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 |
|---|---|---|---|---|---|---|
CStarAlgebra.tendsto_mul_right_approximateUnit | Mathlib/Analysis/CStarAlgebra/ApproximateUnit.lean | /-- This shows `CStarAlgebra.approximateUnit` is a one-sided approximate unit, but this is marked
`private` because it is only used to prove `CStarAlgebra.increasingApproximateUnit`. -/
private lemma tendsto_mul_right_approximateUnit (m : A) :
Tendsto (· * m) (approximateUnit A) (𝓝 m) | case intro.intro
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y => 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : Co... | rw [cfc_mul _ _ m (continuousOn_id' _ |>.mul hg') (continuousOn_id' _),
cfc_mul _ _ m (continuousOn_id' _) hg', cfc_id' .., hm₁.star_eq] | case intro.intro
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y => 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : Co... | 882c5b595b3c7757 |
ProbabilityTheory.IndepFun.variance_add | Mathlib/Probability/Variance.lean | theorem IndepFun.variance_add [IsProbabilityMeasure μ] {X Y : Ω → ℝ} (hX : MemLp X 2 μ)
(hY : MemLp Y 2 μ) (h : IndepFun X Y μ) : variance (X + Y) μ = variance X μ + variance Y μ :=
calc
variance (X + Y) μ = μ[fun a => X a ^ 2 + Y a ^ 2 + 2 * X a * Y a] - μ[X + Y] ^ 2 | case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X Y : Ω → ℝ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
h : IndepFun X Y μ
⊢ Integrable (fun a => X a ^ 2) μ | exact hX.integrable_sq | no goals | 6996a4817b022b25 |
AddCircle.norm_coe_mul | Mathlib/Analysis/Normed/Group/AddCircle.lean | theorem norm_coe_mul (x : ℝ) (t : ℝ) :
‖(↑(t * x) : AddCircle (t * p))‖ = |t| * ‖(x : AddCircle p)‖ | case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z | constructor | case inr.e_a.h.mp
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) → ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z
case inr.e_a.h.mpr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
h... | 1dd8380c8d5f2dbd |
Algebra.IsAlgebraic.algHom_bijective | Mathlib/RingTheory/Algebraic/Basic.lean | theorem algHom_bijective [NoZeroSMulDivisors K L] [Algebra.IsAlgebraic K L] (f : L →ₐ[K] L) :
Function.Bijective f | K : Type u_1
L : Type u_2
inst✝⁴ : CommRing K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : NoZeroSMulDivisors K L
inst✝ : Algebra.IsAlgebraic K L
f : L →ₐ[K] L
⊢ Function.Bijective ⇑f | refine ⟨f.injective, fun b ↦ ?_⟩ | K : Type u_1
L : Type u_2
inst✝⁴ : CommRing K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : NoZeroSMulDivisors K L
inst✝ : Algebra.IsAlgebraic K L
f : L →ₐ[K] L
b : L
⊢ ∃ a, f a = b | bc6d7d9235472cc9 |
HasFPowerSeriesWithinOnBall.congr | Mathlib/Analysis/Analytic/Basic.lean | lemma HasFPowerSeriesWithinOnBall.congr {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F}
{s : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p s x r)
(h' : EqOn g f (s ∩ EMetric.ball x r)) (h'' : g x = f x) :
HasFPowerSeriesWithinOnBall g p s x r | case h.e'_6.inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' ... | simpa using h'' | no goals | a2bff156809360a8 |
PrimeSpectrum.isBasis_basic_opens | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem isBasis_basic_opens : TopologicalSpace.Opens.IsBasis (Set.range (@basicOpen R _)) | case h.e'_3
R : Type u
inst✝ : CommSemiring R
⊢ Set.range (SetLike.coe ∘ basicOpen) = Set.range fun r => ↑(basicOpen r) | rfl | no goals | 58c55a541f3aa057 |
RootPairing.Polarization_apply | Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean | @[simp]
lemma Polarization_apply (x : M) :
P.Polarization x = ∑ i, P.coroot' i x • P.coroot i | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
x : M
⊢ P.Polarization x = ∑ i : ι, (P.coroot' i) x • P.coroot i | simp [Polarization] | no goals | 211a1c000907a11d |
Choose.choose_modEq_choose_mod_mul_choose_div | Mathlib/Data/Nat/Choose/Lucas.lean | theorem choose_modEq_choose_mod_mul_choose_div :
choose n k ≡ choose (n % p) (k % p) * choose (n / p) (k / p) [ZMOD p] | n k p : ℕ
inst✝ : Fact (Nat.Prime p)
decompose : (X + 1) ^ n = (X + 1) ^ (n % p) * (X ^ p + 1) ^ (n / p)
h_iff : ∀ x ∈ range (n % p + 1) ×ˢ range (n / p + 1), k = x.1 + p * x.2 ↔ (k % p, k / p) = x
⊢ (∑ x ∈ range (n % p + 1),
∑ x_1 ∈ range (n / p + 1), if k = x + p * x_1 then 1 * ↑((n % p).choose x * (n / p).choo... | rw [← sum_product', sum_congr rfl (fun a ha ↦ if_congr (h_iff a ha) rfl rfl), sum_ite_eq] | n k p : ℕ
inst✝ : Fact (Nat.Prime p)
decompose : (X + 1) ^ n = (X + 1) ^ (n % p) * (X ^ p + 1) ^ (n / p)
h_iff : ∀ x ∈ range (n % p + 1) ×ˢ range (n / p + 1), k = x.1 + p * x.2 ↔ (k % p, k / p) = x
⊢ (if (k % p, k / p) ∈ range (n % p + 1) ×ˢ range (n / p + 1) then
1 * ↑((n % p).choose (k % p, k / p).1 * (n / p).c... | 33a40e7a05a91053 |
Matroid.IsCircuit.strong_multi_elimination_insert | Mathlib/Data/Matroid/Circuit.lean | /-- A version of `Matroid.IsCircuit.strong_multi_elimination` that is phrased using insertion. -/
lemma IsCircuit.strong_multi_elimination_insert (x : ι → α) (I : ι → Set α) (z : α)
(hxI : ∀ i, x i ∉ I i) (hC : ∀ i, M.IsCircuit (insert (x i) (I i)))
(hJx : M.IsCircuit (J ∪ range x)) (hzJ : z ∈ J) (hzI : ∀ i, z ... | case inr
α : Type u_1
M : Matroid α
ι : Type u_2
J : Set α
x : ι → α
I : ι → Set α
z : α
hxI : ∀ (i : ι), x i ∉ I i
hC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))
hJx : M.IsCircuit (J ∪ range x)
hzJ : z ∈ J
hzI : ∀ (i : ι), z ∉ I i
hι : Nonempty ι
hC' : ∀ (i : ι), M.closure (I i) = M.closure (insert (x i) (I i))
⊢ (J... | rw [union_diff_distrib, union_comm] | case inr
α : Type u_1
M : Matroid α
ι : Type u_2
J : Set α
x : ι → α
I : ι → Set α
z : α
hxI : ∀ (i : ι), x i ∉ I i
hC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))
hJx : M.IsCircuit (J ∪ range x)
hzJ : z ∈ J
hzI : ∀ (i : ι), z ∉ I i
hι : Nonempty ι
hC' : ∀ (i : ι), M.closure (I i) = M.closure (insert (x i) (I i))
⊢ ra... | b2f1fee5a2272693 |
Profinite.NobelingProof.GoodProducts.square_commutes | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem square_commutes : SumEval C ho ∘ Sum.inl =
ModuleCat.ofHom (πs C o) ∘ eval (π C (ord I · < o)) | case h.h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
ho : o < Ordinal.type fun x1 x2 => x1 < x2
l : ↑(GoodProducts (π C fun x => ord I x < o))
x✝ : ↑C
⊢ ((πs C o) (Products.eval (π C fun x => ord I x < o) ↑l)) x✝ =
(eval (π C fun x => ord I x < o) l) (ProjRestrict C ... | rfl | no goals | 101913396b6fcdd4 |
tprod_const | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | theorem tprod_const [T2Space G] (a : G) : ∏' _ : β, a = a ^ (Nat.card β) | case inr.inr
β : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : CommGroup G
inst✝¹ : IsTopologicalGroup G
inst✝ : T2Space G
a : G
hβ : Infinite β
ha : a ≠ 1
⊢ ∏' (x : β), a = 1 | apply tprod_eq_one_of_not_multipliable | case inr.inr.h
β : Type u_2
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : CommGroup G
inst✝¹ : IsTopologicalGroup G
inst✝ : T2Space G
a : G
hβ : Infinite β
ha : a ≠ 1
⊢ ¬Multipliable fun b => a | 4441dbfe0b3c80d5 |
Finset.powerset_inter | Mathlib/Data/Finset/Sups.lean | @[simp] lemma powerset_inter (s t : Finset α) : (s ∩ t).powerset = s.powerset ⊼ t.powerset | case h
α : Type u_2
inst✝ : DecidableEq α
s t u : Finset α
⊢ u ⊆ s ∩ t ↔ ∃ a ⊆ s, ∃ b ⊆ t, a ∩ b = u | refine ⟨fun h ↦ ⟨_, inter_subset_left (s₂ := u), _, inter_subset_left (s₂ := u), ?_⟩, ?_⟩ | case h.refine_1
α : Type u_2
inst✝ : DecidableEq α
s t u : Finset α
h : u ⊆ s ∩ t
⊢ s ∩ u ∩ (t ∩ u) = u
case h.refine_2
α : Type u_2
inst✝ : DecidableEq α
s t u : Finset α
⊢ (∃ a ⊆ s, ∃ b ⊆ t, a ∩ b = u) → u ⊆ s ∩ t | daf8cc6916631354 |
List.reverseAux_reverseAux_nil | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem reverseAux_reverseAux_nil (as bs : List α) : reverseAux (reverseAux as bs) [] = reverseAux bs as | case cons
α : Type u_1
a : α
as : List α
ih : ∀ (bs : List α), (as.reverseAux bs).reverseAux [] = bs.reverseAux as
bs : List α
⊢ ((a :: as).reverseAux bs).reverseAux [] = bs.reverseAux (a :: as) | simp [reverseAux, ih] | no goals | cfbd371923c8716c |
Int.testBit_ofBits_lt | Mathlib/.lake/packages/batteries/Batteries/Data/Int.lean | theorem testBit_ofBits_lt {f : Fin n → Bool} (h : i < n) :
(ofBits f).testBit i = f ⟨i, h⟩ | case isFalse
n i : Nat
f : Fin n → Bool
h : i < n
h✝ : ¬2 * Nat.ofBits f < 2 ^ n
⊢ (subNatNat (Nat.ofBits f) (2 ^ n)).testBit i = f ⟨i, h⟩ | have hlt := Nat.ofBits_lt_two_pow f | case isFalse
n i : Nat
f : Fin n → Bool
h : i < n
h✝ : ¬2 * Nat.ofBits f < 2 ^ n
hlt : Nat.ofBits f < 2 ^ n
⊢ (subNatNat (Nat.ofBits f) (2 ^ n)).testBit i = f ⟨i, h⟩ | 1bb303e4d077688c |
Int.emod_sub_cancel | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem emod_sub_cancel (x y : Int): (x - y) % y = x % y | x y : Int
⊢ (x - y) % y = x % y | by_cases h : y = 0 | case pos
x y : Int
h : y = 0
⊢ (x - y) % y = x % y
case neg
x y : Int
h : ¬y = 0
⊢ (x - y) % y = x % y | e73e001e91089284 |
approxOrderOf.image_pow_subset_of_coprime | Mathlib/NumberTheory/WellApproximable.lean | theorem image_pow_subset_of_coprime (hm : 0 < m) (hmn : n.Coprime m) :
(fun (y : A) => y ^ m) '' approxOrderOf A n δ ⊆ approxOrderOf A n (m * δ) | case h.e'_4
A : Type u_1
inst✝ : SeminormedCommGroup A
m n : ℕ
δ : ℝ
hm : 0 < m
hmn : n.Coprime m
a : A
ha : a ∈ approxOrderOf A n δ
b : A
hab : a ∈ ball b δ
hb : b ^ m ∈ {u | orderOf u = n}
⊢ ball (b ^ m) (↑m • δ) = ball (b ^ m) (m • δ) | simp only [nsmul_eq_mul, Algebra.id.smul_eq_mul] | no goals | 169e467f3ef445f1 |
String.atEnd_of_valid | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem atEnd_of_valid (cs : List Char) (cs' : List Char) :
atEnd ⟨cs ++ cs'⟩ ⟨utf8Len cs⟩ ↔ cs' = [] | cs cs' : List Char
⊢ { data := cs ++ cs' }.atEnd { byteIdx := utf8Len cs } = true ↔ cs' = [] | rw [atEnd_iff] | cs cs' : List Char
⊢ { data := cs ++ cs' }.endPos ≤ { byteIdx := utf8Len cs } ↔ cs' = [] | 4aad2583b3ca35ff |
HahnSeries.SummableFamily.hsum_powerSeriesFamily_mul | Mathlib/RingTheory/HahnSeries/HEval.lean | theorem hsum_powerSeriesFamily_mul (hx : 0 < x.orderTop) (a b : PowerSeries R) :
(powerSeriesFamily hx (a * b)).hsum =
((powerSeriesFamily hx a).mul (powerSeriesFamily hx b)).hsum | case coeff.h
Γ : Type u_1
R : Type u_3
V : Type u_4
inst✝³ : LinearOrderedCancelAddCommMonoid Γ
inst✝² : CommRing R
inst✝¹ : CommRing V
inst✝ : Algebra R V
x : HahnSeries Γ V
hx : 0 < x.orderTop
a b : PowerSeries R
g : Γ
⊢ ∑
i ∈
(image (fun i => i.1 + i.2) (((powerSeriesFamily hx a).mul (powerSeriesFamily... | refine (Finset.sum_of_injOn (fun x => ⟨x.1 + x.2, x⟩) (fun _ _ _ _ => by simp_all) ?_ ?_
(fun _ _ => by simp only [smul_mul_smul_comm, pow_add])).symm | case coeff.h.refine_1
Γ : Type u_1
R : Type u_3
V : Type u_4
inst✝³ : LinearOrderedCancelAddCommMonoid Γ
inst✝² : CommRing R
inst✝¹ : CommRing V
inst✝ : Algebra R V
x : HahnSeries Γ V
hx : 0 < x.orderTop
a b : PowerSeries R
g : Γ
⊢ Set.MapsTo (fun x => ⟨x.1 + x.2, x⟩) ↑(((powerSeriesFamily hx a).mul (powerSeriesFamily ... | 33d7a832d38b770c |
List.of_findIdx?_eq_none | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem of_findIdx?_eq_none {xs : List α} {p : α → Bool} (w : xs.findIdx? p = none) :
∀ i : Nat, match xs[i]? with | some a => ¬ p a | none => true | case cons.succ
α : Type u_1
p : α → Bool
x : α
xs : List α
ih :
findIdx? p xs = none →
∀ (i : Nat),
match xs[i]? with
| some a => p a = false
| none => True
w : (if p x = true then some 0 else Option.map (fun i => i + 1) (findIdx? p xs)) = none
i : Nat
⊢ match (x :: xs)[i + 1]? with
| some a =... | simp only [getElem?_cons_succ] | case cons.succ
α : Type u_1
p : α → Bool
x : α
xs : List α
ih :
findIdx? p xs = none →
∀ (i : Nat),
match xs[i]? with
| some a => p a = false
| none => True
w : (if p x = true then some 0 else Option.map (fun i => i + 1) (findIdx? p xs)) = none
i : Nat
⊢ match xs[i]? with
| some a => p a = fal... | d56cb53d2789be3d |
Part.Fix.approx_mono' | Mathlib/Control/LawfulFix.lean | theorem approx_mono' {i : ℕ} : Fix.approx f i ≤ Fix.approx f (succ i) | case succ
α : Type u_1
β : α → Type u_2
f : ((a : α) → Part (β a)) →o (a : α) → Part (β a)
n✝ : ℕ
i_ih : approx (⇑f) n✝ ≤ approx (⇑f) n✝.succ
i✝ : α
⊢ approx (⇑f) (n✝ + 1) i✝ ≤ approx (⇑f) (n✝ + 1).succ i✝ | apply f.monotone | case succ.a
α : Type u_1
β : α → Type u_2
f : ((a : α) → Part (β a)) →o (a : α) → Part (β a)
n✝ : ℕ
i_ih : approx (⇑f) n✝ ≤ approx (⇑f) n✝.succ
i✝ : α
⊢ approx (⇑f) n✝ ≤ approx (⇑f) (n✝ + 1) | c85ff91073cb5b69 |
LieSubmodule.toSubmodule_injective | Mathlib/Algebra/Lie/Submodule.lean | theorem toSubmodule_injective :
Function.Injective (toSubmodule : LieSubmodule R L M → Submodule R M) := fun x y h ↦ by
cases x; cases y; congr
| R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
x y : LieSubmodule R L M
h : ↑x = ↑y
⊢ x = y | cases x | case mk
R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
y : LieSubmodule R L M
toSubmodule✝ : Submodule R M
lie_mem✝ : ∀ {x : L} {m : M}, m ∈ toSubmodule✝.carrier → ⁅x, m⁆ ∈ toSubmodule✝.carrier
h : ↑{ toSubmodule := toSubmodu... | a4c2d625cc10c6e5 |
Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | theorem circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq {t₁ t₂ : Triangle ℝ P}
{i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃)
