf0ε _ _) (hf0ε _ _)).trans (by rwa [sub_zero])]
|
no goals
|
7bd5745f003136ce
|
CoxeterSystem.length_mul_simple_ne
|
Mathlib/GroupTheory/Coxeter/Length.lean
|
theorem length_mul_simple_ne (w : W) (i : B) : ℓ (w * s i) ≠ ℓ w
|
case inr
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
eq : cs.length (w * cs.simple i) = cs.length w
length_mod_two : cs.length w % 2 = (cs.length w + 1) % 2
odd : cs.length w % 2 = 1
⊢ False
|
rw [odd, Nat.succ_mod_two_eq_zero_iff.mpr odd] at length_mod_two
|
case inr
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
eq : cs.length (w * cs.simple i) = cs.length w
length_mod_two : 1 = 0
odd : cs.length w % 2 = 1
⊢ False
|
97a180c6ccbec595
|
ContDiff.sum
|
Mathlib/Analysis/Calculus/ContDiff/Operations.lean
|
theorem ContDiff.sum {ι : Type*} {f : ι → E → F} {s : Finset ι}
(h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x) : ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : WithTop ℕ∞
ι : Type u_3
f : ι → E → F
s : Finset ι
h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x
⊢ ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x
|
simp only [← contDiffOn_univ] at *
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : WithTop ℕ∞
ι : Type u_3
f : ι → E → F
s : Finset ι
h : ∀ i ∈ s, ContDiffOn 𝕜 n (fun x => f i x) univ
⊢ ContDiffOn 𝕜 n (fun x => ∑ i ∈ s, f i x) univ
|
b7f57f2f31cd5611
|
MeasureTheory.VectorMeasure.map_apply
|
Mathlib/MeasureTheory/VectorMeasure/Basic.lean
|
theorem map_apply {f : α → β} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) :
v.map f s = v (f ⁻¹' s)
|
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ ↑{ measureOf' := fun s => if MeasurableSet s then ↑v (f ⁻¹' s) else 0, empty' := ⋯, not_measurable' := ⋯,
m_iUnion' := ⋯ }
s =
↑v (f ⁻¹' s)
|
exact if_pos hs
|
no goals
|
6a65e07b1b9e8503
|
Nat.all_congr
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Fold.lean
|
theorem all_congr {n m : Nat} (w : n = m) (f : (i : Nat) → i < n → Bool) : all n f = all m (fun i h => f i (by omega))
|
n : Nat
f : (i : Nat) → i < n → Bool
⊢ n.all f = n.all fun i h => f i ⋯
|
rfl
|
no goals
|
2fa39c1067621dda
|
SimpleGraph.edgeSet_fromEdgeSet
|
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
theorem edgeSet_fromEdgeSet : (fromEdgeSet s).edgeSet = s \ { e | e.IsDiag }
|
V : Type u
s : Set (Sym2 V)
⊢ (fromEdgeSet s).edgeSet = s \ {e | e.IsDiag}
|
ext e
|
case h
V : Type u
s : Set (Sym2 V)
e : Sym2 V
⊢ e ∈ (fromEdgeSet s).edgeSet ↔ e ∈ s \ {e | e.IsDiag}
|
718852e4f7197f8b
|
Pi.image_update_segment
|
Mathlib/Analysis/Convex/Segment.lean
|
theorem image_update_segment (i : ι) (x₁ x₂ : π i) (y : ∀ i, π i) :
update y i '' [x₁ -[𝕜] x₂] = [update y i x₁ -[𝕜] update y i x₂]
|
𝕜 : Type u_1
ι : Type u_5
π : ι → Type u_6
inst✝³ : OrderedSemiring 𝕜
inst✝² : (i : ι) → AddCommMonoid (π i)
inst✝¹ : (i : ι) → Module 𝕜 (π i)
inst✝ : DecidableEq ι
i : ι
x₁ x₂ : π i
y : (i : ι) → π i
a : 𝕜 × 𝕜
ha : a ∈ {p | 0 ≤ p.1 ∧ 0 ≤ p.2 ∧ p.1 + p.2 = 1}
⊢ update y i (a.1 • x₁ + a.2 • x₂) = a.1 • update y i x₁ + a.2 • update y i x₂
|
