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
|
|---|---|---|---|---|---|---|
gaussSum_mul
|
Mathlib/NumberTheory/GaussSum.lean
|
/-- A formula for the product of two Gauss sums with the same additive character. -/
lemma gaussSum_mul {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R']
(χ φ : MulChar R R') (ψ : AddChar R R') :
gaussSum χ ψ * gaussSum φ ψ = ∑ t : R, ∑ x : R, χ x * φ (t - x) * ψ t
|
case i_surj
R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
χ φ : MulChar R R'
ψ : AddChar R R'
x : R
⊢ ∀ b ∈ univ, ∃ a, ∃ (_ : a ∈ univ), a + x = b
|
exact fun b _ ↦ ⟨b - x, mem_univ _, by rw [sub_add_cancel]⟩
|
no goals
|
c2a4995a365e510c
|
WeierstrassCurve.natDegree_coeff_preΨ'
|
Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
|
private lemma natDegree_coeff_preΨ' (n : ℕ) :
(W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = expCoeff n
|
case odd.left.refine_2
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
fun {m n} {p q} => coeff_mul_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
fun {m n} {p} => coeff_pow_of_natDegree_le
m : ℕ
h₁ :
(W.preΨ' (m + 1)).natDegree ≤ WeierstrassCurve.expDegree (m + 1) ∧
(W.preΨ' (m + 1)).coeff (WeierstrassCurve.expDegree (m + 1)) = ↑(WeierstrassCurve.expCoeff (m + 1))
h₂ :
(W.preΨ' (m + 2)).natDegree ≤ WeierstrassCurve.expDegree (m + 2) ∧
(W.preΨ' (m + 2)).coeff (WeierstrassCurve.expDegree (m + 2)) = ↑(WeierstrassCurve.expCoeff (m + 2))
h₃ :
(W.preΨ' (m + 3)).natDegree ≤ WeierstrassCurve.expDegree (m + 3) ∧
(W.preΨ' (m + 3)).coeff (WeierstrassCurve.expDegree (m + 3)) = ↑(WeierstrassCurve.expCoeff (m + 3))
h₄ :
(W.preΨ' (m + 4)).natDegree ≤ WeierstrassCurve.expDegree (m + 4) ∧
(W.preΨ' (m + 4)).coeff (WeierstrassCurve.expDegree (m + 4)) = ↑(WeierstrassCurve.expCoeff (m + 4))
⊢ (if Even m then 1 else W.Ψ₂Sq ^ 2).natDegree ≤ if Even m then 0 else 2 * 3
|
split_ifs <;>
simp only [apply_ite natDegree, natDegree_one.le, dp W.natDegree_Ψ₂Sq_le]
|
no goals
|
9ededa4b854063c6
|
MeasureTheory.FiniteMeasure.map_fst_prod
|
Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
|
@[simp] lemma map_fst_prod : (μ.prod ν).map Prod.fst = ν univ • μ
|
case h
α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : FiniteMeasure α
ν : FiniteMeasure β
s✝ : Set α
a✝ : MeasurableSet s✝
⊢ ↑((μ.prod ν).map Prod.fst) s✝ = ↑(ν univ • μ) s✝
|
simp
|
no goals
|
83ce0c5517f6d9c1
|
convexIndependent_iff_finset
|
Mathlib/Analysis/Convex/Independent.lean
|
theorem convexIndependent_iff_finset {p : ι → E} :
ConvexIndependent 𝕜 p ↔
∀ (s : Finset ι) (x : ι), p x ∈ convexHull 𝕜 (s.image p : Set E) → x ∈ s
|
case refine_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
h : ∀ (s : Finset ι) (x : ι), p x ∈ (convexHull 𝕜) ↑(image p s) → x ∈ s
s : Set ι
x : ι
hx : p x ∈ ⋃ t, ⋃ (_ : ↑t ⊆ p '' s), (convexHull 𝕜) ↑t
hp : Injective p
⊢ x ∈ s
|
simp_rw [Set.mem_iUnion] at hx
|
case refine_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
h : ∀ (s : Finset ι) (x : ι), p x ∈ (convexHull 𝕜) ↑(image p s) → x ∈ s
s : Set ι
x : ι
hp : Injective p
hx : ∃ i, ∃ (_ : ↑i ⊆ p '' s), p x ∈ (convexHull 𝕜) ↑i
⊢ x ∈ s
|
04d80ee248bee17b
|
Finset.toLeft_disjSum
|
Mathlib/Data/Finset/Sum.lean
|
@[simp] lemma toLeft_disjSum : (s.disjSum t).toLeft = s
|
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
⊢ (s.disjSum t).toLeft = s
|
ext x
|
case h
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
x : α
⊢ x ∈ (s.disjSum t).toLeft ↔ x ∈ s
|
caef727a42445544
|
Finsupp.mul_prod_erase'
|
Mathlib/Algebra/BigOperators/Finsupp.lean
|
theorem mul_prod_erase' (f : α →₀ M) (y : α) (g : α → M → N) (hg : ∀ i : α, g i 0 = 1) :
g y (f y) * (erase y f).prod g = f.prod g
|
case neg
α : Type u_1
M : Type u_8
N : Type u_10
inst✝¹ : Zero M
inst✝ : CommMonoid N
f : α →₀ M
y : α
g : α → M → N
hg : ∀ (i : α), g i 0 = 1
hyf : y ∉ f.support
⊢ g y (f y) * (erase y f).prod g = f.prod g
|
rw [not_mem_support_iff.mp hyf, hg y, erase_of_not_mem_support hyf, one_mul]
|
no goals
|
b036a31ae1b6fc3f
|
Ideal.isPrime_map_C_iff_isPrime
|
Mathlib/RingTheory/Polynomial/Basic.lean
|
theorem isPrime_map_C_iff_isPrime (P : Ideal R) :
IsPrime (map (C : R →+* R[X]) P : Ideal R[X]) ↔ IsPrime P
|
case mpr.mem_or_mem'
R : Type u
inst✝ : CommRing R
P : Ideal R
h : P.IsPrime
f g : R[X]
⊢ f * g ∈ map C P → f ∈ map C P ∨ g ∈ map C P
|
simp only [mem_map_C_iff]
|
case mpr.mem_or_mem'
R : Type u
inst✝ : CommRing R
P : Ideal R
h : P.IsPrime
f g : R[X]
⊢ (∀ (n : ℕ), (f * g).coeff n ∈ P) → (∀ (n : ℕ), f.coeff n ∈ P) ∨ ∀ (n : ℕ), g.coeff n ∈ P
|
851bc056ada1d54f
|
MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative
|
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
theorem of_symmDiff_compl_positive_negative {s : SignedMeasure α} {i j : Set α}
(hi : MeasurableSet i) (hj : MeasurableSet j) (hi' : 0 ≤[i] s ∧ s ≤[iᶜ] 0)
(hj' : 0 ≤[j] s ∧ s ≤[jᶜ] 0) : s (i ∆ j) = 0 ∧ s (iᶜ ∆ jᶜ) = 0
|
case left.hA
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i j : Set α
hi : MeasurableSet i
hj : MeasurableSet j
hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j
hj' :
(∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧
∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1
⊢ MeasurableSet (jᶜ ∩ i)
|
exact hj.compl.inter hi
|
no goals
|
36f2a6a398061eba
|
coeff_minpolyDiv_sub_pow_mem_span
|
Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean
|
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
Submodule.span R ((x ^ ·) '' Set.Iio i)
|
case succ.refine_1.h.a
R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : IsIntegral R x
i : ℕ
IH :
i ≤ (minpolyDiv R x).natDegree →
(minpolyDiv R x).coeff ((minpolyDiv R x).natDegree - i) - x ^ i ∈
Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : i + 1 ≤ (minpolyDiv R x).natDegree
⊢ 1 ∈ (fun x_1 => x ^ x_1) '' Set.Iio (i + 1)
|
exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
|
no goals
|
36fcc4f5170c6e08
|
MvQPF.Cofix.bisim_aux
|
Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
|
theorem Cofix.bisim_aux {α : TypeVec n} (r : Cofix F α → Cofix F α → Prop) (h' : ∀ x, r x x)
(h : ∀ x y, r x y →
appendFun id (Quot.mk r) <$$> Cofix.dest x = appendFun id (Quot.mk r) <$$> Cofix.dest y) :
∀ x y, r x y → x = y
|
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h' : ∀ (x : Cofix F α), r x x
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
x y : (P F).M α
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M α → (P F).M α → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : (P F).M α
r'ab : r' a b
h₀ :
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)
h₁ : ∀ (u v : (P F).M α), Mcongr u v → Quot.mk r' u = Quot.mk r' v
c : (P F).M α
⊢ ∀ (a : (P F).M α),
r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →
Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)
|
intro d rcd
|
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h' : ∀ (x : Cofix F α), r x x
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
x y : (P F).M α
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M α → (P F).M α → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : (P F).M α
r'ab : r' a b
h₀ :
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)
h₁ : ∀ (u v : (P F).M α), Mcongr u v → Quot.mk r' u = Quot.mk r' v
c d : (P F).M α
rcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)
⊢ Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr d)
|
0af76fe163f43a26
|
Finset.maximal_iff_forall_insert
|
Mathlib/Order/CompleteLattice/Finset.lean
|
theorem maximal_iff_forall_insert (hP : ∀ ⦃s t⦄, P t → s ⊆ t → P s) :
Maximal P s ↔ P s ∧ ∀ x ∉ s, ¬ P (insert x s)
|
α : Type u_2
inst✝ : DecidableEq α
P : Finset α → Prop
s : Finset α
hP : ∀ ⦃s t : Finset α⦄, P t → s ⊆ t → P s
⊢ Maximal P s ↔ P s ∧ ∀ x ∉ s, ¬P (insert x s)
|
simp only [Maximal, and_congr_right_iff]
|
α : Type u_2
inst✝ : DecidableEq α
P : Finset α → Prop
s : Finset α
hP : ∀ ⦃s t : Finset α⦄, P t → s ⊆ t → P s
⊢ P s → ((∀ ⦃y : Finset α⦄, P y → s ≤ y → y ≤ s) ↔ ∀ x ∉ s, ¬P (insert x s))
|
9f02cf77d2e1d69f
|
Complex.Gamma_ne_zero
|
Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
|
theorem Gamma_ne_zero {s : ℂ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0
|
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
h_im : ¬s.im = 0
⊢ sin (↑π * s) ≠ 0
|
rw [Complex.sin_ne_zero_iff]
|
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
h_im : ¬s.im = 0
⊢ ∀ (k : ℤ), ↑π * s ≠ ↑k * ↑π
|
a4fbdcf127ba8136
|
Metric.cthickening_eq_iInter_cthickening'
|
Mathlib/Topology/MetricSpace/Thickening.lean
|
theorem cthickening_eq_iInter_cthickening' {δ : ℝ} (s : Set ℝ) (hsδ : s ⊆ Ioi δ)
(hs : ∀ ε, δ < ε → (s ∩ Ioc δ ε).Nonempty) (E : Set α) :
cthickening δ E = ⋂ ε ∈ s, cthickening ε E
|
case h₂
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
⊢ ⋂ ε ∈ s, cthickening ε E ⊆ cthickening δ E
|
unfold cthickening
|
case h₂
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
⊢ ⋂ ε ∈ s, {x | infEdist x E ≤ ENNReal.ofReal ε} ⊆ {x | infEdist x E ≤ ENNReal.ofReal δ}
|
6f877c5032a5d223
|
mem_leftCoset_leftCoset
|
Mathlib/GroupTheory/Coset/Basic.lean
|
theorem mem_leftCoset_leftCoset {a : α} (ha : a • (s : Set α) = s) : a ∈ s
|
α : Type u_1
inst✝ : Monoid α
s : Submonoid α
a : α
ha : a • ↑s = ↑s
⊢ a ∈ a • ↑s
|
exact mem_own_leftCoset s a
|
no goals
|
02d577c82cbff820
|
List.get?_set_of_lt'
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem get?_set_of_lt' (a : α) {m n} (l : List α) (h : m < length l) :
(set l m a).get? n = if m = n then some a else l.get? n
|
α : Type u_1
a : α
m n : Nat
l : List α
h : m < l.length
⊢ (l.set m a).get? n = if m = n then some a else l.get? n
|
simp [getElem?_set]
|
α : Type u_1
a : α
m n : Nat
l : List α
h : m < l.length
⊢ (if m = n then if m < l.length then some a else none else l[n]?) = if m = n then some a else l[n]?