(h₁ : t₁.points i₁ = t₂.points i₁) (h₃ : t₁.points i₃ = t₂.points i₃)
(h₂ : (2 : ℤ) • ∡ (t₁.points i₁) (t₁.points i₂) (t₁.points i₃) =
(2 : ... | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁ : t₁.points i₁ = t₂.points i₁
h₃ : t₁.po... | rw [t₁.circumsphere_eq_of_dist_of_oangle h₁₂ h₁₃ h₂₃,
t₂.circumsphere_eq_of_dist_of_oangle h₁₂ h₁₃ h₂₃,
Real.Angle.tan_eq_of_two_zsmul_eq h₂, Real.Angle.abs_sin_eq_of_two_zsmul_eq h₂, h₁, h₃] | no goals | 58b55a61bf9e6c2b |
compl_beattySeq | Mathlib/NumberTheory/Rayleigh.lean | theorem compl_beattySeq {r s : ℝ} (hrs : r.IsConjExponent s) :
{beattySeq r k | k}ᶜ = {beattySeq' s k | k} | case pos
r s : ℝ
hrs : r.IsConjExponent s
j : ℤ
h₁ : j ∉ {x | ∃ k, beattySeq r k = x}
h₂ : j ∈ {x | ∃ k, beattySeq' s k = x}
⊢ j ∈ {x | ∃ k, beattySeq r k = x}ᶜ ↔ j ∈ {x | ∃ k, beattySeq' s k = x} | simp only [Set.mem_compl_iff, h₁, h₂, not_false_eq_true] | no goals | e07c1c3e414c7df6 |
OnePoint.nhdsWithin_compl_infty_eq | Mathlib/Topology/Compactification/OnePoint.lean | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | X : Type u_1
inst✝ : TopologicalSpace X
⊢ 𝓝[≠] ∞ = map some (coclosedCompact X) | refine (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) ?_ ?_ | case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
⊢ ∀ (i : Set (OnePoint X)), ∞ ∈ i ∧ IsOpen i → ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ i ∩ {∞}ᶜ
case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
⊢ ∀ (i' : Set X), IsClosed i' ∧ IsCompact i' → ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' i'ᶜ | 57112e0951fe8484 |
QuaternionAlgebra.star_smul' | Mathlib/Algebra/Quaternion.lean | theorem star_smul' [Monoid S] [DistribMulAction S R] (s : S) (a : ℍ[R,c₁,0,c₃]) :
star (s • a) = s • star a :=
QuaternionAlgebra.ext (by simp) (smul_neg _ _).symm (smul_neg _ _).symm (smul_neg _ _).symm
| S : Type u_1
R : Type u_3
c₁ c₃ : R
inst✝² : CommRing R
inst✝¹ : Monoid S
inst✝ : DistribMulAction S R
s : S
a : ℍ[R,c₁,c₃]
⊢ (star (s • a)).re = (s • star a).re | simp | no goals | 9f6f8beaf9e688d3 |
Array.size_attach | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem size_attach {L : Array α} : L.attach.size = L.size | α : Type u_1
L : Array α
⊢ L.attach.size = L.size | cases L | case mk
α : Type u_1
toList✝ : List α
⊢ { toList := toList✝ }.attach.size = { toList := toList✝ }.size | 740f491d707d51ee |
CategoryTheory.Limits.biproduct.map_eq | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | theorem biproduct.map_eq [HasFiniteBiproducts C] {f g : J → C} {h : ∀ j, f j ⟶ g j} :
biproduct.map h = ∑ j : J, biproduct.π f j ≫ h j ≫ biproduct.ι g j | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
inst✝ : HasFiniteBiproducts C
f g : J → C
h : (j : J) → f j ⟶ g j
⊢ map h = ∑ j : J, π f j ≫ h j ≫ ι g j | ext | case w.w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
inst✝ : HasFiniteBiproducts C
f g : J → C
h : (j : J) → f j ⟶ g j
j✝¹ j✝ : J
⊢ ι f j✝ ≫ map h ≫ π g j✝¹ = ι f j✝ ≫ (∑ j : J, π f j ≫ h j ≫ ι g j) ≫ π g j✝¹ | cf703403d1633014 |
Monoid.le_minOrder_iff_forall_subgroup | Mathlib/GroupTheory/Order/Min.lean | @[to_additive]
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
n ≤ minOrder α ↔ ∀ ⦃s : Subgroup α⦄, s ≠ ⊥ → (s : Set α).Finite → n ≤ Nat.card s | α : Type u_1
inst✝ : Group α
n : ℕ∞
⊢ (∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ ↑(orderOf a)) ↔ ∀ ⦃s : Subgroup α⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s) | refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩ | case refine_1
α : Type u_1
inst✝ : Group α
n : ℕ∞
h : ∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ ↑(orderOf a)
s : Subgroup α
hs : s ≠ ⊥
hs' : (↑s).Finite
⊢ n ≤ ↑(Nat.card ↥s)
case refine_2
α : Type u_1
inst✝ : Group α
n : ℕ∞
h : ∀ ⦃s : Subgroup α⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)
a : α
ha : a ≠ 1
ha' : IsOfFinOr... | e885fe81e49a4b03 |
Subalgebra.op_adjoin | Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean | theorem op_adjoin (s : Set A) :
(Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s) | R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s) | apply toSubsemiring_injective | case a
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op.toSubsemiring = (Algebra.adjoin R (MulOpposite.unop ⁻¹' s)).toSubsemiring | 1440a73c0917844e |
Polynomial.Chebyshev.S_sq_add_S_sq | Mathlib/RingTheory/Polynomial/Chebyshev.lean | theorem S_sq_add_S_sq (n : ℤ) : S R n ^ 2 + S R (n + 1) ^ 2 - X * S R n * S R (n + 1) = 1 | case hn
R : Type u_1
inst✝ : CommRing R
n : ℕ
ih : S R (-↑n) ^ 2 + S R (-↑n + 1) ^ 2 - X * S R (-↑n) * S R (-↑n + 1) = 1
h₁ : S R (-↑n - 1) = X * S R (-↑n) - S R (-↑n + 1)
⊢ S R (-↑n - 1) ^ 2 + S R (-↑n - 1 + 1) ^ 2 - X * S R (-↑n - 1) * S R (-↑n - 1 + 1) = 1 | linear_combination (norm := ring_nf) (S R (-1 - n) - S R (1 - n)) * h₁ + ih | no goals | c157673ed3e5d8d9 |
Turing.TM2to1.tr_supports | Mathlib/Computability/TuringMachine.lean | theorem tr_supports {S} (ss : TM2.Supports M S) : TM1.Supports (tr M) (trSupp M S) :=
⟨Finset.mem_biUnion.2 ⟨_, ss.1, Finset.mem_insert.2 <| Or.inl rfl⟩, fun l' h ↦ by
suffices ∀ (q) (_ : TM2.SupportsStmt S q) (_ : ∀ x ∈ trStmts₁ q, x ∈ trSupp M S),
TM1.SupportsStmt (trSupp M S) (trNormal q) ∧
∀ l... | K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
l' : Λ' K Γ Λ σ
h : l' ∈ trSupp M S
⊢ TM1.SupportsStmt (trSupp M S) (tr M l') | suffices ∀ (q) (_ : TM2.SupportsStmt S q) (_ : ∀ x ∈ trStmts₁ q, x ∈ trSupp M S),
TM1.SupportsStmt (trSupp M S) (trNormal q) ∧
∀ l' ∈ trStmts₁ q, TM1.SupportsStmt (trSupp M S) (tr M l') by
rcases Finset.mem_biUnion.1 h with ⟨l, lS, h⟩
have :=
this _ (ss.2 l lS) fun x hx ↦ Finset.mem_biUnion.2 ⟨_, lS, Fi... | K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
l' : Λ' K Γ Λ σ
h : l' ∈ trSupp M S
⊢ ∀ (q : TM2.Stmt Γ Λ σ),
TM2.SupportsStmt S q →
(∀ x ∈ trStmts₁ q, x ∈ trSupp M S) →
TM1.SupportsStmt (trSup... | a6dbda729b9e1134 |
Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm | Mathlib/Algebra/Polynomial/CancelLeads.lean | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff
h : p.natDegree ≤ q.natDegree
hq : 0 < q.natDegree
⊢ (p.cancelLeads q).natDegree < q.natDegree | by_cases hp : p = 0 | case pos
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff
h : p.natDegree ≤ q.natDegree
hq : 0 < q.natDegree
hp : p = 0
⊢ (p.cancelLeads q).natDegree < q.natDegree
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : p.leadingCoeff * q.leadingCoeff = q.l... | be925249834bc69d |