simp only [← update_smul, ← update_add, Convex.combo_self ha.2.2]
|
no goals
|
79696759074d7b20
|
CStarModule.inner_smul_right_real
|
Mathlib/Analysis/CStarAlgebra/Module/Defs.lean
|
@[simp]
lemma inner_smul_right_real {z : ℝ} {x y : E} : ⟪x, z • y⟫ = z • ⟪x, y⟫
|
A : Type u_1
E : Type u_2
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℂ A
inst✝⁶ : Module ℂ E
inst✝⁵ : PartialOrder A
inst✝⁴ : SMul Aᵐᵒᵖ E
inst✝³ : Norm A
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarModule ℂ A
z : ℝ
x y : E
⊢ inner x (z •> y) = z •> inner x y
|
have h₁ : z • y = (z : ℂ) • y := by simp
|
A : Type u_1
E : Type u_2
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℂ A
inst✝⁶ : Module ℂ E
inst✝⁵ : PartialOrder A
inst✝⁴ : SMul Aᵐᵒᵖ E
inst✝³ : Norm A
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarModule ℂ A
z : ℝ
x y : E
h₁ : z •> y = ↑z •> y
⊢ inner x (z •> y) = z •> inner x y
|
292cfc3b7c127b19
|
Real.cos_bound
|
Mathlib/Data/Complex/Trigonometric.lean
|
theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = ‖Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)‖
|
x : ℝ
hx : |x| ≤ 1
⊢ cexp (↑x * I) + cexp (-↑x * I) - (2 - ↑x ^ 2) =
cexp (↑x * I) - ∑ m ∈ range 4, (↑x * I) ^ m / ↑m.factorial +
(cexp (-↑x * I) - ∑ m ∈ range 4, (-↑x * I) ^ m / ↑m.factorial)
|
simp only [neg_mul, pow_succ, pow_zero, sum_range_succ, range_zero, sum_empty,
Nat.factorial, Nat.cast_succ, zero_add, mul_one, Nat.mul_one, mul_neg, neg_neg]
|
x : ℝ
hx : |x| ≤ 1
⊢ cexp (↑x * I) + cexp (-(↑x * I)) - (2 - 1 * ↑x * ↑x) =
cexp (↑x * I) -
(1 / (↑0 + 1) + 1 * (↑x * I) / (↑0 + 1) + 1 * (↑x * I) * (↑x * I) / (↑0 + 1 + 1) +
1 * (↑x * I) * (↑x * I) * (↑x * I) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) +
(cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(1 * (↑x * I)) / (↑0 + 1) + 1 * (↑x * I) * (↑x * I) / (↑0 + 1 + 1) +
-(1 * (↑x * I) * (↑x * I) * (↑x * I)) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)))
|
fc89f3e7e59ce086
|
ProbabilityTheory.Kernel.integral_fn_integral_add_comp
|
Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean
|
theorem integral_fn_integral_add_comp ⦃f g : γ → E⦄ (F : E → E')
(hf : Integrable f ((η ∘ₖ κ) a)) (hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, F (∫ y, f y + g y ∂η x) ∂κ a = ∫ x, F (∫ y, f y ∂η x + ∫ y, g y ∂η x) ∂κ a
|
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝² : NormedSpace ℝ E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace ℝ E'
f g : γ → E
F : E → E'
hf : Integrable f ((η ∘ₖ κ) a)
hg : Integrable g ((η ∘ₖ κ) a)
⊢ ∫ (x : β), F (∫ (y : γ), f y + g y ∂η x) ∂κ a = ∫ (x : β), F (∫ (y : γ), f y ∂η x + ∫ (y : γ), g y ∂η x) ∂κ a
|
refine integral_congr_ae ?_
|
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝² : NormedSpace ℝ E
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace ℝ E'
f g : γ → E
F : E → E'
hf : Integrable f ((η ∘ₖ κ) a)
hg : Integrable g ((η ∘ₖ κ) a)
⊢ (fun x => F (∫ (y : γ), f y + g y ∂η x)) =ᶠ[ae (κ a)] fun x => F (∫ (y : γ), f y ∂η x + ∫ (y : γ), g y ∂η x)
|
c2605788d01690c3
|
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