|
1b679c7b7ad7a355
|
Submonoid.closure_eq_one_union
|
Mathlib/Algebra/Group/Submonoid/Basic.lean
|
/-- The `Submonoid.closure` of a set is the union of `{1}` and its `Subsemigroup.closure`. -/
lemma closure_eq_one_union (s : Set M) :
closure s = {(1 : M)} ∪ (Subsemigroup.closure s : Set M)
|
case a.one
M : Type u_1
inst✝ : MulOneClass M
s : Set M
x : M
⊢ 1 ∈ {1} ∪ ↑(Subsemigroup.closure s)
|
exact Or.inl <| by simp
|
no goals
|
2ccb3e3d18b60b77
|
AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen
|
Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
|
theorem isIso_ΓSpec_adjunction_unit_app_basicOpen {X : Scheme} [CompactSpace X]
[QuasiSeparatedSpace X] (f : X.presheaf.obj (op ⊤)) :
IsIso ((ΓSpec.adjunction.unit.app X).c.app (op (PrimeSpectrum.basicOpen f)))
|
X : Scheme
inst✝¹ : CompactSpace ↑↑X.toPresheafedSpace
inst✝ : QuasiSeparatedSpace ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op ⊤))
⊢ IsIso ((ΓSpec.adjunction.unit.app X).c.app (op (PrimeSpectrum.basicOpen f)) ≫ X.presheaf.map (eqToHom ⋯).op)
|
rw [ConcreteCategory.isIso_iff_bijective]
|
X : Scheme
inst✝¹ : CompactSpace ↑↑X.toPresheafedSpace
inst✝ : QuasiSeparatedSpace ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op ⊤))
⊢ Function.Bijective
⇑(ConcreteCategory.hom
((ΓSpec.adjunction.unit.app X).c.app (op (PrimeSpectrum.basicOpen f)) ≫ X.presheaf.map (eqToHom ⋯).op))
|
02965fe549346ba3
|
RootPairing.isReduced_iff
|
Mathlib/LinearAlgebra/RootSystem/Reduced.lean
|
lemma isReduced_iff : P.IsReduced ↔ ∀ i j : ι, i ≠ j →
¬ LinearIndependent R ![P.root i, P.root j] → P.root i = - P.root j
|
case refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
h : ∀ (i j : ι), i ≠ j → ¬LinearIndependent R ![P.root i, P.root j] → P.root i = -P.root j
i j : ι
hLin : ¬LinearIndependent R ![P.root i, P.root j]
⊢ P.root i = P.root j ∨ P.root i = -P.root j
|
rcases eq_or_ne i j with rfl | h'
|
case refine_2.inl
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
h : ∀ (i j : ι), i ≠ j → ¬LinearIndependent R ![P.root i, P.root j] → P.root i = -P.root j
i : ι
hLin : ¬LinearIndependent R ![P.root i, P.root i]
⊢ P.root i = P.root i ∨ P.root i = -P.root i
case refine_2.inr
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
h : ∀ (i j : ι), i ≠ j → ¬LinearIndependent R ![P.root i, P.root j] → P.root i = -P.root j
i j : ι
hLin : ¬LinearIndependent R ![P.root i, P.root j]
h' : i ≠ j
⊢ P.root i = P.root j ∨ P.root i = -P.root j
|
5f51830d5df7e580
|
List.eq_replicate_or_eq_replicate_append_cons
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem eq_replicate_or_eq_replicate_append_cons {α : Type _} (l : List α) :
(l = []) ∨ (∃ n a, l = replicate n a ∧ 0 < n) ∨
(∃ n a b l', l = replicate n a ++ b :: l' ∧ 0 < n ∧ a ≠ b)
|
case nil
α : Type u_1
⊢ [] = [] ∨ (∃ n a, [] = replicate n a ∧ 0 < n) ∨ ∃ n a b l', [] = replicate n a ++ b :: l' ∧ 0 < n ∧ a ≠ b
|
simp
|
no goals
|
1321b019df9b3ed1
|
List.prod_erase_of_comm
|
Mathlib/Algebra/BigOperators/Group/List/Basic.lean
|
@[to_additive]
lemma prod_erase_of_comm [DecidableEq M] (ha : a ∈ l) (comm : ∀ x ∈ l, ∀ y ∈ l, x * y = y * x) :
a * (l.erase a).prod = l.prod
|
case cons.inr.intro
M : Type u_4
inst✝¹ : Monoid M
l✝ : List M
a : M
inst✝ : DecidableEq M
b : M
l : List M
ih : a ∈ l → (∀ (x : M), x ∈ l → ∀ (y : M), y ∈ l → x * y = y * x) → a * (l.erase a).prod = l.prod
ha : a ∈ b :: l
comm : ∀ (x : M), x ∈ b :: l → ∀ (y : M), y ∈ b :: l → x * y = y * x
ne : a ≠ b
h : a ∈ l
⊢ a * ((b :: l).erase a).prod = (b :: l).prod
|
rw [List.erase, beq_false_of_ne ne.symm, List.prod_cons, List.prod_cons, ← mul_assoc,
comm a ha b (l.mem_cons_self b), mul_assoc,
ih h fun x hx y hy ↦ comm _ (List.mem_cons_of_mem b hx) _ (List.mem_cons_of_mem b hy)]
|
no goals
|
8f450a4084d77ffa
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftRight.twoPowShift_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean
|
theorem twoPowShift_eq (aig : AIG α) (target : TwoPowShiftTarget aig w) (lhs : BitVec w)
(rhs : BitVec target.n) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, target.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < target.n), ⟦aig, target.rhs.get idx hidx, assign⟧ = rhs.getLsbD idx) :
∀ (idx : Nat) (hidx : idx < w),
⟦
(twoPowShift aig target).aig,
(twoPowShift aig target).vec.get idx hidx,
assign
⟧
=
(lhs >>> (rhs &&& BitVec.twoPow target.n target.pow)).getLsbD idx
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := { n := n, lhs := lvec, rhs := rvec, pow := pow }.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < { n := n, lhs := lvec, rhs := rvec, pow := pow }.n),
⟦assign, { aig := aig, ref := { n := n, lhs := lvec, rhs := rvec, pow := pow }.rhs.get idx hidx }⟧ = rhs.getLsbD idx
h✝ : pow < n
hg :
RefVec.ite (blastShiftRightConst aig { vec := lvec, distance := 2 ^ pow }).aig
{ discr := (rvec.cast ⋯).get pow h✝, lhs := (blastShiftRightConst aig { vec := lvec, distance := 2 ^ pow }).vec,
rhs := lvec.cast ⋯ } =
res
hif1 : rhs.getLsbD pow = true
⊢ 2 ^ pow % 2 ^ n = 2 ^ pow
|
apply Nat.mod_eq_of_lt
|
case h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := { n := n, lhs := lvec, rhs := rvec, pow := pow }.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < { n := n, lhs := lvec, rhs := rvec, pow := pow }.n),
⟦assign, { aig := aig, ref := { n := n, lhs := lvec, rhs := rvec, pow := pow }.rhs.get idx hidx }⟧ = rhs.getLsbD idx
h✝ : pow < n
hg :
RefVec.ite (blastShiftRightConst aig { vec := lvec, distance := 2 ^ pow }).aig
{ discr := (rvec.cast ⋯).get pow h✝, lhs := (blastShiftRightConst aig { vec := lvec, distance := 2 ^ pow }).vec,
rhs := lvec.cast ⋯ } =
res
hif1 : rhs.getLsbD pow = true
⊢ 2 ^ pow < 2 ^ n
|
a2117c7fa85ff087
|
Vector.zipWith_append
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Zip.lean
|
theorem zipWith_append (f : α → β → γ)
(l : Vector α n) (la : Vector α m) (l' : Vector β n) (lb : Vector β m) :
zipWith f (l ++ la) (l' ++ lb) = zipWith f l l' ++ zipWith f la lb
|
case mk
α : Type u_1
β : Type u_2
γ : Type u_3
m : Nat
f : α → β → γ
la : Vector α m
lb : Vector β m
l : Array α
l' : Vector β l.size
⊢ zipWith f ({ toArray := l, size_toArray := ⋯ } ++ la) (l' ++ lb) =
zipWith f { toArray := l, size_toArray := ⋯ } l' ++ zipWith f la lb
|
rcases l' with ⟨l', h⟩
|
case mk.mk
α : Type u_1
β : Type u_2
γ : Type u_3
m : Nat
f : α → β → γ
la : Vector α m
lb : Vector β m
l : Array α
l' : Array β
h : l'.size = l.size
⊢ zipWith f ({ toArray := l, size_toArray := ⋯ } ++ la) ({ toArray := l', size_toArray := h } ++ lb) =
zipWith f { toArray := l, size_toArray := ⋯ } { toArray := l', size_toArray := h } ++ zipWith f la lb
|
4f97035c25aaaf53
|
Complex.Gamma_mul_Gamma_eq_betaIntegral
|
Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
|
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t
|
case e_f.h
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
∫ (x : ℝ) in Ioi 0,
∫ (t_1 : ℝ) in 0 ..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
(∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
x : ℝ
hx : x ∈ Ioi 0
y : ℝ
⊢ ↑(rexp (-y)) * ↑y ^ (s - 1) * (↑(rexp (-(x - y))) * ↑(x - y) ^ (t - 1)) =
↑(rexp (-x)) * (↑y ^ (s - 1) * (↑x - ↑y) ^ (t - 1))
|
push_cast
|
case e_f.h
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
∫ (x : ℝ) in Ioi 0,
∫ (t_1 : ℝ) in 0 ..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
(∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
x : ℝ
hx : x ∈ Ioi 0
y : ℝ
⊢ cexp (-↑y) * ↑y ^ (s - 1) * (cexp (-(↑x - ↑y)) * (↑x - ↑y) ^ (t - 1)) =
cexp (-↑x) * (↑y ^ (s - 1) * (↑x - ↑y) ^ (t - 1))
|
18eb59fc7415b248
|
TendstoLocallyUniformlyOn.differentiableOn
|
Mathlib/Analysis/Complex/LocallyUniformLimit.lean
|
theorem _root_.TendstoLocallyUniformlyOn.differentiableOn [φ.NeBot]
(hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U)
(hU : IsOpen U) : DifferentiableOn ℂ f U
|
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
⊢ DifferentiableWithinAt ℂ f U x
|
obtain ⟨K, ⟨hKx, hK⟩, hKU⟩ := (compact_basis_nhds x).mem_iff.mp (hU.mem_nhds hx)
|
case intro.intro.intro
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
K : Set ℂ
hKU : K ⊆ U
hKx : K ∈ 𝓝 x
hK : IsCompact K
⊢ DifferentiableWithinAt ℂ f U x
|
4179b164980029db
|
List.take_eq_take
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean
|
theorem take_eq_take :
∀ {l : List α} {m n : Nat}, l.take m = l.take n ↔ min m l.length = min n l.length
| [], m, n => by simp [Nat.min_zero]
| _ :: xs, 0, 0 => by simp
| x :: xs, m + 1, 0 => by simp [Nat.zero_min, succ_min_succ]
| x :: xs, 0, n + 1 => by simp [Nat.zero_min, succ_min_succ]
| x :: xs, m + 1, n + 1 => by simp [succ_min_succ, take_eq_take]
|
α : Type u_1
x : α
xs : List α
n : Nat
⊢ take 0 (x :: xs) = take (n + 1) (x :: xs) ↔ min 0 (x :: xs).length = min (n + 1) (x :: xs).length
|
simp [Nat.zero_min, succ_min_succ]
|
no goals
|
0c82902b146db8e8
|
ProbabilityTheory.measure_limsup_eq_one
|
Mathlib/Probability/BorelCantelli.lean
|
theorem measure_limsup_eq_one {s : ℕ → Set Ω} (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ)
(hs' : (∑' n, μ (s n)) = ∞) : μ (limsup s atTop) = 1
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ⊤