Polynomial.Monic.nextCoeff_multiset_prod | Mathlib/Algebra/Polynomial/Monic.lean | theorem Monic.nextCoeff_multiset_prod (t : Multiset ι) (f : ι → R[X]) (h : ∀ i ∈ t, Monic (f i)) :
nextCoeff (t.map f).prod = (t.map fun i => nextCoeff (f i)).sum | case refine_2
R : Type u
ι : Type y
inst✝ : CommSemiring R
t✝ : Multiset ι
f : ι → R[X]
a : ι
t : Multiset ι
ih : (∀ i ∈ t, (f i).Monic) → (Multiset.map f t).prod.nextCoeff = (Multiset.map (fun i => (f i).nextCoeff) t).sum
ht : ∀ i ∈ a ::ₘ t, (f i).Monic
⊢ (Multiset.map f (a ::ₘ t)).prod.nextCoeff = (Multiset.map (fun ... | rw [Multiset.map_cons, Multiset.prod_cons, Multiset.map_cons, Multiset.sum_cons,
Monic.nextCoeff_mul, ih] | case refine_2
R : Type u
ι : Type y
inst✝ : CommSemiring R
t✝ : Multiset ι
f : ι → R[X]
a : ι
t : Multiset ι
ih : (∀ i ∈ t, (f i).Monic) → (Multiset.map f t).prod.nextCoeff = (Multiset.map (fun i => (f i).nextCoeff) t).sum
ht : ∀ i ∈ a ::ₘ t, (f i).Monic
⊢ ∀ i ∈ t, (f i).Monic
case refine_2.hp
R : Type u
ι : Type y
in... | 12e7b67c521e8aff |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.size_ofArray_fold_fn | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem size_ofArray_fold_fn {n : Nat} (assignments : Array Assignment)
(cOpt : Option (DefaultClause n)) :
(ofArray_fold_fn assignments cOpt).size = assignments.size | case h_2
n : Nat
assignments : Array Assignment
cOpt✝ : Option (DefaultClause n)
c✝ : DefaultClause n
⊢ (match c✝.isUnit with
| none => assignments
| some (l, true) => assignments.modify l.val addPosAssignment
| some (l, false) => assignments.modify l.val addNegAssignment).size =
assignments.size | split <;> simp [Array.size_modify] | no goals | bd5cba9349edfdee |
Ordnode.Valid'.map_aux | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.map_aux {β} [Preorder β] {f : α → β} (f_strict_mono : StrictMono f) {t a₁ a₂}
(h : Valid' a₁ t a₂) :
Valid' (Option.map f a₁) (map f t) (Option.map f a₂) ∧ (map f t).size = t.size | case node.intro.intro.sz.right.right
α : Type u_1
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α → β
f_strict_mono : StrictMono f
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂
t_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option... | exact t_r_valid.sz | no goals | ff36f5e3d5092df5 |
Int.bitwise_bit | Mathlib/Data/Int/Bitwise.lean | theorem bitwise_bit (f : Bool → Bool → Bool) (a m b n) :
bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) | case negSucc.negSucc
f : Bool → Bool → Bool
a b : Bool
m n : ℕ
⊢ (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n)+1]
else ↑(Nat.bitwise (fun x y => f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n))) =
bit (f a b)
(bif f true true then -[Nat.bitwise (fun x y => !f (... | by_cases h : f true true <;> simp +decide [h] | no goals | 6d457d78c3d19bdd |
HasFPowerSeriesWithinAt.comp | Mathlib/Analysis/Analytic/Composition.lean | theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
{p : FormalMultilinearSeries 𝕜 E F} {x : E} {t : Set F} {s : Set E}
(hg : HasFPowerSeriesWithinAt g q t (f x)) (hf : HasFPowerSeriesWithinAt f p s x)
(hs : Set.MapsTo f s t) : HasFPowerSeriesWithinAt (g ∘ f) (... | case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q :... | refine
⟨le_trans (min_le_right rf' r) (FormalMultilinearSeries.le_comp_radius_of_summable q p r hr),
min_pos, fun {y} h'y hy ↦ ?_⟩ | case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q :... | 919c288392263270 |
PadicInt.norm_int_lt_one_iff_dvd | Mathlib/NumberTheory/Padics/PadicIntegers.lean | theorem norm_int_lt_one_iff_dvd (k : ℤ) : ‖(k : ℤ_[p])‖ < 1 ↔ (p : ℤ) ∣ k :=
suffices ‖(k : ℚ_[p])‖ < 1 ↔ ↑p ∣ k by rwa [norm_intCast_eq_padic_norm]
padicNormE.norm_int_lt_one_iff_dvd k
| p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
this : ‖↑k‖ < 1 ↔ ↑p ∣ k
⊢ ‖↑k‖ < 1 ↔ ↑p ∣ k | rwa [norm_intCast_eq_padic_norm] | no goals | c3237d04872b89b4 |
Computability.decode_encodePosNum | Mathlib/Computability/Encoding.lean | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n | ⊢ ∀ (n : PosNum), decodePosNum (encodePosNum n) = n | intro n | n : PosNum
⊢ decodePosNum (encodePosNum n) = n | d4afd0220e747d57 |
Algebra.FinitePresentation.ker_fg_of_mvPolynomial | Mathlib/RingTheory/FinitePresentation.lean | theorem ker_fg_of_mvPolynomial {n : ℕ} (f : MvPolynomial (Fin n) R →ₐ[R] A)
(hf : Function.Surjective f) [FinitePresentation R A] : f.toRingHom.ker.FG | case intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomial (Fin m) R)
hs : Ideal.span ... | obtain ⟨_, ⟨x, rfl⟩, y, hy, rfl⟩ := AddSubmonoid.mem_sup.mp this | case intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomi... | 37c27acde7e67205 |
Subgroup.Normal.quotient_commutative_iff_commutator_le | Mathlib/GroupTheory/Abelianization.lean | theorem Subgroup.Normal.quotient_commutative_iff_commutator_le {N : Subgroup G} [N.Normal] :
Std.Commutative (· * · : G ⧸ N → _ → _) ↔ _root_.commutator G ≤ N | case mpr.comm.intro.intro.a
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hGN : _root_.commutator G ≤ N
x y : G
⊢ ⁅x, y⁆ ∈ closure (commutatorSet G) | exact Subgroup.subset_closure (commutator_mem_commutatorSet x y) | no goals | d61010dbb1cc2ad5 |
LinearPMap.dExt_iff | Mathlib/LinearAlgebra/LinearPMap.lean | theorem dExt_iff {f g : E →ₗ.[R] F} :
f = g ↔
∃ _domain_eq : f.domain = g.domain,
∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y :=
⟨fun EQ =>
EQ ▸
⟨rfl, fun x y h => by
congr
exact mod_cast h⟩,
fun ⟨deq, feq⟩ => dExt deq feq⟩
| R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f g : E →ₗ.[R] F
EQ : f = g
x y : ↥f.domain
h : ↑x = ↑y
⊢ ↑f x = ↑f y | congr | case e_a
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f g : E →ₗ.[R] F
EQ : f = g
x y : ↥f.domain
h : ↑x = ↑y
⊢ x = y | 6a38ee861edcbe44 |
Set.finite_iUnion_iff | Mathlib/Data/Set/Finite/Lattice.lean | /-- An indexed union of pairwise disjoint sets is finite iff all sets are finite, and all but
finitely many are empty. -/
lemma finite_iUnion_iff {ι : Type*} {s : ι → Set α} (hs : Pairwise fun i j ↦ Disjoint (s i) (s j)) :
(⋃ i, s i).Finite ↔ (∀ i, (s i).Finite) ∧ {i | (s i).Nonempty}.Finite where
mp h | case mk.mk.a
α : Type u
ι : Type u_1
s : ι → Set α
hs : Pairwise fun i j => Disjoint (s i) (s j)
h : (⋃ i, s i).Finite
u : ↑{i | (s i).Nonempty} → ↑(⋃ i, s i) := fun i => ⟨Exists.choose ⋯, ⋯⟩
i : ι
hi : i ∈ {i | (s i).Nonempty}
j : ι
hj : j ∈ {i | (s i).Nonempty}
hij : u ⟨i, hi⟩ = u ⟨j, hj⟩
⊢ ↑(u ⟨j, hj⟩) ∈ s ↑⟨j, hj⟩ | exact hj.choose_spec | no goals | 11fa585e2b7f3d1f |
List.foldl_const | Mathlib/Logic/Function/Iterate.lean | theorem foldl_const (f : α → α) (a : α) (l : List β) :
l.foldl (fun b _ ↦ f b) a = f^[l.length] a | α : Type u