⊢ μ (limsup s atTop) = 1
|
have : IsProbabilityMeasure μ := hs.isProbabilityMeasure
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ⊤
this : IsProbabilityMeasure μ
⊢ μ (limsup s atTop) = 1
|
b8c76ecaf1a3d1c2
|
MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent
|
Mathlib/RingTheory/MvPowerSeries/Basic.lean
|
theorem coeff_eq_zero_of_constantCoeff_nilpotent
{f : MvPowerSeries σ R} {m : ℕ} (hf : constantCoeff σ R f ^ m = 0)
{d : σ →₀ ℕ} {n : ℕ} (hn : m + degree d ≤ n) : coeff R d (f ^ n) = 0
|
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : (constantCoeff σ R) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + d.degree ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := Finset.filter (fun i => k i = 0) (range n)
hs_def : s = Finset.filter (fun i => k i = 0) (range n)
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff R (k i)) f = (constantCoeff σ R) f
hs'' : ∀ i ∈ s, k i = 0
⊢ m + #(range n \ Finset.filter (fun i => k i = 0) (range n)) ≤ m + d.degree
|
simp only [add_comm m, Nat.add_le_add_iff_right, ← hk.1,
← sum_sdiff (hs), sum_eq_zero (s := s) hs'', add_zero]
|
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : (constantCoeff σ R) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + d.degree ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := Finset.filter (fun i => k i = 0) (range n)
hs_def : s = Finset.filter (fun i => k i = 0) (range n)
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff R (k i)) f = (constantCoeff σ R) f
hs'' : ∀ i ∈ s, k i = 0
⊢ #(range n \ Finset.filter (fun i => k i = 0) (range n)) ≤ (∑ x ∈ range n \ s, k x).degree
|
cf9ab574f0bd652b
|
Finsupp.range_linearCombination
|
Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean
|
theorem range_linearCombination : LinearMap.range (linearCombination R v) = span R (range v)
|
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
⊢ LinearMap.range (linearCombination R v) = span R (Set.range v)
|
ext x
|
case h
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
x : M
⊢ x ∈ LinearMap.range (linearCombination R v) ↔ x ∈ span R (Set.range v)
|
28c3b4ab3746ce9c
|
MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm
|
Mathlib/MeasureTheory/Function/LpSpace/Basic.lean
|
theorem ae_tendsto_of_cauchy_eLpNorm [CompleteSpace E] {f : ℕ → α → E}
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) :
∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l)
|
case pos
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ‖(f n - f m) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ CauchySeq fun n => f n x
|
refine cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) ?_ ?_
|
case pos.refine_1
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ‖(f n - f m) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n x) (f m x) ≤ (fun n => (B n).toReal) N
case pos.refine_2
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ‖(f n - f m) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ Tendsto (fun n => (B n).toReal) atTop (𝓝 0)
|
5f2619b40c9c366d
|
FormalMultilinearSeries.ofScalars_radius_eq_top_of_tendsto
|
Mathlib/Analysis/Analytic/OfScalars.lean
|
theorem ofScalars_radius_eq_top_of_tendsto (hc : ∀ᶠ n in atTop, c n ≠ 0)
(hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)) : (ofScalars E c).radius = ⊤
|
case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Summable fun n => ‖‖c n‖ * ↑r' ^ n‖
|
apply summable_of_ratio_test_tendsto_lt_one zero_lt_one (hc.mp (Eventually.of_forall ?_))
|
case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Tendsto (fun n => ‖‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖‖ / ‖‖‖c n‖ * ↑r' ^ n‖‖) atTop (𝓝 0)
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ ∀ (x : ℕ), c x ≠ 0 → ‖‖c x‖ * ↑r' ^ x‖ ≠ 0
|
fed995db04eea233
|
Finset.exists_subsuperset_card_eq
|
Mathlib/Data/Finset/Card.lean
|
/-- Given a subset `s` of a set `t`, of sizes at most and at least `n` respectively, there exists a
set `u` of size `n` which is both a superset of `s` and a subset of `t`. -/
lemma exists_subsuperset_card_eq (hst : s ⊆ t) (hsn : #s ≤ n) (hnt : n ≤ #t) :
∃ u, s ⊆ u ∧ u ⊆ t ∧ #u = n
|
α : Type u_1
s t : Finset α
n : ℕ
hst : s ⊆ t
hsn : #s ≤ n
hnt : n ≤ #t
k : ℕ
a✝ : k < #t
hnk : n ≤ k
u : Finset α
hu₁ : s ⊆ u
hu₂ : u ⊆ t
hu₃ : #u = k + 1
⊢ (u \ s).Nonempty
|
rw [← card_pos, card_sdiff hu₁]
|
α : Type u_1
s t : Finset α
n : ℕ
hst : s ⊆ t
hsn : #s ≤ n
hnt : n ≤ #t
k : ℕ
a✝ : k < #t
hnk : n ≤ k
u : Finset α
hu₁ : s ⊆ u
hu₂ : u ⊆ t
hu₃ : #u = k + 1
⊢ 0 < #u - #s
|
ea6bc578b42c3acb
|
List.findIdx_subtype
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem findIdx_subtype {p : α → Prop} {l : List { x // p x }}
{f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
l.findIdx f = l.unattach.findIdx g
|
α : Type u_1
p : α → Prop
l : List { x // p x }
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
⊢ findIdx f l = findIdx g (map (fun x => x.val) l)
|
induction l with
| nil => simp
| cons a l ih =>
simp [ih, hf, findIdx_cons]
|
no goals
|
3d0a173f80838bde
|
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos
|
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0
|
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
fun n =>
MeasureTheory.SignedMeasure.findExistsOneDivLT s
(i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
⊢ ∃ k, 1 ≤ bdd k ∧ 1 / ↑(bdd k) < ↑s E
|
rw [tendsto_atTop_atTop] at h₄
|
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ :=
fun n =>
MeasureTheory.SignedMeasure.findExistsOneDivLT s
(i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : ∀ (b : ℝ), ∃ i, ∀ (a : ℕ), i ≤ a → b ≤ ↑(bdd a)
A_meas : MeasurableSet A
E : Set α
hE₁ : MeasurableSet E
hE₂ : E ⊆ A
hE₃ : ↑0 E < ↑s E
⊢ ∃ k, 1 ≤ bdd k ∧ 1 / ↑(bdd k) < ↑s E
|
37ee21864bec50c9
|
List.pairwise_replicate
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Pairwise.lean
|
theorem pairwise_replicate {n : Nat} {a : α} :
(replicate n a).Pairwise R ↔ n ≤ 1 ∨ R a a
|
case zero
α : Type u_1
R : α → α → Prop
a : α
⊢ Pairwise R (replicate 0 a) ↔ 0 ≤ 1 ∨ R a a
|
simp
|
no goals
|
2d4c357c8a7b913e
|
weightedVSub_mem_vectorSpan_pair
|
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
theorem weightedVSub_mem_vectorSpan_pair {p : ι → P} (h : AffineIndependent k p) {w w₁ w₂ : ι → k}
{s : Finset ι} (hw : ∑ i ∈ s, w i = 0) (hw₁ : ∑ i ∈ s, w₁ i = 1)
(hw₂ : ∑ i ∈ s, w₂ i = 1) :
s.weightedVSub p w ∈
vectorSpan k ({s.affineCombination k p w₁, s.affineCombination k p w₂} : Set P) ↔
∃ r : k, ∀ i ∈ s, w i = r * (w₁ i - w₂ i)
|
case refine_1.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 0
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
r : k
hr : (s.weightedVSub p) (r • (w₁ - w₂) - w) = 0
i : ι
hi : i ∈ s
hw' : ∑ j ∈ s, (r • (w₁ - w₂) - w) j = 0
hr' : (r • (w₁ - w₂) - w) i = 0
⊢ r • (w₁ i - w₂ i) - w i = 0
|
exact hr'
|
no goals
|
6e1484c0d97194ef
|
Std.DHashMap.Internal.List.isEmpty_replaceEntry
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem isEmpty_replaceEntry [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} :
(replaceEntry k v l).isEmpty = l.isEmpty
|
case nil
α : Type u
β : α → Type v
inst✝ : BEq α
k : α
v : β k
⊢ (replaceEntry k v []).isEmpty = [].isEmpty
|
simp
|
no goals
|
f9d09d9271745f83
|
Real.cos_nat_mul_two_pi_add_pi
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
theorem cos_nat_mul_two_pi_add_pi (n : ℕ) : cos (n * (2 * π) + π) = -1
|
n : ℕ
⊢ cos (↑n * (2 * π) + π) = -1
|
simpa only [cos_zero] using (cos_periodic.nat_mul n).add_antiperiod_eq cos_antiperiodic
|
no goals
|
2d62e007bd9e70e9
|
List.Ico.trichotomy
|
Mathlib/Data/List/Intervals.lean
|
theorem trichotomy (n a b : ℕ) : n < a ∨ b ≤ n ∨ n ∈ Ico a b
|
case pos
n a b : ℕ
h₁ : ¬n < a
h₂ : n ∈ Ico a b
⊢ b ≤ n ∨ n ∈ Ico a b
|
right
|
case pos.h
n a b : ℕ
h₁ : ¬n < a
h₂ : n ∈ Ico a b
⊢ n ∈ Ico a b
|
1ef8cb21636fcea9
|
Mathlib.Meta.NormNum.isRat_lt_true
|
Mathlib/Tactic/NormNum/Ineq.lean
|
theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (R := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) := pos_invOf_of_invertible_cast da
have hb : 0 < ⅟(db : α) := pos_invOf_of_invertible_cast db
have h := (mul_lt_mul_of_pos_left · hb) <| mul_lt_mul_of_pos_right h ha
rw [← mul_assoc, Int.commute_cast] at h
simp? at h says simp only [Int.cast_mul, Int.cast_natCast, mul_invOf_cancel_right'] at h
rwa [Int.commute_cast] at h
|
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h✝ : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
h : ↑(num✝¹ * ↑db) * ⅟↑db * ⅟↑da < ⅟↑db * ((fun x => ↑x) (num✝ * ↑da) * ⅟↑da)
⊢ ↑num✝¹ * ⅟↑da < ↑num✝ * ⅟↑db
|
simp? at h says simp only [Int.cast_mul, Int.cast_natCast, mul_invOf_cancel_right'] at h
|
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h✝ : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
h : ↑num✝¹ * ⅟↑da < ⅟↑db * ↑num✝
⊢ ↑num✝¹ * ⅟↑da < ↑num✝ * ⅟↑db
|
97c6649e3f6d6fbe
|
exists_sum_eq_one_iff_pairwise_coprime
|
Mathlib/RingTheory/Coprime/Lemmas.lean
|
theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
(∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) = 1) ↔
Pairwise (IsCoprime on fun i : t ↦ s i)
|
case cons.mp.intro.refine_2.refine_1.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ t \ {x} = (insert a t \ {x}) \ {a}
|
rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
|
no goals
|
8a28cdbc75233b6c
|
MeasureTheory.Measure.MeasureDense.indicatorConstLp_subset_closure
|
Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
|
theorem Measure.MeasureDense.indicatorConstLp_subset_closure (h𝒜 : μ.MeasureDense 𝒜) (c : E) :
{indicatorConstLp p hs hμs c | (s : Set X) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)} ⊆
closure {indicatorConstLp p (h𝒜.measurable s hs) hμs c |
(s : Set X) (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞)}
|
case inl.intro.intro.intro.intro.intro
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ⊤)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ⊤
t : Set X
ht : t ∈ 𝒜
hμt : μ t ≠ ⊤
⊢ indicatorConstLp p ⋯ hμt 0 = indicatorConstLp p ms hμs 0
|
simp_rw [indicatorConstLp]
|
case inl.intro.intro.intro.intro.intro
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ⊤)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ⊤
t : Set X
ht : t ∈ 𝒜
hμt : μ t ≠ ⊤
⊢ MemLp.toLp (t.indicator fun x => 0) ⋯ = MemLp.toLp (s.indicator fun x => 0) ⋯
|
3507838922f74cb4
|
IsLocallyConstant.of_germ_isConstant
|
Mathlib/Topology/Germ.lean
|
/-- If the germ of `f` w.r.t. each `𝓝 x` is constant, `f` is locally constant. -/
lemma IsLocallyConstant.of_germ_isConstant (h : ∀ x : X, (f : Germ (𝓝 x) Y).IsConstant) :
IsLocallyConstant f
|
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), (↑f).IsConstant
s : Set Y
⊢ IsOpen (f ⁻¹' s)
|
rw [isOpen_iff_mem_nhds]
|
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), (↑f).IsConstant
s : Set Y
⊢ ∀ x ∈ f ⁻¹' s, f ⁻¹' s ∈ 𝓝 x
|
cda63f65afa76d70
|
mem_parallelepiped_iff
|
Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean
|
theorem mem_parallelepiped_iff (v : ι → E) (x : E) :
x ∈ parallelepiped v ↔ ∃ t ∈ Icc (0 : ι → ℝ) 1, x = ∑ i, t i • v i
|
ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
x : E
⊢ x ∈ parallelepiped v ↔ ∃ t ∈ Icc 0 1, x = ∑ i : ι, t i • v i
|
simp [parallelepiped, eq_comm]
|
no goals
|
39b1e0a4954ba12e
|
IntermediateField.AdjoinSimple.trace_gen_eq_zero
|
Mathlib/RingTheory/Trace/Basic.lean
|
theorem trace_gen_eq_zero {x : L} (hx : ¬IsIntegral K x) :
Algebra.trace K K⟮x⟯ (AdjoinSimple.gen K x) = 0
|
case h
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
hx : ¬IsIntegral K x
⊢ ¬∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)
|
contrapose! hx
|
case h
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
hx : ∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)
⊢ IsIntegral K x
|
e3d62924677daba0
|
Matrix.nonsing_inv_nonsing_inv
|
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
theorem nonsing_inv_nonsing_inv (h : IsUnit A.det) : A⁻¹⁻¹ = A :=
calc
A⁻¹⁻¹ = 1 * A⁻¹⁻¹
|
n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h : IsUnit A.det
⊢ 1 * A⁻¹⁻¹ = A * A⁻¹ * A⁻¹⁻¹
|
rw [A.mul_nonsing_inv h]
|
no goals
|
2a08da7f87c7ce96
|
cardinal_eq_of_mem_nhds_zero
|
Mathlib/Topology/Algebra/Module/Cardinality.lean
|
/-- In a topological vector space over a nontrivially normed field, any neighborhood of zero has
the same cardinality as the whole space.
See also `cardinal_eq_of_mem_nhds`. -/
lemma cardinal_eq_of_mem_nhds_zero
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : s ∈ 𝓝 (0 : E)) : #s = #E
|
E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
this : Tendsto (fun n => (c ^ n)⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun n => (c ^ n)⁻¹ • x) atTop (𝓝 (0 • x))
|
exact Tendsto.smul_const this x
|
no goals
|
892055d13be3c7d1
|
linearIndependent_sum
|
Mathlib/LinearAlgebra/LinearIndependent/Basic.lean
|
theorem linearIndependent_sum {v : ι ⊕ ι' → M} :
LinearIndependent R v ↔
LinearIndependent R (v ∘ Sum.inl) ∧
LinearIndependent R (v ∘ Sum.inr) ∧
Disjoint (Submodule.span R (range (v ∘ Sum.inl)))
(Submodule.span R (range (v ∘ Sum.inr)))
|
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι ⊕ ι' → M
hl : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • (v ∘ Sum.inl) i = 0 → ∀ i ∈ s, g i = 0
hr : ∀ (s : Finset ι') (g : ι' → R), ∑ i ∈ s, g i • (v ∘ Sum.inr) i = 0 → ∀ i ∈ s, g i = 0
hlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))
s : Finset (ι ⊕ ι')
g : ι ⊕ ι' → R
hg : ∑ i ∈ s, g i • v i = 0
i : ι ⊕ ι'
hi : i ∈ s
⊢ ∑ i ∈ s.preimage Sum.inl ⋯, (fun x => g x • v x) (Sum.inl i) +
∑ i ∈ s.preimage Sum.inr ⋯, (fun x => g x • v x) (Sum.inr i) =
0
|
rw [Finset.sum_preimage' (g := fun x => g x • v x),
Finset.sum_preimage' (g := fun x => g x • v x), ← Finset.sum_union, ← Finset.filter_or]
|
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι ⊕ ι' → M
hl : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • (v ∘ Sum.inl) i = 0 → ∀ i ∈ s, g i = 0
hr : ∀ (s : Finset ι') (g : ι' → R), ∑ i ∈ s, g i • (v ∘ Sum.inr) i = 0 → ∀ i ∈ s, g i = 0
hlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))
s : Finset (ι ⊕ ι')
g : ι ⊕ ι' → R
hg : ∑ i ∈ s, g i • v i = 0
i : ι ⊕ ι'
hi : i ∈ s
⊢ ∑ x ∈ Finset.filter (fun a => a ∈ range Sum.inl ∨ a ∈ range Sum.inr) s, g x • v x = 0
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι ⊕ ι' → M
hl : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • (v ∘ Sum.inl) i = 0 → ∀ i ∈ s, g i = 0
hr : ∀ (s : Finset ι') (g : ι' → R), ∑ i ∈ s, g i • (v ∘ Sum.inr) i = 0 → ∀ i ∈ s, g i = 0
hlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))
s : Finset (ι ⊕ ι')
g : ι ⊕ ι' → R
hg : ∑ i ∈ s, g i • v i = 0
i : ι ⊕ ι'
hi : i ∈ s
⊢ Disjoint (Finset.filter (fun x => x ∈ range Sum.inl) s) (Finset.filter (fun x => x ∈ range Sum.inr) s)
|
2518c65a54e13190
|
WeierstrassCurve.b₈_of_isCharThreeJNeZeroNF
|
Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean
|
theorem b₈_of_isCharThreeJNeZeroNF : W.b₈ = 4 * W.a₂ * W.a₆
|
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharThreeJNeZeroNF
⊢ W.b₈ = 4 * W.a₂ * W.a₆
|
simp
|
no goals
|
25031e2af685ec8b
|
refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
|
Mathlib/Topology/Compactness/Paracompact.lean
|
theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [WeaklyLocallyCompactSpace X]
[SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
{s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
(∀ a, c a ∈ s ∧ p (c a) (r a)) ∧
(s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a ↦ B (c a) (r a)
|
case refine_3.mk.mk.intro.intro
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n => K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find x + 1)
Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s)
this✝¹ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 ↑x
r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x
hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x)
hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (K n)ᶜ
hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x
T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s)
hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n)
x✝ : X
this✝ : (⋃ k, ⋃ (_ : k ≤ K'.find x✝ + 2), range (Sigma.mk k)).Finite
k : ℕ
c : ↑(Kdiff (k + 1) ∩ s)
hc : c ∈ T' k
x : X
hxB : x ∈ B (↑c) (r k c)
hxK : x ∈ interior (K (K'.find x✝ + 3))
this : x ∉ K k
⊢ k ≤ K'.find x✝ + 2
|
contrapose! this with hnk
|
case refine_3.mk.mk.intro.intro
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n => K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find x + 1)
Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s)
this✝ : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 ↑x
r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x
hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x)
hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (K n)ᶜ
hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x
T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s)
hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n)
x✝ : X
this : (⋃ k, ⋃ (_ : k ≤ K'.find x✝ + 2), range (Sigma.mk k)).Finite
k : ℕ
c : ↑(Kdiff (k + 1) ∩ s)
hc : c ∈ T' k
x : X
hxB : x ∈ B (↑c) (r k c)
hxK : x ∈ interior (K (K'.find x✝ + 3))
hnk : K'.find x✝ + 2 < k
⊢ x ∈ K k
|
d7d8360667f4b953
|
IsSeparable.of_algebra_isSeparable_of_isSeparable
|
Mathlib/FieldTheory/SeparableDegree.lean
|
theorem IsSeparable.of_algebra_isSeparable_of_isSeparable [Algebra E K] [IsScalarTower F E K]
[Algebra.IsSeparable F E] {x : K} (hsep : IsSeparable E x) : IsSeparable F x
|
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E
x : K
hsep : IsSeparable E x
f : E[X] := minpoly E x
hf : f = minpoly E x
⊢ IsSeparable F x
|
let E' : IntermediateField F E := adjoin F f.coeffs
|
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E
x : K
hsep : IsSeparable E x
f : E[X] := minpoly E x
hf : f = minpoly E x
E' : IntermediateField F E := adjoin F ↑f.coeffs
⊢ IsSeparable F x
|
6901ddfd7b743413
|
List.nodup_permutations
|
Mathlib/Data/List/Permutation.lean
|
theorem nodup_permutations (s : List α) (hs : Nodup s) : Nodup s.permutations
|
case cons.right.intro.mk.intro.mk.inr.inl
α : Type u_1
s : List α
x : α
l : List α
h : ∀ a' ∈ l, x ≠ a'