β : Type v
f : α → α
a : α
l : List β
⊢ foldl (fun b x => f b) a l = f^[l.length] a | induction l generalizing a with
| nil => rfl
| cons b l H => rw [length_cons, foldl, iterate_succ_apply, H] | no goals | 82d647f485f0ad66 |
isProperMap_iff_ultrafilter | Mathlib/Topology/Maps/Proper/Basic.lean | /-- Characterization of proper maps by ultrafilters. -/
lemma isProperMap_iff_ultrafilter : IsProperMap f ↔ Continuous f ∧
∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous f
⊢ (∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ) ↔
∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x | constructor <;> intro H | case mp
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : Continuous f
H : ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
⊢ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
case mpr
X : Type u_1
Y : Type u_2
in... | 3f42244004d66b9d |
Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type*) [CommRing R]
[IsDomain R] : algebraMap _ (RatFunc R) (cyclotomic n R) =
∏ i ∈ n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ μ i.fst | n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =
∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1 | rcases n.eq_zero_or_pos with (rfl | hpos) | case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic 0 R) =
∏ i ∈ Nat.divisorsAntidiagonal 0, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1
case inr
n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hpos : n > 0
⊢ (algebraMap R[X] (RatFunc R)) (cycl... | 657390a55a58edcb |
Nat.iSup_lt_succ' | Mathlib/Data/Nat/Lattice.lean | theorem iSup_lt_succ' (u : ℕ → α) (n : ℕ) : ⨆ k < n + 1, u k = u 0 ⊔ ⨆ k < n, u (k + 1) | α : Type u_1
inst✝ : CompleteLattice α
u : ℕ → α
n : ℕ
⊢ ⨆ k, ⨆ (_ : k < n + 1), u k = u 0 ⊔ ⨆ k, ⨆ (_ : k < n), u (k + 1) | rw [← sup_iSup_nat_succ] | α : Type u_1
inst✝ : CompleteLattice α
u : ℕ → α
n : ℕ
⊢ (⨆ (_ : 0 < n + 1), u 0) ⊔ ⨆ i, ⨆ (_ : i + 1 < n + 1), u (i + 1) = u 0 ⊔ ⨆ k, ⨆ (_ : k < n), u (k + 1) | 381864dc8c4d3882 |
List.nodup_permutations'Aux_iff | Mathlib/Data/List/Permutation.lean | theorem nodup_permutations'Aux_iff {s : List α} {x : α} : Nodup (permutations'Aux x s) ↔ x ∉ s | case intro.mk.a.inr.inl
α : Type u_1
s : List α
x : α
h : Injective (permutations'Aux x s).get
H : x ∈ s
n : ℕ
hk : n < s.length
hk' : s.get ⟨n, hk⟩ = x
kl : n < (permutations'Aux x s).length
k1l : n + 1 < (permutations'Aux x s).length
hl : (insertIdx n x s).length = (insertIdx (n + 1) x s).length
hn : n < (insertIdx n... | rw [get_insertIdx_self _ _ _ hk.le, get_insertIdx_of_lt _ _ _ _ (Nat.lt_succ_self _) hk, hk'] | no goals | beb3d6b4848d8597 |
Real.tendsto_harmonic_sub_log | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | lemma tendsto_harmonic_sub_log :
Tendsto (fun n : ℕ ↦ harmonic n - log n) atTop (𝓝 eulerMascheroniConstant) | ⊢ eulerMascheroniSeq' =ᶠ[atTop] fun n => ↑(harmonic n) - log ↑n | filter_upwards [eventually_ne_atTop 0] with n hn | case h
n : ℕ
hn : n ≠ 0
⊢ eulerMascheroniSeq' n = ↑(harmonic n) - log ↑n | 7c3b53a7104901b8 |
exists_nhds_square | Mathlib/Topology/Constructions.lean | theorem exists_nhds_square {s : Set (X × X)} {x : X} (hx : s ∈ 𝓝 (x, x)) :
∃ U : Set X, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s | X : Type u
inst✝ : TopologicalSpace X
s : Set (X × X)
x : X
hx : s ∈ 𝓝 (x, x)
⊢ ∃ U, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s | simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and_assoc, and_left_comm] using hx | no goals | 3d0c39ec7bd8ea00 |
LinearMap.ortho_smul_right | Mathlib/LinearAlgebra/SesquilinearForm.lean | theorem ortho_smul_right {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₂} {ha : a ≠ 0} :
IsOrtho B x y ↔ IsOrtho B x (a • y) | K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁]... | dsimp only [IsOrtho] | K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁]... | a33e5d231fee9035 |
ContMDiffFiberwiseLinear.locality_aux₁ | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | theorem ContMDiffFiberwiseLinear.locality_aux₁
(n : WithTop ℕ∞) (e : PartialHomeomorph (B × F) (B × F))
(h : ∀ p ∈ e.source, ∃ s : Set (B × F), IsOpen s ∧ p ∈ s ∧
∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u)
(hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : F →L[𝕜] F)) u)
(... | case intro.intro.refine_2
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpac... | rw [← hesu, e.restr_source_inter] | case intro.intro.refine_2
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpac... | 70cae96477ef715a |
Monoid.exponent_min | Mathlib/GroupTheory/Exponent.lean | theorem exponent_min (m : ℕ) (hpos : 0 < m) (hm : m < exponent G) : ∃ g : G, g ^ m ≠ 1 | G : Type u
inst✝ : Monoid G
m : ℕ
hpos : 0 < m
hm : m < exponent G
⊢ ∃ g, g ^ m ≠ 1 | by_contra! h | G : Type u
inst✝ : Monoid G
m : ℕ
hpos : 0 < m
hm : m < exponent G
h : ∀ (g : G), g ^ m = 1
⊢ False | d416baf409c8a971 |
Std.DHashMap.Internal.Raw₀.size_modify | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem size_modify (h : m.1.WF) {k : α} {f : β k → β k} :
(m.modify k f).1.size = m.1.size | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
h : m.val.WF
k : α
f : β k → β k
⊢ (m.modify k f).val.size = m.val.size | simp_to_model [modify] using List.length_modifyKey | no goals | 80628700a83a3777 |
CategoryTheory.Functor.eval_section_surjective_of_surjective | Mathlib/CategoryTheory/CofilteredSystem.lean | theorem eval_section_surjective_of_surjective (i : J) :
(fun s : F.sections => s.val i).Surjective := fun x => by
let s : Set (F.obj i) := {x}
haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
refine... | J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
⊢ ∃ a, (fun s => ↑s i) a = x | haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x) | J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this : ∀ (j : J), Nonempty ((F.toPreimages s).obj j)
⊢... | a4a32b2b370542c3 |
CategoryTheory.extensive_regular_generate_coherent | Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean | theorem extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) | case h.h.h.refine_2.transitive
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y : C
T S✝ : Sieve Y
a✝¹ : (coherentCoverage C).Saturate Y T
a✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, T.arrows f → (coherentCoverage C).Saturate Y_1 (Sieve.pullback f S✝)
a_ih✝¹ : T ∈... | apply Coverage.Saturate.transitive Y T<;> [assumption; assumption] | no goals | 82f80bbd286c526f |
Set.exists_equiv_range_eq_graphOn_univ | Mathlib/Data/Set/Function.lean | /-- **Line test** for equivalences.