h' : Pairwise (fun x1 x2 => x1 ≠ x2) l
IH : l.permutations'.Nodup
as : List α
ha : as ~ l
bs : List α
hb : bs ~ l
H : as ≠ bs
a : List α
ha' : a ∈ permutations'Aux x as
hb' : a ∈ permutations'Aux x bs
n : ℕ
hn✝ : n < (permutations'Aux x as).length
m : ℕ
hm✝ : m < (permutations'Aux x bs).length
hm' : insertIdx m x bs = a
hl : as.length = bs.length
hn : n ≤ as.length
hm : m ≤ bs.length
hx : (insertIdx n x as)[m] = x
hx' : (insertIdx m x bs)[n] = x
ht : n = m
hn' : insertIdx m x as = a
⊢ False
|
rw [← hm'] at hn'
|
case cons.right.intro.mk.intro.mk.inr.inl
α : Type u_1
s : List α
x : α
l : List α
h : ∀ a' ∈ l, x ≠ a'
h' : Pairwise (fun x1 x2 => x1 ≠ x2) l
IH : l.permutations'.Nodup
as : List α
ha : as ~ l
bs : List α
hb : bs ~ l
H : as ≠ bs
a : List α
ha' : a ∈ permutations'Aux x as
hb' : a ∈ permutations'Aux x bs
n : ℕ
hn✝ : n < (permutations'Aux x as).length
m : ℕ
hm✝ : m < (permutations'Aux x bs).length
hm' : insertIdx m x bs = a
hl : as.length = bs.length
hn : n ≤ as.length
hm : m ≤ bs.length
hx : (insertIdx n x as)[m] = x
hx' : (insertIdx m x bs)[n] = x
ht : n = m
hn' : insertIdx m x as = insertIdx m x bs
⊢ False
|
25ee05be5844251a
|
IsPrimePow.factorization_minFac_ne_zero
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
lemma IsPrimePow.factorization_minFac_ne_zero {n : ℕ} (hn : IsPrimePow n) :
n.factorization n.minFac ≠ 0
|
n : ℕ
hn : IsPrimePow n
⊢ Nat.Prime n.minFac ∧ n.minFac ∣ n ∧ n ≠ 0
|
exact ⟨n.minFac_prime hn.ne_one, n.minFac_dvd, hn.ne_zero⟩
|
no goals
|
f87a3b63f3edf769
|
Finset.shatters_univ
|
Mathlib/Combinatorics/SetFamily/Shatter.lean
|
@[simp] lemma shatters_univ [Fintype α] : 𝒜.Shatters univ ↔ 𝒜 = univ
|
α : Type u_1
inst✝¹ : DecidableEq α
𝒜 : Finset (Finset α)
inst✝ : Fintype α
⊢ image (fun t => univ ∩ t) 𝒜 = univ ↔ 𝒜 = univ
|
simp_rw [univ_inter, image_id']
|
no goals
|
d7004e2093b0cbc2
|
Polynomial.roots_expand_pow_map_iterateFrobenius
|
Mathlib/FieldTheory/Perfect.lean
|
theorem roots_expand_pow_map_iterateFrobenius :
(expand R (p ^ n) f).roots.map (iterateFrobenius R p n) = p ^ n • f.roots
|
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
p n : ℕ
inst✝¹ : ExpChar R p
f : R[X]
inst✝ : PerfectRing R p
⊢ Multiset.map (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots = p ^ n • f.roots
|
simp_rw [← coe_iterateFrobeniusEquiv, roots_expand_pow, Multiset.map_nsmul,
Multiset.map_map, comp_apply, RingEquiv.apply_symm_apply, map_id']
|
no goals
|
28317504e6e10568
|
MeasureTheory.Measure.haarScalarFactor_smul
|
Mathlib/MeasureTheory/Measure/Haar/Unique.lean
|
@[to_additive (attr := simp) addHaarScalarFactor_smul]
lemma haarScalarFactor_smul [LocallyCompactSpace G] (μ' μ : Measure G) [IsHaarMeasure μ]
[IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] {c : ℝ≥0} :
haarScalarFactor (c • μ') μ = c • haarScalarFactor μ' μ
|
case intro.mk.intro.intro
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
g : G → ℝ
g_cont : Continuous g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
int_g_ne_zero : ∫ (x : G), g x ∂μ ≠ 0
⊢ (c • μ').haarScalarFactor μ = c • μ'.haarScalarFactor μ
|
apply NNReal.coe_injective
|
case intro.mk.intro.intro.a
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
g : G → ℝ
g_cont : Continuous g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
int_g_ne_zero : ∫ (x : G), g x ∂μ ≠ 0
⊢ ↑((c • μ').haarScalarFactor μ) = ↑(c • μ'.haarScalarFactor μ)
|
25416b28ad97352c
|
Filter.frequently_lt_of_lt_limsSup
|
Mathlib/Order/LiminfLimsup.lean
|
theorem frequently_lt_of_lt_limsSup {f : Filter α} [ConditionallyCompleteLinearOrder α] {a : α}
(hf : f.IsCobounded (· ≤ ·)
|
α : Type u_1
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder α
a : α
hf : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) f) _auto✝
h : ¬∃ᶠ (n : α) in f, a < n
⊢ f.limsSup ≤ a
|
simp only [not_frequently, not_lt] at h
|
α : Type u_1
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder α
a : α
hf : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) f) _auto✝
h : ∀ᶠ (x : α) in f, x ≤ a
⊢ f.limsSup ≤ a
|
d8988798e121de64
|
MeasureTheory.FiniteMeasure.continuous_mass
|
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
theorem continuous_mass : Continuous fun μ : FiniteMeasure Ω ↦ μ.mass
|
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
⊢ Continuous fun μ => μ.testAgainstNN 1
|
exact continuous_testAgainstNN_eval 1
|
no goals
|
42bdd81967bc02f9
|
HomologicalComplex₂.D₁_totalShift₂XIso_hom
|
Mathlib/Algebra/Homology/TotalComplexShift.lean
|
@[reassoc]
lemma D₁_totalShift₂XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁') :
((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
y.negOnePow • ((K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁')
|
case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : n₀' + 1 = n₁'
⊢ n₀ + 1 = n₁
|
omega
|
no goals
|
95c2874255c2e5ba
|
CategoryTheory.Sheaf.isLocallySurjective_iff_epi
|
Mathlib/CategoryTheory/Sites/LocallySurjective.lean
|
lemma isLocallySurjective_iff_epi {F G : Sheaf J (Type w)} (φ : F ⟶ G)
[HasSheafify J (Type w)] :
IsLocallySurjective φ ↔ Epi φ
|
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
φ : F ⟶ G
inst✝ : HasSheafify J (Type w)
a✝ : Epi φ
⊢ IsLocallySurjective φ
|
have := epi_of_epi_fac (Sheaf.toImage_ι φ)
|
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
φ : F ⟶ G
inst✝ : HasSheafify J (Type w)
a✝ : Epi φ
this : Epi (imageι φ)
⊢ IsLocallySurjective φ
|
1d3f917e21094b8e
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
|
theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.1)
(h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true) :
have hsize' : (Array.modify acc.1 l.1.1 (addAssignment l.snd)).size = n
|
case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
Eq.mpr (id (congrArg (fun _a => _a = n) (Array.size_modify acc.fst l.fst.val (addAssignment l.snd)))) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j1 j2 : Fin (List.length acc.snd.fst)
j1_eq_i : (List.get acc.snd.fst j1).fst.val = ↑i
j2_eq_i : (List.get acc.snd.fst j2).fst.val = ↑i
j1_eq_true : (List.get acc.snd.fst j1).snd = true
j2_eq_false : (List.get acc.snd.fst j2).snd = false
h1 : acc.fst[↑i] = both
h2 : f.assignments[↑i] = unassigned
h3 : ∀ (k : Fin (List.length acc.snd.fst)), k ≠ j1 → k ≠ j2 → (List.get acc.snd.fst k).fst.val ≠ ↑i
j1_succ_in_bounds : ↑j1 + 1 < (l :: acc.snd.fst).length
j2_succ_in_bounds : ↑j2 + 1 < (l :: acc.snd.fst).length
j1_succ : Fin (l :: acc.snd.fst).length := ⟨↑j1 + 1, j1_succ_in_bounds⟩
j2_succ : Fin (l :: acc.snd.fst).length := ⟨↑j2 + 1, j2_succ_in_bounds⟩
⊢ let_fun i_lt_assignments_size := ⋯;
let_fun i_lt_f_assignments_size := ⋯;
let assignments_i := (acc.fst.modify l.fst.val (addAssignment l.snd))[↑i];
let fassignments_i := f.assignments[↑i];
(assignments_i = fassignments_i ∧ ∀ (l_1 : Literal (PosFin n)), l_1 ∈ l :: acc.snd.fst → l_1.fst.val ≠ ↑i) ∨
(∃ j,
((l :: acc.snd.fst).get j).fst.val = ↑i ∧
assignments_i = addAssignment ((l :: acc.snd.fst).get j).snd fassignments_i ∧
¬hasAssignment ((l :: acc.snd.fst).get j).snd fassignments_i = true ∧
∀ (k : Fin (l :: acc.snd.fst).length), k ≠ j → ((l :: acc.snd.fst).get k).fst.val ≠ ↑i) ∨
∃ j1 j2,
((l :: acc.snd.fst).get j1).fst.val = ↑i ∧
((l :: acc.snd.fst).get j2).fst.val = ↑i ∧
((l :: acc.snd.fst).get j1).snd = true ∧
((l :: acc.snd.fst).get j2).snd = false ∧
assignments_i = both ∧
fassignments_i = unassigned ∧
∀ (k : Fin (l :: acc.snd.fst).length), k ≠ j1 → k ≠ j2 → ((l :: acc.snd.fst).get k).fst.val ≠ ↑i
|
apply Or.inr ∘ Or.inr ∘ Exists.intro j1_succ ∘ Exists.intro j2_succ
|
case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
Eq.mpr (id (congrArg (fun _a => _a = n) (Array.size_modify acc.fst l.fst.val (addAssignment l.snd)))) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j1 j2 : Fin (List.length acc.snd.fst)
j1_eq_i : (List.get acc.snd.fst j1).fst.val = ↑i
j2_eq_i : (List.get acc.snd.fst j2).fst.val = ↑i
j1_eq_true : (List.get acc.snd.fst j1).snd = true
j2_eq_false : (List.get acc.snd.fst j2).snd = false
h1 : acc.fst[↑i] = both
h2 : f.assignments[↑i] = unassigned
h3 : ∀ (k : Fin (List.length acc.snd.fst)), k ≠ j1 → k ≠ j2 → (List.get acc.snd.fst k).fst.val ≠ ↑i
j1_succ_in_bounds : ↑j1 + 1 < (l :: acc.snd.fst).length
j2_succ_in_bounds : ↑j2 + 1 < (l :: acc.snd.fst).length
j1_succ : Fin (l :: acc.snd.fst).length := ⟨↑j1 + 1, j1_succ_in_bounds⟩
j2_succ : Fin (l :: acc.snd.fst).length := ⟨↑j2 + 1, j2_succ_in_bounds⟩
⊢ ((l :: acc.snd.fst).get j1_succ).fst.val = ↑i ∧
((l :: acc.snd.fst).get j2_succ).fst.val = ↑i ∧
((l :: acc.snd.fst).get j1_succ).snd = true ∧
((l :: acc.snd.fst).get j2_succ).snd = false ∧
(acc.fst.modify l.fst.val (addAssignment l.snd))[↑i] = both ∧
f.assignments[↑i] = unassigned ∧
∀ (k : Fin (l :: acc.snd.fst).length), k ≠ j1_succ → k ≠ j2_succ → ((l :: acc.snd.fst).get k).fst.val ≠ ↑i
|
02ca1706b397b254
|
Array.flatMap_eq_foldl
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem flatMap_eq_foldl (f : α → Array β) (l : Array α) :
l.flatMap f = l.foldl (fun acc a => acc ++ f a) #[]
|
α : Type u_1
β : Type u_2
f : α → Array β
l : List α
this :
∀ (l' : List β),