Let `f : α → β × γ` be a homomorphism to a product of monoids. Assume that `f` is surjective on both
factors and that the image of `f` intersects every "vertical line" `{(b, c) | c : γ}` and every
"horizontal line" `{(b, c) | b : β}` at most once. Then the image of `f` is the graph ... | case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β × γ
hf₁ : Surjective (Prod.fst ∘ f)
hf₂ : Surjective (Prod.snd ∘ f)
hf : ∀ (g₁ g₂ : α), (f g₁).1 = (f g₂).1 ↔ (f g₁).2 = (f g₂).2
e₁ : β → γ
he₁ : range f = graphOn e₁ univ
e₂ : γ → β
he₂ : range (⇑(prodComm β γ) ∘ f) = graphOn e₂ univ
he₁₂ : ∀ (h : β) (... | exact ⟨
{ toFun := e₁
invFun := e₂
left_inv := fun h ↦ by rw [← he₁₂]
right_inv := fun i ↦ by rw [he₁₂] }, he₁⟩ | no goals | e623d6d79db3116f |
Set.Subsingleton.minimal_mem_iff | Mathlib/Order/Minimal.lean | theorem Set.Subsingleton.minimal_mem_iff (h : s.Subsingleton) : Minimal (· ∈ s) x ↔ x ∈ s | α : Type u_1
x : α
s : Set α
inst✝ : Preorder α
h : s.Subsingleton
⊢ Minimal (fun x => x ∈ s) x ↔ x ∈ s | obtain (rfl | ⟨x, rfl⟩) := h.eq_empty_or_singleton <;> simp | no goals | 61c1a58f51ea16ba |
Array.mapFinIdx_eq_push_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/MapIdx.lean | theorem mapFinIdx_eq_push_iff {l : Array α} {b : β} {f : (i : Nat) → α → (h : i < l.size) → β} :
l.mapFinIdx f = l₂.push b ↔
∃ (l₁ : Array α) (a : α) (w : l = l₁.push a),
l₁.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₂ ∧ b = f (l.size - 1) a (by simp [w]) | case mp
α : Type u_1
β : Type u_2
l₂ : Array β
l : Array α
b : β
f : (i : Nat) → α → i < l.size → β
⊢ (∃ l₁' l₂' w, (l₁'.mapFinIdx fun i a h => f i a ⋯) = l₂ ∧ (l₂'.mapFinIdx fun i a h => f (i + l₁'.size) a ⋯) = #[b]) →
∃ l₁ a w, (l₁.mapFinIdx fun i a_1 h => f i a_1 ⋯) = l₂ ∧ b = f (l.size - 1) a ⋯ | rintro ⟨l₁, l₂, rfl, rfl, h₂⟩ | case mp.intro.intro.intro.intro
α : Type u_1
β : Type u_2
b : β
l₁ l₂ : Array α
f : (i : Nat) → α → i < (l₁ ++ l₂).size → β
h₂ : (l₂.mapFinIdx fun i a h => f (i + l₁.size) a ⋯) = #[b]
⊢ ∃ l₁_1 a w,
((l₁_1.mapFinIdx fun i a_1 h => f i a_1 ⋯) = l₁.mapFinIdx fun i a h => f i a ⋯) ∧ b = f ((l₁ ++ l₂).size - 1) a ⋯ | 439910e5a37c1f8d |
MeasurableEmbedding.rnDeriv_map | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | lemma _root_.MeasurableEmbedding.rnDeriv_map (hf : MeasurableEmbedding f)
(μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
(fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν | α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝¹ : SigmaFinite (map f ν)
this✝ : SigmaFinite (map f (μ.singularPart ν))
this : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_add :
(fun x... | refine h_add.trans ((Measure.rnDeriv_add' _ _ _).trans ?_).symm | α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝¹ : SigmaFinite (map f ν)
this✝ : SigmaFinite (map f (μ.singularPart ν))
this : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_add :
(fun x... | ea80b5c7d85148f7 |
ComplexShape.Embedding.f_eq_of_r_eq_some | Mathlib/Algebra/Homology/Embedding/Basic.lean | lemma f_eq_of_r_eq_some {i : ι} {i' : ι'} (hi : e.r i' = some i) :
e.f i = i' | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i : ι
i' : ι'
hi : e.r i' = some i
⊢ e.f i = i' | by_cases h : ∃ (k : ι), e.f k = i' | case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i : ι
i' : ι'
hi : e.r i' = some i
h : ∃ k, e.f k = i'
⊢ e.f i = i'
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i : ι
i' : ι'
hi : e.r i' = some i
h : ¬∃ k, e.f k = i'
⊢ e.f... | 36befa20b79054b1 |
Matrix.fromBlocks_zero | Mathlib/Data/Matrix/Block.lean | theorem fromBlocks_zero [Zero α] : fromBlocks (0 : Matrix n l α) 0 0 (0 : Matrix o m α) = 0 | case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
inst✝ : Zero α
i : n ⊕ o
j : l ⊕ m
⊢ fromBlocks 0 0 0 0 i j = 0 i j | rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl | no goals | e4b7562999fc9629 |
MeasureTheory.IsSetSemiring.disjointOfUnion_props | Mathlib/MeasureTheory/SetSemiring.lean | theorem disjointOfUnion_props (hC : IsSetSemiring C) (h1 : ↑J ⊆ C) :
∃ K : Set α → Finset (Set α),
PairwiseDisjoint J K
∧ (∀ i ∈ J, ↑(K i) ⊆ C)
∧ PairwiseDisjoint (⋃ x ∈ J, (K x : Set (Set α))) id
∧ (∀ j ∈ J, ⋃₀ K j ⊆ j)
∧ (∀ j ∈ J, ∅ ∉ K j)
∧ ⋃₀ J = ⋃₀ (⋃ x ∈ J, (K x : Set (Set ... | case cons.intro.intro.intro.intro.intro.intro
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃... | use K1 | case h
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀... | defbf789029f4cb3 |
List.prod_isUnit_iff | Mathlib/Algebra/BigOperators/Group/List/Lemmas.lean | theorem prod_isUnit_iff {α : Type*} [CommMonoid α] {L : List α} :
IsUnit L.prod ↔ ∀ m ∈ L, IsUnit m | case nil
α : Type u_8
inst✝ : CommMonoid α
h : IsUnit [].prod
⊢ ∀ m ∈ [], IsUnit m | exact fun m' h' => False.elim (not_mem_nil m' h') | no goals | 8f8c6d61a54e9772 |
ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
{f : D → E} {g : D → F} {N : WithTop ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
(hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : n ≤ N) :
‖iteratedFDerivWithin 𝕜 n (fun y => ... | 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G... | refine ContinuousLinearMap.opNorm_le_bound _ (by positivity) fun x => ?_ | 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G... | c6fa542bdff00851 |
Array.eraseP_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Erase.lean | theorem eraseP_map (f : β → α) (l : Array β) : (map f l).eraseP p = map f (l.eraseP (p ∘ f)) | β : Type u_1
α : Type u_2
p : α → Bool
f : β → α
l : Array β
⊢ (map f l).eraseP p = map f (l.eraseP (p ∘ f)) | rcases l with ⟨l⟩ | case mk
β : Type u_1
α : Type u_2
p : α → Bool
f : β → α
l : List β
⊢ (map f { toList := l }).eraseP p = map f ({ toList := l }.eraseP (p ∘ f)) | 3c683efe6b8c7af3 |
MeasureTheory.lintegral_iSup_directed | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
(h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
⊢ ∀ᵐ (x : α) ∂μ, p x fun i => f i x | filter_upwards [] with x i j | case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
x : α
i j : β
⊢ ∃ z, (fun i => f i x) i ≤ (fun i => f i x) z ∧ (fun i =... | be3dd78af53e2dbc |
IsPrimitiveRoot.not_exists_int_prime_dvd_sub_of_prime_pow_ne_two | Mathlib/NumberTheory/Cyclotomic/Rat.lean | theorem not_exists_int_prime_dvd_sub_of_prime_pow_ne_two
[hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (htwo : p ^ (k + 1) ≠ 2) :
¬(∃ n : ℤ, (p : 𝓞 K) ∣ (hζ.toInteger - n : 𝓞 K)) | case pos
p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
n : ℤ
x : 𝓞 K
h : hζ.toInteger - ↑n = ↑↑p * x
pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasis
hdim : pB.dim = ↑p ^ k * (↑p - 1)
hk ... | exact htwo.symm.lt_of_le hp.1.two_le | no goals | 020fe74a3b14d73d |
Real.smul_map_diagonal_volume_pi | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | theorem smul_map_diagonal_volume_pi [DecidableEq ι] {D : ι → ℝ} (h : det (diagonal D) ≠ 0) :
ENNReal.ofReal (abs (det (diagonal D))) • Measure.map (toLin' (diagonal D)) volume =
volume | ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ofReal |∏ i : ι, D i| * volume (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi s) = ∏ x : ι, volume (s x)
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (M... | swap | ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ Measurable ⇑(toLin' (Matrix.diagonal D))
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), Meas... | f356868f3a872858 |
Metric.Sigma.isOpen_iff | Mathlib/Topology/MetricSpace/Gluing.lean | theorem isOpen_iff (s : Set (Σi, E i)) :
IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s | case mpr.intro.intro
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
s : Set ((i : ι) × E i)
H : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s
i : ι
x : E i
hx : x ∈ Sigma.mk i ⁻¹' s
ε : ℝ
εpos : ε > 0
hε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s
y : E i
hy : y ∈ ball x ε
⊢ y ... | apply hε ⟨i, y⟩ | case mpr.intro.intro
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
s : Set ((i : ι) × E i)
H : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s
i : ι
x : E i
hx : x ∈ Sigma.mk i ⁻¹' s
ε : ℝ
εpos : ε > 0
hε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s
y : E i
hy : y ∈ ball x ε
⊢ di... | 7b8d60884f4e1cdd |
Nat.max_le_of_le_of_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem max_le_of_le_of_le {a b c : Nat} : a ≤ c → b ≤ c → max a b ≤ c | case inl
a b c : Nat
a✝¹ : a ≤ c
a✝ : b ≤ c
h : a ≤ b
⊢ max a b ≤ c | rw [Nat.max_eq_right h] | case inl
a b c : Nat
a✝¹ : a ≤ c
a✝ : b ≤ c
h : a ≤ b
⊢ b ≤ c | c24f049c080ef4a9 |
Cardinal.mk_perm_eq_two_power | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem mk_perm_eq_two_power : #(Equiv.Perm α) = 2 ^ #α | α : Type u
inst✝ : Infinite α
⊢ #(Perm α) = 2 ^ #α | rw [mk_perm_eq_self_power, power_self_eq (aleph0_le_mk α)] | no goals | 3d4955e7f47c2151 |
geometric_hahn_banach_open_open | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
(ht₃ : IsOpen t) (disj : Disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b | E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
a₀ : E
ha₀ : a₀ ∈ s
ht₁ : Convex ℝ ∅
ht₃ : IsOpen ∅
disj : Disjoint s ∅
a : E
_ha : a ∈ s
⊢ 0 a < 1 | norm_num | no goals | a127ebc8b2d8721c |
WittVector.mulN_coeff | Mathlib/RingTheory/WittVector/MulP.lean | theorem mulN_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) :
(x * n).coeff k = aeval x.coeff (wittMulN p n k) | case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
n : ℕ
ih : ∀ (k : ℕ), (x * ↑n).coeff k = (aeval x.coeff) (wittMulN p n k)
k : ℕ
⊢ peval (wittAdd p k) ![(x * ↑n).coeff, x.coeff] =
(aeval fun i => (aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)) (wittAdd p k) | apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl | p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
n : ℕ
ih : ∀ (k : ℕ), (x * ↑n).coeff k = (aeval x.coeff) (wittMulN p n k)
k : ℕ
⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] = fun i => (aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i) | 1ea0f91cf5cc7a73 |
List.getElem_mapIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean | theorem getElem_mapIdx {l : List α} {f : Nat → α → β} {i : Nat} {h : i < (l.mapIdx f).length} :
(l.mapIdx f)[i] = f i (l[i]'(by simpa using h)) | α : Type u_1
β : Type u_2
l : List α
f : Nat → α → β
i : Nat
h : i < (mapIdx f l).length
⊢ some (mapIdx f l)[i] = some (f i l[i]) | rw [← getElem?_eq_getElem, getElem?_mapIdx, getElem?_eq_getElem (by simpa using h)] | α : Type u_1
β : Type u_2
l : List α
f : Nat → α → β
i : Nat
h : i < (mapIdx f l).length
⊢ Option.map (f i) (some l[i]) = some (f i l[i]) | d2755ac6b5cd5c59 |
Ordinal.isOpen_singleton_iff | Mathlib/SetTheory/Ordinal/Topology.lean | theorem isOpen_singleton_iff : IsOpen ({a} : Set Ordinal) ↔ ¬IsLimit a | case refine_1.intro.intro.intro
a : Ordinal.{u}
h : IsOpen {a}
ha : a.IsLimit
b c : Ordinal.{u}
hbc : a ∈ Set.Ioo b c
hbc' : Set.Ioo b c ⊆ {a}
⊢ False | have hba := ha.succ_lt hbc.1 | case refine_1.intro.intro.intro
a : Ordinal.{u}
h : IsOpen {a}
ha : a.IsLimit
b c : Ordinal.{u}
hbc : a ∈ Set.Ioo b c
hbc' : Set.Ioo b c ⊆ {a}
hba : succ b < a
⊢ False | c638dc4be42bbf41 |
Finset.prod_dite_eq' | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem prod_dite_eq' [DecidableEq α] (s : Finset α) (a : α) (b : ∀ x : α, x = a → β) :
∏ x ∈ s, (if h : x = a then b x h else 1) = ite (a ∈ s) (b a rfl) 1 | case neg.hnc
α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
a : α
b : (x : α) → x = a → β
h : a ∉ s
x✝ : α
a✝ : x✝ ∈ s
⊢ ¬x✝ = a | rintro rfl | case neg.hnc
α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
x✝ : α
a✝ : x✝ ∈ s
b : (x : α) → x = x✝ → β
h : x✝ ∉ s
⊢ False | 54ed43a0c8f53b8c |
Option.get_attach | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Attach.lean | theorem get_attach {o : Option α} (h : o.attach.isSome = true) :
o.attach.get h = ⟨o.get (by simpa using h), by simp⟩ | case none
α : Type u_1
h : none.attach.isSome = true
⊢ none.attach.get h = ⟨none.get ⋯, ⋯⟩ | simp at h | no goals | 4c6e8d0a413cc9e7 |
MeasureTheory.condExp_indep_eq | Mathlib/Probability/ConditionalExpectation.lean | theorem condExp_indep_eq (hle₁ : m₁ ≤ m) (hle₂ : m₂ ≤ m) [SigmaFinite (μ.trim hle₂)]
(hf : StronglyMeasurable[m₁] f) (hindp : Indep m₁ m₂ μ) : μ[f|m₂] =ᵐ[μ] fun _ => μ[f] | case pos.refine_5
Ω : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m : MeasurableSpace Ω
μ : Measure Ω
f : Ω → E
hle₁ : m₁ ≤ m
hle₂ : m₂ ≤ m
inst✝ : SigmaFinite (μ.trim hle₂)
hf : StronglyMeasurable f
hindp : Indep m₁ m₂ μ
hfint : MemLp f 1 μ
s : Set Ω
hms ... | rwa [← integral_congr_ae huv, ←
(setIntegral_congr_ae (hle₂ _ hms) _ : ∫ x in s, u x ∂μ = ∫ x in s, v x ∂μ)] | Ω : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m : MeasurableSpace Ω
μ : Measure Ω
f : Ω → E
hle₁ : m₁ ≤ m
hle₂ : m₂ ≤ m
inst✝ : SigmaFinite (μ.trim hle₂)
hf : StronglyMeasurable f
hindp : Indep m₁ m₂ μ
hfint : MemLp f 1 μ
s : Set Ω
hms : MeasurableSet s
... | 653df51a31c82027 |
isBigO_norm_Icc_restrict_atTop | Mathlib/Analysis/Fourier/PoissonSummation.lean | theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) | E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), 0 ⊔ -2 * R < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 + 0 ⊔ -2 * R ≤ x ∧ d - R ≤... | linarith | no goals | aa895dee78ab3fae |
PNat.Coprime.coprime_dvd_left | Mathlib/Data/PNat/Prime.lean | theorem Coprime.coprime_dvd_left {m k n : ℕ+} : m ∣ k → k.Coprime n → m.Coprime n | m k n : ℕ+
⊢ ↑m ∣ ↑k → k.Coprime n → m.Coprime n | repeat rw [← coprime_coe] | m k n : ℕ+
⊢ ↑m ∣ ↑k → (↑k).Coprime ↑n → (↑m).Coprime ↑n | 3a086300da92b6da |
Std.Sat.AIG.mkAtomCached_miss_aig | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean | theorem mkAtomCached_miss_aig (aig : AIG α) (hcache : aig.cache.get? (.atom var) = none) :
(aig.mkAtomCached var).aig.decls = aig.decls.push (.atom var) | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
var : α
aig : AIG α
hcache : aig.cache.get? (Decl.atom var) = none
⊢ (match aig.cache.get? (Decl.atom var) with
| some hit =>
{ aig := { decls := aig.decls, cache := aig.cache, invariant := ⋯ }, ref := { gate := hit.idx, hgate := ⋯ } }
| none ... | split <;> simp_all | no goals | 5450a04ceea6c7f5 |
Stream'.WSeq.length_eq_map | Mathlib/Data/Seq/WSeq.lean | theorem length_eq_map (s : WSeq α) : length s = Computation.map List.length (toList s) | α : Type u
s : WSeq α
⊢ Computation.IsBisimulation fun c1 c2 =>
∃ l s,
c1 =
Computation.corec
(fun x =>
match x with
| (n, s) =>
match Seq.destruct s with
| none => Sum.inl n
| some (none, s') => Sum.inr (n, s')
... | intro s1 s2 h | α : Type u
s : WSeq α
s1 s2 : Computation ℕ
h :
∃ l s,
s1 =
Computation.corec
(fun x =>
match x with
| (n, s) =>
match Seq.destruct s with
| none => Sum.inl n
| some (none, s') => Sum.inr (n, s')
| some (some val, s'... | 106c9ead6d923014 |
maximalIdeal_isPrincipal_of_isDedekindDomain | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | theorem maximalIdeal_isPrincipal_of_isDedekindDomain [IsLocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal | R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn ... | by_contra! h' | R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
n : ℕ
hn ... | e801db6f819d1797 |
MvQPF.Fix.rec_eq | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | theorem Fix.rec_eq {β : Type u} (g : F (append1 α β) → β) (x : F (append1 α (Fix F α))) :
Fix.rec g (Fix.mk x) = g (appendFun id (Fix.rec g) <$$> x) | n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : F (α ::: β) → β
x : F (α ::: Fix F α)
⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x) | have : recF g ∘ fixToW = Fix.rec g := by
apply funext
apply Quotient.ind
intro x
apply recF_eq_of_wEquiv
apply wrepr_equiv | n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : F (α ::: β) → β
x : F (α ::: Fix F α)