(List.foldl (fun acc a => acc ++ (f a).toList) l' l).toArray = List.foldl (fun acc a => acc ++ f a) l'.toArray l
⊢ (List.foldl (fun acc a => acc ++ (f a).toList) [] l).toArray = List.foldl (fun acc a => acc ++ f a) #[] l
|
simpa using this []
|
no goals
|
f50a943587d55d11
|
MeasureTheory.lintegral_tsum
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) :
∫⁻ a, ∑' i, f i a ∂μ = ∑' i, ∫⁻ a, f i a ∂μ
|
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
b : Finset β
⊢ AEMeasurable (fun a => ∑ i ∈ b, f i a) μ
|
exact Finset.aemeasurable_sum _ fun i _ => hf i
|
no goals
|
a9b7b8322de28144
|
Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
|
Mathlib/RingTheory/Filtration.lean
|
theorem Stable.exists_pow_smul_eq_of_ge (h : F.Stable) :
∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀
|
case h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
h : F.Stable
n₀ : ℕ
hn₀ : ∀ (k : ℕ), F.N (n₀ + k) = I ^ k • F.N n₀
⊢ ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀
|
intro n hn
|
case h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
h : F.Stable
n₀ : ℕ
hn₀ : ∀ (k : ℕ), F.N (n₀ + k) = I ^ k • F.N n₀
n : ℕ
hn : n ≥ n₀
⊢ F.N n = I ^ (n - n₀) • F.N n₀
|
c03dafbfc1423a4d
|
Cardinal.mk_finsupp_lift_of_infinite
|
Mathlib/SetTheory/Cardinal/Finsupp.lean
|
theorem mk_finsupp_lift_of_infinite (α : Type u) (β : Type v) [Infinite α] [Zero β] [Nontrivial β] :
#(α →₀ β) = max (lift.{v} #α) (lift.{u} #β)
|
case a.h₂
α : Type u
β : Type v
inst✝² : Infinite α
inst✝¹ : Zero β
inst✝ : Nontrivial β
inhabited_h : Inhabited α
⊢ lift.{max v u, v} #β ≤ lift.{max v u, max v u} #(α →₀ β)
|
exact lift_mk_le.{u}.2 ⟨⟨_, Finsupp.single_injective default⟩⟩
|
no goals
|
cbf4d08f0ffb8987
|
FormalMultilinearSeries.ofScalarsSum_op
|
Mathlib/Analysis/Analytic/OfScalars.lean
|
theorem ofScalarsSum_op [T2Space E] (x : E) :
ofScalarsSum c (MulOpposite.op x) = MulOpposite.op (ofScalarsSum c x)
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : T2Space E
x : E
⊢ ofScalarsSum c (MulOpposite.op x) = MulOpposite.op (ofScalarsSum c x)
|
simp [ofScalars, ofScalars_sum_eq, ← MulOpposite.op_pow, ← MulOpposite.op_smul, tsum_op]
|
no goals
|
179ea3d1c4be08b4
|
LinearMap.polar_weak_closed
|
Mathlib/Analysis/LocallyConvex/Polar.lean
|
theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
(B.polar s)
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x : E
x✝ : x ∈ s
⊢ IsClosed {y | ‖(B x) y‖ ≤ 1}
|
exact isClosed_le (WeakBilin.eval_continuous B.flip x).norm continuous_const
|
no goals
|
db8d40f8cb36d3e1
|
Array.mapM_eq_foldlM_push
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Monadic.lean
|
theorem mapM_eq_foldlM_push [Monad m] [LawfulMonad m] (f : α → m β) (l : Array α) :
mapM f l = l.foldlM (fun acc a => return (acc.push (← f a))) #[]
|
case mk
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
l : List α
k : List β
⊢ (fun a => a.reverse.toArray) <$> List.foldlM (fun acc a => (fun a => a :: acc) <$> f a) k l =
List.foldlM (fun acc a => acc.push <$> f a) k.reverse.toArray l
|
induction l generalizing k with
| nil => simp
| cons a as ih =>
simp [ih, List.foldlM_cons]
|
no goals
|
d0bfdbb80f558ce1
|
SimplexCategoryGenRel.isSplitMono_P_δ
|
Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/EpiMono.lean
|
/-- All `P_δ` are split monos as composition of such. -/
lemma isSplitMono_P_δ {x y : SimplexCategoryGenRel} {m : x ⟶ y} (hm : P_δ m) :
IsSplitMono m
|
case of.δ
x y : SimplexCategoryGenRel
m : x ⟶ y
n✝ : ℕ
i✝ : Fin (n✝ + 2)
⊢ IsSplitMono (δ i✝)
|
infer_instance
|
no goals
|
dcc88895a7959856
|
Homeomorph.residual_map_eq
|
Mathlib/Topology/Baire/BaireMeasurable.lean
|
theorem Homeomorph.residual_map_eq (h : α ≃ₜ β) : (residual α).map h = residual β
|
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
h : α ≃ₜ β
⊢ ∀ s ∈ residual α, ⇑h '' s ∈ residual β
|
simp_rw [← preimage_symm]
|
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
h : α ≃ₜ β
⊢ ∀ s ∈ residual α, ⇑h.symm ⁻¹' s ∈ residual β
|
49fe750364f191b5
|
Matrix.cons_mul
|
Mathlib/Data/Matrix/Notation.lean
|
theorem cons_mul [Fintype n'] (v : n' → α) (A : Fin m → n' → α) (B : Matrix n' o' α) :
of (vecCons v A) * B = of (vecCons (v ᵥ* B) (of.symm (of A * B)))
|
case a.refine_1
α : Type u
m : ℕ
n' : Type uₙ
o' : Type uₒ
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : Fintype n'
v : n' → α
A : Fin m → n' → α
B : Matrix n' o' α
i : Fin m.succ
j : o'
⊢ (of (vecCons v A) * B) 0 j = of (vecCons (v ᵥ* B) (of.symm (of A * B))) 0 j
|
rfl
|
no goals
|
b404ae237eacf4e2
|
Associates.le_of_count_ne_zero
|
Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean
|
theorem le_of_count_ne_zero {m p : Associates α} (h0 : m ≠ 0) (hp : Irreducible p) :
count p m.factors ≠ 0 → p ≤ m
|
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
m p : Associates α
h0 : m ≠ 0
hp : Irreducible p
a✝ : Nontrivial α
⊢ 0 < p.count m.factors → p ≤ m
|
intro h
|
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
m p : Associates α
h0 : m ≠ 0
hp : Irreducible p
a✝ : Nontrivial α
h : 0 < p.count m.factors
⊢ p ≤ m
|
5a1b8803456126b4
|
AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux
|
Mathlib/Analysis/Complex/OpenMapping.lean
|
theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt ℂ f z₀) :
(∀ᶠ z in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ map f (𝓝 z₀)
|
case intro.intro.intro
f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
h : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
R : ℝ
hR : 0 < R
h1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀
h2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z
ρ : ℝ
hρ : ρ > 0
h4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀
h3 : DiffContOnCl ℂ f (ball z₀ ρ)
r : ℝ := ρ ⊓ R
hr : 0 < r
h5 : closedBall z₀ r ⊆ closedBall z₀ ρ
h6 : DiffContOnCl ℂ f (ball z₀ r)
h7 : ∀ z ∈ sphere z₀ r, f z ≠ f z₀
h8 : (sphere z₀ r).Nonempty
h9 : ContinuousOn (fun x => ‖f x - f z₀‖) (sphere z₀ r)
⊢ ∃ i, 0 < i ∧ ball (f z₀) i ⊆ f '' closedBall z₀ R
|
obtain ⟨x, hx, hfx⟩ := (isCompact_sphere z₀ r).exists_isMinOn h8 h9
|
case intro.intro.intro.intro.intro
f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
h : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
R : ℝ
hR : 0 < R
h1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀
h2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z
ρ : ℝ
hρ : ρ > 0
h4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀
h3 : DiffContOnCl ℂ f (ball z₀ ρ)
r : ℝ := ρ ⊓ R
hr : 0 < r
h5 : closedBall z₀ r ⊆ closedBall z₀ ρ
h6 : DiffContOnCl ℂ f (ball z₀ r)
h7 : ∀ z ∈ sphere z₀ r, f z ≠ f z₀
h8 : (sphere z₀ r).Nonempty
h9 : ContinuousOn (fun x => ‖f x - f z₀‖) (sphere z₀ r)
x : ℂ
hx : x ∈ sphere z₀ r
hfx : IsMinOn (fun x => ‖f x - f z₀‖) (sphere z₀ r) x
⊢ ∃ i, 0 < i ∧ ball (f z₀) i ⊆ f '' closedBall z₀ R
|
3b877a39b266ab74
|
TopCat.range_pullback_to_prod
|
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
theorem range_pullback_to_prod {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) :
Set.range (prod.lift (pullback.fst f g) (pullback.snd f g)) =
{ x | (Limits.prod.fst ≫ f) x = (Limits.prod.snd ≫ g) x }
|
case h.mp.intro
X Y Z : TopCat
f : X ⟶ Z
g : Y ⟶ Z
y : ↑(pullback f g)
⊢ (ConcreteCategory.hom (prod.lift (pullback.fst f g) (pullback.snd f g))) y ∈
{x | (ConcreteCategory.hom (prod.fst ≫ f)) x = (ConcreteCategory.hom (prod.snd ≫ g)) x}
|
simp only [← ConcreteCategory.comp_apply, Set.mem_setOf_eq]
|
case h.mp.intro
X Y Z : TopCat
f : X ⟶ Z
g : Y ⟶ Z
y : ↑(pullback f g)
⊢ (ConcreteCategory.hom (prod.lift (pullback.fst f g) (pullback.snd f g) ≫ prod.fst ≫ f)) y =
(ConcreteCategory.hom (prod.lift (pullback.fst f g) (pullback.snd f g) ≫ prod.snd ≫ g)) y
|
4b3b4b2af80f752f
|
CategoryTheory.Triangulated.Subcategory.mem_W_iff_of_distinguished
|
Mathlib/CategoryTheory/Triangulated/Subcategory.lean
|
lemma mem_W_iff_of_distinguished
[S.P.IsClosedUnderIsomorphisms] (T : Triangle C) (hT : T ∈ distTriang C) :
S.W T.mor₁ ↔ S.P T.obj₃
|
C : Type u_1
inst✝⁶ : Category.{u_2, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
S : Subcategory C
inst✝ : S.P.IsClosedUnderIsomorphisms
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ S.W T.mor₁ ↔ S.P T.obj₃
|
constructor
|
case mp
C : Type u_1
inst✝⁶ : Category.{u_2, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
S : Subcategory C
inst✝ : S.P.IsClosedUnderIsomorphisms
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ S.W T.mor₁ → S.P T.obj₃
case mpr
C : Type u_1
inst✝⁶ : Category.{u_2, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
S : Subcategory C
inst✝ : S.P.IsClosedUnderIsomorphisms
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ S.P T.obj₃ → S.W T.mor₁
|
6c0736afeb5017c5
|
Vector.not_mem_of_not_mem_push
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean
|
theorem not_mem_of_not_mem_push {a b : α} {l : Vector α n} (h : a ∉ l.push b) : a ∉ l
|
α : Type u_1
n : Nat
a b : α
l : Vector α n
h : ¬a ∈ l ∧ ¬a = b
⊢ ¬a ∈ l
|
exact h.1
|
no goals
|
498aeb995c711f10
|
AddCircle.addWellApproximable_ae_empty_or_univ
|
Mathlib/NumberTheory/WellApproximable.lean
|
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
(∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x
|
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n => approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᶠ[ae μ] ∅ ∨ B ↑p =ᶠ[ae μ] univ
p : Nat.Primes
e : Set 𝕊 ≃o Set 𝕊 := Equiv.toOrderIsoSet (AddAction.toPerm (u p))
⊢ (blimsup (⇑e ∘ X) atTop fun n => 0 < n ∧ addOrderOf (u p) ^ 2 ∣ n) = C (addOrderOf (u p))
|
exact blimsup_congr (Eventually.of_forall fun n hn =>
approxAddOrderOf.vadd_eq_of_mul_dvd (δ n) hn.1 hn.2)
|
no goals
|
aeca070d334a419e
|
exists_gt_t2space
|
Mathlib/Topology/ShrinkingLemma.lean
|
theorem exists_gt_t2space (v : PartialRefinement u s (fun w => IsCompact (closure w)))
(hs : IsCompact s) (i : ι) (hi : i ∉ v.carrier) :
∃ v' : PartialRefinement u s (fun w => IsCompact (closure w)),
v < v' ∧ IsCompact (closure (v' i))
|
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w => IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
⊢ IsOpen (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)
|
apply isOpen_biUnion
|
case h
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w => IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
⊢ ∀ i_1 ∈ fun i_2 => i_2 = i → False, IsOpen (v.toFun i_1)
|
b61f3fede2ad34cb
|
List.intercalate_eq_intercalateTR
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Impl.lean
|
theorem intercalate_eq_intercalateTR : @intercalate = @intercalateTR
|
⊢ @intercalate = @intercalateTR
|
funext α sep l
|
case h.h.h
α : Type u_1
sep : List α
l : List (List α)
⊢ sep.intercalate l = sep.intercalateTR l
|
5b334d25ddf61cb1
|
Filter.map_neBot_iff
|
Mathlib/Order/Filter/Map.lean
|
theorem map_neBot_iff (f : α → β) {F : Filter α} : NeBot (map f F) ↔ NeBot F
|
α : Type u_1
β : Type u_2
f : α → β
F : Filter α
⊢ (map f F).NeBot ↔ F.NeBot
|
simp only [neBot_iff, Ne, map_eq_bot_iff]
|
no goals
|
f049c14c81044e2e
|
Bornology.IsVonNBounded.image_multilinear'
|
Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
|
theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by
classical
if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then
exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩
else
let _ : NontriviallyNormedField 𝕜 := ⟨by simpa using h₁⟩
obtain ⟨I, t, ht₀, hft⟩ :
∃ (I : Finset ι) (t : ∀ i, Set (E i)), (∀ i, t i ∈ 𝓝 0) ∧ Set.pi I t ⊆ f ⁻¹' V
|
case intro.intro
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : ∀ (i : ι), IsVonNBounded 𝕜 (eval i '' s)
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := NontriviallyNormedField.mk ⋯
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
i : ι
this : ∀ᶠ (x : 𝕜) in 𝓝 0, x • eval i '' s ⊆ t i
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : 𝕜⦄, x ∈ {y | ‖y‖ < r} → x • eval i '' s ⊆ t i
⊢ ∃ c, c ≠ 0 ∧ ∀ (c' : 𝕜), ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i
|
rcases NormedField.exists_norm_lt 𝕜 hr₀ with ⟨c, hc₀, hc⟩
|
case intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : ∀ (i : ι), IsVonNBounded 𝕜 (eval i '' s)
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := NontriviallyNormedField.mk ⋯
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
i : ι
this : ∀ᶠ (x : 𝕜) in 𝓝 0, x • eval i '' s ⊆ t i
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : 𝕜⦄, x ∈ {y | ‖y‖ < r} → x • eval i '' s ⊆ t i
c : 𝕜
hc₀ : 0 < ‖c‖
hc : ‖c‖ < r
⊢ ∃ c, c ≠ 0 ∧ ∀ (c' : 𝕜), ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i
|
b4505873b047f890
|
Turing.ToPartrec.Code.exists_code
|
Mathlib/Computability/TMConfig.lean
|
theorem exists_code {n} {f : List.Vector ℕ n →. ℕ} (hf : Nat.Partrec' f) :
∃ c : Code, ∀ v : List.Vector ℕ n, c.eval v.1 = pure <$> f v
|
case pos
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ →. ℕ
n✝ : ℕ
f : List.Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n✝
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v₀ : List ℕ
n : ℕ
hm : ∀ m < n, ¬f (m ::ᵥ v) = 0
h : [f (n ::ᵥ v)].headI = 0
h2 :
n.succ :: ↑v ∈
PFun.fix
(fun v =>
(cf.eval v).bind fun y =>
Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
(n :: ↑v)
IH :
∀ (a'' : List ℕ),
(Sum.inr a'' ∈
(cf.eval (n :: ↑v)).bind fun y =>
Part.some
(if y.headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail))) →
∀ (n_1 : ℕ),
a'' = n_1 :: ↑v →
(∀ m < n_1, ¬f (m ::ᵥ v) = 0) →
∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [(n.succ :: ↑v).headI.pred]
⊢ ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [(n.succ :: ↑v).headI.pred]
|
exact ⟨_, ⟨h, @(hm)⟩, rfl⟩
|
no goals
|
01823c87e1740c3b
|
List.lookupAll_eq_nil
|
Mathlib/Data/List/Sigma.lean
|
theorem lookupAll_eq_nil {a : α} :
∀ {l : List (Sigma β)}, lookupAll a l = [] ↔ ∀ b : β a, Sigma.mk a b ∉ l
| [] => by simp
| ⟨a', b⟩ :: l => by
by_cases h : a = a'
· subst a'
simp only [lookupAll_cons_eq, mem_cons, Sigma.mk.inj_iff, heq_eq_eq, true_and, not_or,
false_iff, not_forall, not_and, not_not, reduceCtorEq]
use b
simp
· simp [h, lookupAll_eq_nil]
|
case pos
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
l : List (Sigma β)
b : β a
⊢ lookupAll a (⟨a, b⟩ :: l) = [] ↔ ∀ (b_1 : β a), ⟨a, b_1⟩ ∉ ⟨a, b⟩ :: l
|
simp only [lookupAll_cons_eq, mem_cons, Sigma.mk.inj_iff, heq_eq_eq, true_and, not_or,
false_iff, not_forall, not_and, not_not, reduceCtorEq]
|
case pos
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
l : List (Sigma β)
b : β a
⊢ ∃ x, ¬x = b → ⟨a, x⟩ ∈ l
|
8edbefa7d6f1441c
|
MeasureTheory.L1.setToL1_zero_left
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem setToL1_zero_left (hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C)
(f : α →₁[μ] E) : setToL1 hT f = 0
|
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
C : ℝ
hT : DominatedFinMeasAdditive μ 0 C
f : ↥(Lp E 1 μ)
⊢ ContinuousLinearMap.comp 0 (coeToLp α E ℝ) = setToL1SCLM α E μ hT
|
ext1 f
|
case h
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
C : ℝ
hT : DominatedFinMeasAdditive μ 0 C
f✝ : ↥(Lp E 1 μ)
f : ↥(simpleFunc E 1 μ)
⊢ (ContinuousLinearMap.comp 0 (coeToLp α E ℝ)) f = (setToL1SCLM α E μ hT) f
|
cc8f5a6f789f4fe2
|
mul_eq_zero_add_eq_one_ext_right
|
Mathlib/Algebra/Order/Ring/Idempotent.lean
|
lemma mul_eq_zero_add_eq_one_ext_right (eq : a.1.2 = b.1.2) : a = b
|
case refine_3
α : Type u_1
inst✝ : CommSemiring α
a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
eq : (↑a).2 = (↑b).2
⊢ (↑a).2 + (↑b).1 = 1
|
rw [add_comm, eq, b.2.2]
|
no goals
|
2eb8ae992e68b504
|
AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff
|
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
theorem basicOpen_union_eq_self_iff (s : Set Γ(X, U)) :
⨆ f : s, X.basicOpen (f : Γ(X, U)) = U ↔ Ideal.span s = ⊤
|
X : Scheme
U : X.Opens
hU : IsAffineOpen U
s : Set ↑Γ(X, U)
⊢ PrimeSpectrum.zeroLocus (⋃ i, {↑i}) = ∅ ↔ PrimeSpectrum.zeroLocus s = ∅
|
simp only [Set.iUnion_singleton_eq_range, Subtype.range_val_subtype, Set.setOf_mem_eq]
|
no goals
|
e342cbef83e9eeeb
|
AkraBazziRecurrence.dist_r_b'
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma dist_r_b' : ∀ᶠ n in atTop, ∀ i, ‖(r i n : ℝ) - b i * n‖ ≤ n / log n ^ 2
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∀ (i : α), ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2
|
intro i
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
⊢ ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2
|
9b15e28bb0f75398
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
|
theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.1)
(h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true) :
have hsize' : (Array.modify acc.1 l.1.1 (addAssignment l.snd)).size = n
|
case intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
Eq.mpr (id (congrArg (fun _a => _a = n) (Array.size_modify acc.fst l.fst.val (addAssignment l.snd)))) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j1 j2 : Fin (List.length acc.snd.fst)
j1_eq_i : (List.get acc.snd.fst j1).fst.val = ↑i
j2_eq_i : (List.get acc.snd.fst j2).fst.val = ↑i
j1_eq_true : (List.get acc.snd.fst j1).snd = true
j2_eq_false : (List.get acc.snd.fst j2).snd = false
h1 : acc.fst[↑i] = both
h2 : f.assignments[↑i] = unassigned
h3 : ∀ (k : Fin (List.length acc.snd.fst)), k ≠ j1 → k ≠ j2 → (List.get acc.snd.fst k).fst.val ≠ ↑i
j1_succ_in_bounds : ↑j1 + 1 < (l :: acc.snd.fst).length
j2_succ_in_bounds : ↑j2 + 1 < (l :: acc.snd.fst).length
j1_succ : Fin (l :: acc.snd.fst).length := ⟨↑j1 + 1, j1_succ_in_bounds⟩
j2_succ : Fin (l :: acc.snd.fst).length := ⟨↑j2 + 1, j2_succ_in_bounds⟩
l_ne_i : l.fst.val ≠ ↑i
k : Fin (List.length acc.snd.fst + 1)
k_ne_j1_succ : ¬k = j1_succ
k_ne_j2_succ : ¬k = j2_succ
zero_in_bounds : 0 < (l :: acc.snd.fst).length
k_ne_zero : ¬k = ⟨0, zero_in_bounds⟩
k' : Nat
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
k_eq_succ : k = ⟨k' + 1, k'_succ_in_bounds⟩
⊢ ¬((l :: acc.snd.fst).get k).fst.val = ↑i
|
rw [k_eq_succ]
|
case intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
Eq.mpr (id (congrArg (fun _a => _a = n) (Array.size_modify acc.fst l.fst.val (addAssignment l.snd)))) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j1 j2 : Fin (List.length acc.snd.fst)
j1_eq_i : (List.get acc.snd.fst j1).fst.val = ↑i