this : recF g ∘ fixToW = rec g
⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x) | 016b121b34bf62b7 |
HasProd.of_sigma | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | theorem HasProd.of_sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
(hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) (hg : HasProd g a)
(h : CauchySeq (fun (s : Finset (Σ b : β, γ b)) ↦ ∏ i ∈ s, f i)) :
HasProd f a | α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformGroup α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
hf : ∀ (b : β), HasProd (fun c => f ⟨b, c⟩) (g b)
hg : HasProd g a
h : CauchySeq fun s => ∏ i ∈ s, f i
u : Set α
hu : u ∈ 𝓝 a
s : Finset ((b : β) × γ b)
v : Set α
vu : v ... | exact (A.and (Ici_mem_atTop _)).exists | no goals | fae81e5b072596b2 |
AkraBazziRecurrence.GrowsPolynomially.add | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | protected lemma GrowsPolynomially.add {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hg : GrowsPolynomially g) (hf' : 0 ≤ᶠ[atTop] f) (hg' : 0 ≤ᶠ[atTop] g) :
GrowsPolynomially fun x => f x + g x | case h₂
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄... | exact (hg u hu).1 | no goals | 23da4f580f793a06 |
FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt | Mathlib/RingTheory/Multiplicity.lean | theorem FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt (hf : FiniteMultiplicity a b) {m : ℕ}
(hm : multiplicity a b < m) : ¬a ^ m ∣ b | case hm
α : Type u_1
inst✝ : Monoid α
a b : α
hf : FiniteMultiplicity a b
m : ℕ
hm : multiplicity a b < m
⊢ emultiplicity a b < ↑m | rw [hf.emultiplicity_eq_multiplicity] | case hm
α : Type u_1
inst✝ : Monoid α
a b : α
hf : FiniteMultiplicity a b
m : ℕ
hm : multiplicity a b < m
⊢ ↑(multiplicity a b) < ↑m | 8a0c58939e3c5124 |
FormalMultilinearSeries.derivSeries_eq_zero | Mathlib/Analysis/Analytic/ChangeOrigin.lean | theorem derivSeries_eq_zero {n : ℕ} (hp : p (n + 1) = 0) : p.derivSeries n = 0 | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hp : p (n + 1) = 0
⊢ p.changeOriginSeries 1 n = 0 | apply Finset.sum_eq_zero (fun s hs ↦ ?_) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hp : p (n + 1) = 0
s : { s // s.card = n }
hs : s ∈ Finset.univ
⊢ p.changeOriginSeriesTerm 1... | e92bd3437a4bd85b |
DirichletCharacter.re_log_comb_nonneg | Mathlib/NumberTheory/LSeries/Nonvanishing.lean | private lemma re_log_comb_nonneg {n : ℕ} (hn : 2 ≤ n) {x : ℝ} (hx : 1 < x) (y : ℝ) :
0 ≤ 3 * (-log (1 - (1 : DirichletCharacter ℂ N) n * n ^ (-x : ℂ))).re +
4 * (-log (1 - χ n * n ^ (-(x + I * y)))).re +
(-log (1 - (χ n ^ 2) * n ^ (-(x + 2 * I * y)))).re | case pos
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
ha₀ : 0 ≤ ↑n ^ (-x)
ha₁ : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ 0 ≤
3 * (-log (1 - ↑n ^ (-↑x))).re + 4 * (-log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y)))).re +
(-log (1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))... | convert re_log_comb_nonneg' ha₀ ha₁ hz using 6 | case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
ha₀ : 0 ≤ ↑n ^ (-x)
ha₁ : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ log (1 - ↑n ^ (-↑x)) = log (1 - ↑(↑n ^ (-x)))
case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3
N : ℕ
χ : Dir... | 7c14ed735b43068e |
CategoryTheory.Limits.biprod.total | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | theorem biprod.total : biprod.fst ≫ biprod.inl + biprod.snd ≫ biprod.inr = 𝟙 (X ⊞ Y) | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ fst ≫ inl + snd ≫ inr = 𝟙 (X ⊞ Y) | ext <;> simp [add_comp] | no goals | b80ccbe50d9cc7fd |
MvQPF.Cofix.dest_corec' | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | theorem Cofix.dest_corec' {α : TypeVec.{u} n} {β : Type u}
(g : β → F (α.append1 (Cofix F α ⊕ β))) (x : β) :
Cofix.dest (Cofix.corec' g x) =
appendFun id (Sum.elim _root_.id (Cofix.corec' g)) <$$> g x | n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x : β
⊢ (corec' g x).dest = (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x | rw [Cofix.corec', Cofix.dest_corec] | n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x : β
⊢ (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) ∘ dest) g)) <$$>
Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) ∘ dest) g (Sum.inr x) =
(TypeVec.id ::: Sum.elim _... | be80bc82f7ec558d |
MvPowerSeries.le_weightedOrder | Mathlib/RingTheory/MvPowerSeries/Order.lean | theorem le_weightedOrder {n : ℕ∞} (h : ∀ d : σ →₀ ℕ, weight w d < n → coeff R d f = 0) :
n ≤ f.weightedOrder w | case top.h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
h : ∀ (d : σ →₀ ℕ), ↑((weight w) d) < ⊤ → (coeff R d) f = 0
d : σ →₀ ℕ
⊢ (coeff R d) f = (coeff R d) 0 | exact h d (ENat.coe_lt_top _) | no goals | e9609904000f25c2 |
FiniteInter.finiteInterClosure_insert | Mathlib/Data/Set/Constructions.lean | theorem finiteInterClosure_insert {A : Set α} (cond : FiniteInter S) (P)
(H : P ∈ finiteInterClosure (insert A S)) : P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q | case basic
α : Type u_1
S : Set (Set α)
A : Set α
cond : FiniteInter S
P s✝ : Set α
h : s✝ ∈ insert A S
⊢ s✝ ∈ S ∨ ∃ Q ∈ S, s✝ = A ∩ Q | cases h | case basic.inl
α : Type u_1
S : Set (Set α)
A : Set α
cond : FiniteInter S
P s✝ : Set α
h✝ : s✝ = A
⊢ s✝ ∈ S ∨ ∃ Q ∈ S, s✝ = A ∩ Q
case basic.inr
α : Type u_1
S : Set (Set α)
A : Set α
cond : FiniteInter S
P s✝ : Set α
h✝ : s✝ ∈ S
⊢ s✝ ∈ S ∨ ∃ Q ∈ S, s✝ = A ∩ Q | 299bf10b6982fec7 |
ContinuousLinearMap.exists_mul_lt_of_lt_opNorm | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | theorem exists_mul_lt_of_lt_opNorm (f : E →SL[σ₁₂] F) {r : ℝ} (hr₀ : 0 ≤ r) (hr : r < ‖f‖) :
∃ x, r * ‖x‖ < ‖f x‖ | case intro
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_6
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂]... | exact f.exists_mul_lt_apply_of_lt_opNNNorm hr | no goals | 60a5919e60f1c4e2 |
Real.arccos_eq_arctan | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | theorem arccos_eq_arctan {x : ℝ} (h : 0 < x) : arccos x = arctan (√(1 - x ^ 2) / x) | x : ℝ
h : 0 < x
⊢ arccos x = arctan (√(1 - x ^ 2) / x) | rw [arccos, eq_comm] | x : ℝ
h : 0 < x
⊢ arctan (√(1 - x ^ 2) / x) = π / 2 - arcsin x | f0b304be4e2c2b7b |
MeasureTheory.IsStoppingTime.measurableSpace_le | Mathlib/Probability/Process/Stopping.lean | theorem measurableSpace_le [IsCountablyGenerated (atTop : Filter ι)] [IsDirected ι (· ≤ ·)]
(hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m | case inr.intro
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Te... | exact MeasurableSet.iUnion fun i => f.le (seq i) _ (hs (seq i)) | no goals | 558f46e9446c7fd1 |
Real.arctan_add_arctan_lt_pi_div_two | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | lemma arctan_add_arctan_lt_pi_div_two {x y : ℝ} (h : x * y < 1) : arctan x + arctan y < π / 2 | case inr
x y : ℝ
h : x * y < 1
hy : 0 < y
⊢ arctan x + arctan y < π / 2 | rw [← lt_div_iff₀ hy, ← inv_eq_one_div] at h | case inr
x y : ℝ
h : x < y⁻¹
hy : 0 < y
⊢ arctan x + arctan y < π / 2 | 61d35e5c47383b65 |
Function.eq_update_self_iff | Mathlib/Logic/Function/Basic.lean | @[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b | α : Sort u
β : α → Sort v
inst✝ : DecidableEq α
f : (a : α) → β a
a : α
b : β a
⊢ f = update f a b ↔ f a = b | simp [eq_update_iff] | no goals | aca3e5c9b472e2c1 |
CategoryTheory.Presheaf.isLocallyInjective_of_whisker | Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.lean | lemma isLocallyInjective_of_whisker (hH : CoverPreserving J K H)
[H.IsCoverDense K] [IsLocallyInjective J (whiskerLeft H.op f)] : IsLocallyInjective K f where
equalizerSieve_mem {X} a b h | case h.hS.refine_1
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{u_6, u_1} C
inst✝⁵ : Category.{u_7, u_2} D
inst✝⁴ : Category.{u_8, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA... | simpa using hq | no goals | 30c94ee735ff7949 |
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