j2_eq_i : (List.get acc.snd.fst j2).fst.val = ↑i
j1_eq_true : (List.get acc.snd.fst j1).snd = true
j2_eq_false : (List.get acc.snd.fst j2).snd = false
h1 : acc.fst[↑i] = both
h2 : f.assignments[↑i] = unassigned
h3 : ∀ (k : Fin (List.length acc.snd.fst)), k ≠ j1 → k ≠ j2 → (List.get acc.snd.fst k).fst.val ≠ ↑i
j1_succ_in_bounds : ↑j1 + 1 < (l :: acc.snd.fst).length
j2_succ_in_bounds : ↑j2 + 1 < (l :: acc.snd.fst).length
j1_succ : Fin (l :: acc.snd.fst).length := ⟨↑j1 + 1, j1_succ_in_bounds⟩
j2_succ : Fin (l :: acc.snd.fst).length := ⟨↑j2 + 1, j2_succ_in_bounds⟩
l_ne_i : l.fst.val ≠ ↑i
k : Fin (List.length acc.snd.fst + 1)
k_ne_j1_succ : ¬k = j1_succ
k_ne_j2_succ : ¬k = j2_succ
zero_in_bounds : 0 < (l :: acc.snd.fst).length
k_ne_zero : ¬k = ⟨0, zero_in_bounds⟩
k' : Nat
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
k_eq_succ : k = ⟨k' + 1, k'_succ_in_bounds⟩
⊢ ¬((l :: acc.snd.fst).get ⟨k' + 1, k'_succ_in_bounds⟩).fst.val = ↑i
|
02ca1706b397b254
|
CategoryTheory.IsFiltered.bowtie
|
Mathlib/CategoryTheory/Filtered/Basic.lean
|
theorem bowtie {j₁ j₂ k₁ k₂ : C} (f₁ : j₁ ⟶ k₁) (g₁ : j₁ ⟶ k₂) (f₂ : j₂ ⟶ k₁) (g₂ : j₂ ⟶ k₂) :
∃ (s : C) (α : k₁ ⟶ s) (β : k₂ ⟶ s), f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = g₂ ≫ β
|
case intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ k₁ k₂ : C
f₁ : j₁ ⟶ k₁
g₁ : j₁ ⟶ k₂
f₂ : j₂ ⟶ k₁
g₂ : j₂ ⟶ k₂
t : C
k₁t : k₁ ⟶ t
k₂t : k₂ ⟶ t
ht : f₁ ≫ k₁t = g₁ ≫ k₂t
s : C
ts : t ⟶ s
hs : (f₂ ≫ k₁t) ≫ ts = (g₂ ≫ k₂t) ≫ ts
⊢ ∃ s α β, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = g₂ ≫ β
|
simp_rw [Category.assoc] at hs
|
case intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ k₁ k₂ : C
f₁ : j₁ ⟶ k₁
g₁ : j₁ ⟶ k₂
f₂ : j₂ ⟶ k₁
g₂ : j₂ ⟶ k₂
t : C
k₁t : k₁ ⟶ t
k₂t : k₂ ⟶ t
ht : f₁ ≫ k₁t = g₁ ≫ k₂t
s : C
ts : t ⟶ s
hs : f₂ ≫ k₁t ≫ ts = g₂ ≫ k₂t ≫ ts
⊢ ∃ s α β, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = g₂ ≫ β
|
9fbfd17cd6a1fcb5
|
RingHom.locally_localizationAwayPreserves
|
Mathlib/RingTheory/RingHom/Locally.lean
|
/-- If `P` is preserved by localization away, then so is `Locally P`. -/
lemma locally_localizationAwayPreserves (hPl : LocalizationAwayPreserves P) :
LocalizationAwayPreserves (Locally P)
|
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
⊢ LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => Locally fun {R S} [CommRing R] [CommRing S] => P
|
introv R hf
|
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
hf : Locally (fun {R S} [CommRing R] [CommRing S] => P) f
⊢ Locally (fun {R S} [CommRing R] [CommRing S] => P) (IsLocalization.Away.map R' S' f r)
|
2508d7210fe7161f
|
PiTensorProduct.bddBelow_projectiveSemiNormAux
|
Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean
|
theorem bddBelow_projectiveSemiNormAux (x : ⨂[𝕜] i, E i) :
BddBelow (Set.range (fun (p : lifts x) ↦ projectiveSeminormAux p.1))
|
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : ⨂[𝕜] (i : ι), E i
⊢ BddBelow (Set.range fun p => projectiveSeminormAux ↑p)
|
existsi 0
|
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : ⨂[𝕜] (i : ι), E i
⊢ 0 ∈ lowerBounds (Set.range fun p => projectiveSeminormAux ↑p)
|
1114b06691be3409
|
ENNReal.lintegral_Lp_add_le
|
Mathlib/MeasureTheory/Integral/MeanInequalities.lean
|
theorem lintegral_Lp_add_le {p : ℝ} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
(hp1 : 1 ≤ p) :
(∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
(∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
|
have hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
|
917d466c6455bcc8
|
Surreal.Multiplication.mul_right_le_of_equiv
|
Mathlib/SetTheory/Surreal/Multiplication.lean
|
theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)
(h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y
|
x₁ x₂ y : PGame
h₁ : x₁.Numeric
h₂ : x₂.Numeric
h₁₂ : IH24 x₁ x₂ y
h₂₁ : IH24 x₂ x₁ y
he : x₁ ≈ x₂
⊢ x₁ * y ≤ x₂ * y
|
have he' := neg_equiv_neg_iff.2 he
|
x₁ x₂ y : PGame
h₁ : x₁.Numeric
h₂ : x₂.Numeric
h₁₂ : IH24 x₁ x₂ y
h₂₁ : IH24 x₂ x₁ y
he : x₁ ≈ x₂
he' : -x₁ ≈ -x₂
⊢ x₁ * y ≤ x₂ * y
|
e1812a149e6bada9
|
ZMod.val_cast_eq_val_of_lt
|
Mathlib/Data/ZMod/Basic.lean
|
theorem val_cast_eq_val_of_lt {m n : ℕ} [nzm : NeZero m] {a : ZMod m}
(h : a.val < n) : (a.cast : ZMod n).val = a.val
|
m n : ℕ
nzm : NeZero m
a : ZMod m
h : a.val < n
⊢ NeZero n
|
constructor
|
case out
m n : ℕ
nzm : NeZero m
a : ZMod m
h : a.val < n
⊢ n ≠ 0
|
4f9773579b5fa7c2
|
QuasispectrumRestricts.cfc
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
|
theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalContinuousFunctionalCalculus R p where
predicate_zero := h0
compactSpace_quasispectrum a
|
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(σₙ R a), R)₀
⊢ σₙ R ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) g) = range ⇑g
|
rw [nonUnitalStarAlgHom_apply]
|
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module S A
inst✝¹¹ : IsScalarTower S A A
inst✝¹⁰ : SMulCommClass S A A
inst✝⁹ : Algebra R S
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : StarModule R S
inst✝⁵ : ContinuousSMul R S
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus S q
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
g : C(↑(σₙ R a), R)₀
⊢ σₙ R
((cfcₙHom ⋯)
({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
(g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))) =
range ⇑g
|
3af5d542f22c6b37
|
LipschitzOnWith.extend_real
|
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
theorem LipschitzOnWith.extend_real {f : α → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → ℝ, LipschitzWith K g ∧ EqOn f g s
|
case inr
α : Type u
inst✝ : PseudoMetricSpace α
f : α → ℝ
s : Set α
K : ℝ≥0
hf : LipschitzOnWith K f s
hs : s.Nonempty
this : Nonempty ↑s
g : α → ℝ := fun y => ⨅ x, f ↑x + ↑K * dist y ↑x
B : ∀ (y : α), BddBelow (range fun x => f ↑x + ↑K * dist y ↑x)
E : EqOn f g s
x y : α
z : ↑s
⊢ g x ≤ f ↑z + ↑K * dist y ↑z + ↑K * dist x y
|
calc
g x ≤ f z + K * dist x z := ciInf_le (B x) _
_ ≤ f z + K * dist y z + K * dist x y := by
rw [add_assoc, ← mul_add, add_comm (dist y z)]
gcongr
apply dist_triangle
|
no goals
|
509beb34bd5ad882
|
Besicovitch.card_le_multiplicity
|
Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
|
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
(h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E
|
case h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ s.card ∈ {N | ∃ s, s.card = N ∧ (∀ c ∈ s, ‖c‖ ≤ 2) ∧ ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖}
|
simp only [mem_setOf_eq, Ne]
|
case h₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ ∃ s_1, s_1.card = s.card ∧ (∀ c ∈ s_1, ‖c‖ ≤ 2) ∧ ∀ c ∈ s_1, ∀ d ∈ s_1, ¬c = d → 1 ≤ ‖c - d‖
|
48761ca441d21fc8
|
Nat.eq_mul_of_div_eq_left
|
Mathlib/Data/Nat/Init.lean
|
protected lemma eq_mul_of_div_eq_left (H1 : b ∣ a) (H2 : a / b = c) : a = c * b
|
a b c : ℕ
H1 : b ∣ a
H2 : a / b = c
⊢ a = c * b
|
rw [Nat.mul_comm, Nat.eq_mul_of_div_eq_right H1 H2]
|
no goals
|
4802a3e7fa74afdc
|
not_differentiableAt_norm_zero
|
Mathlib/Analysis/Calculus/FDeriv/Norm.lean
|
theorem not_differentiableAt_norm_zero [Nontrivial E] :
¬DifferentiableAt ℝ (‖·‖) (0 : E)
|
case intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
hx : 0 < ‖x‖
h : DifferentiableAt ℝ (fun x => ‖x‖) 0
⊢ False
|
have : DifferentiableAt ℝ (fun t : ℝ ↦ ‖t • x‖) 0 := DifferentiableAt.comp _ (by simpa) (by simp)
|
case intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
hx : 0 < ‖x‖
h : DifferentiableAt ℝ (fun x => ‖x‖) 0
this : DifferentiableAt ℝ (fun t => ‖t • x‖) 0
⊢ False
|
0274b6a9eb508011
|
sphere_prod
|
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
lemma sphere_prod (x : α × β) (r : ℝ) :
sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r
|
case inr.inr.h.mk.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
r : ℝ
hr : 0 < r
x' : α
y' : β
⊢ dist x' x.1 = r → (dist y' x.2 ≤ dist x' x.1 ↔ dist y' x.2 ≤ r)
case inr.inr.h.mk.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
r : ℝ
hr : 0 < r
x' : α
y' : β
⊢ dist y' x.2 = r → (dist x' x.1 ≤ dist y' x.2 ↔ dist x' x.1 ≤ r)
|
all_goals rintro rfl; rfl
|
no goals
|
949a8c74f6143514
|
Module.Baer.extensionOfMax_le
|
Mathlib/Algebra/Module/Injective.lean
|
theorem extensionOfMax_le (h : Module.Baer R Q) {y : N} :
extensionOfMax i f ≤ extensionOfMaxAdjoin i f h y :=
⟨le_sup_left, fun x x' EQ => by
symm
change ExtensionOfMaxAdjoin.extensionToFun i f h _ = _
rw [ExtensionOfMaxAdjoin.extensionToFun_wd i f h x' x 0 (by simp [EQ]), map_zero,
add_zero]⟩
|
R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
x : ↥(extensionOfMax i f).domain
x' : ↥(extensionOfMaxAdjoin i f h y).domain
EQ : ↑x = ↑x'
⊢ ↑(extensionOfMaxAdjoin i f h y).toLinearPMap x' = ↑(extensionOfMax i f).toLinearPMap x
|
change ExtensionOfMaxAdjoin.extensionToFun i f h _ = _
|
R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
x : ↥(extensionOfMax i f).domain
x' : ↥(extensionOfMaxAdjoin i f h y).domain
EQ : ↑x = ↑x'
⊢ ExtensionOfMaxAdjoin.extensionToFun i f h x' = ↑(extensionOfMax i f).toLinearPMap x
|
907d63c4d7960ae0
|
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