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 |
|---|---|---|---|---|---|---|
ModularGroup.g_eq_of_c_eq_one | Mathlib/NumberTheory/Modular.lean | theorem g_eq_of_c_eq_one (hc : g 1 0 = 1) : g = T ^ g 0 0 * S * T ^ g 1 1 | g : SL(2, ℤ)
hc : ↑g 1 0 = 1
hg : ↑g 0 1 = ↑g 0 0 * ↑g 1 1 - 1
⊢ !![↑g 0 0, ↑g 0 1; ↑g 1 0, ↑g 1 1] = ↑(T ^ ↑g 0 0 * S * T ^ ↑g 1 1) | simp only [hg, sub_eq_add_neg, hc, coe_mul, coe_T_zpow, coe_S, mul_fin_two, mul_zero, mul_one,
zero_add, one_mul, add_zero, zero_mul] | no goals | 87cab62460b6b82b |
Stirling.factorial_isEquivalent_stirling | Mathlib/Analysis/SpecialFunctions/Stirling.lean | /-- **Stirling's Formula**, formulated in terms of `Asymptotics.IsEquivalent`. -/
lemma factorial_isEquivalent_stirling :
(fun n ↦ n ! : ℕ → ℝ) ~[atTop] fun n ↦ Real.sqrt (2 * n * π) * (n / exp 1) ^ n | case refine_2
⊢ Tendsto ((fun n => ↑n !) / fun n => √(2 * ↑n * π) * (↑n / rexp 1) ^ n) atTop (𝓝 1) | have : sqrt π ≠ 0 := by positivity | case refine_2
this : √π ≠ 0
⊢ Tendsto ((fun n => ↑n !) / fun n => √(2 * ↑n * π) * (↑n / rexp 1) ^ n) atTop (𝓝 1) | bd93a102ceca7c20 |
intervalIntegral.intervalIntegrable_rpow' | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b | a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
hderiv : ∀ x ∈ Set.Ioo 0 c, HasDerivAt (fun x => x ^ (r + 1) / (r + 1)) (x ^ r) x
⊢ ContinuousOn (fun x => x ^ (r + 1) / (r + 1)) (Set.Icc 0 c) | refine (continuousOn_id.rpow_const ?_).div_const _ | a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
hderiv : ∀ x ∈ Set.Ioo 0 c, HasDerivAt (fun x => x ^ (r + 1) / (r + 1)) (x ^ r) x
⊢ ∀ x ∈ Set.Icc 0 c, id x ≠ 0 ∨ 0 ≤ r + 1 | 03dcc4f722838526 |
Batteries.RBNode.Balanced.append | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean | theorem Balanced.append {l r : RBNode α}
(hl : l.Balanced c₁ n) (hr : r.Balanced c₂ n) :
(l.append r).RedRed (c₁ = black → c₂ ≠ black) n | α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝³ x✝² a✝⁴ : RBNode α
x✝¹ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node black a✝⁴ x✝¹ b).Balanced c₁ n
c₁✝² : RBColor
n✝ : Nat
c₂✝² : RBColor
ha : a✝⁴.Balanced c₁✝² n✝
hb : b.Balanced c₂✝² n✝
hr : (node black c y✝ d✝).Balanced c₂ (n✝ + 1)
c₁✝¹ c₂✝¹ : RBColor
hc : c.B... | cases H _ _ _ rfl | no goals | b3ecc8a07b6c9ec6 |
AlgebraicGeometry.Scheme.IdealSheafData.isClosed_support | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma isClosed_support (I : IdealSheafData X) : IsClosed I.support | X : Scheme
I : X.IdealSheafData
U : ↑X.affineOpens
⊢ IsClosed (Subtype.val ⁻¹' I.support) | refine ⟨(X.zeroLocus (U := U.1) (I.ideal U))ᶜ, (X.zeroLocus_isClosed _).isOpen_compl, ?_⟩ | X : Scheme
I : X.IdealSheafData
U : ↑X.affineOpens
⊢ Subtype.val ⁻¹' (X.zeroLocus ↑(I.ideal U))ᶜ = (Subtype.val ⁻¹' I.support)ᶜ | ee5d35732d6710dc |
HurwitzZeta.hasSum_int_evenKernel₀ | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | /-- Modified version of `hasSum_int_evenKernel` omitting the constant term at `∞`. -/
lemma hasSum_int_evenKernel₀ (a : ℝ) {t : ℝ} (ht : 0 < t) :
HasSum (fun n : ℤ ↦ if n + a = 0 then 0 else rexp (-π * (n + a) ^ 2 * t))
(evenKernel a t - if (a : UnitAddCircle) = 0 then 1 else 0) | a t : ℝ
ht : 0 < t
this✝ : (a : Prop) → Decidable a
h : ¬∃ n, ↑n = a
this : ∀ (n : ℤ), ↑n + a ≠ 0
⊢ HasSum (fun n => if ↑n + a = 0 then 0 else rexp (-π * (↑n + a) ^ 2 * t)) (evenKernel (↑a) t - 0) | simpa [this] using hasSum_int_evenKernel a ht | no goals | 18fe5d7edd31b1c5 |
Nat.compare_def_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Compare.lean | theorem compare_def_le (a b : Nat) :
compare a b = if a ≤ b then if b ≤ a then .eq else .lt else .gt | a b : Nat
hge : ¬a < b
hgt : b < a
⊢ Ordering.gt = if a ≤ b then if b ≤ a then Ordering.eq else Ordering.lt else Ordering.gt | simp [Nat.le_of_lt hgt, Nat.not_le.2 hgt] | no goals | 3aacc245f4a53309 |
Filter.coprodᵢ_cocompact | Mathlib/Topology/Compactness/Compact.lean | theorem Filter.coprodᵢ_cocompact {X : ι → Type*} [∀ d, TopologicalSpace (X d)] :
(Filter.coprodᵢ fun d => Filter.cocompact (X d)) = Filter.cocompact (∀ d, X d) | ι : Type u_1
X : ι → Type u_3
inst✝ : (d : ι) → TopologicalSpace (X d)
s : Set ((i : ι) → X i)
K : (i : ι) → Set (X i)
hKc : ∀ (i : ι), IsCompact (K i)
htK : ∀ (i : ι), s ⊆ eval i ⁻¹' K i
⊢ ∃ t, IsCompact t ∧ s ⊆ t | exact ⟨Set.pi univ K, isCompact_univ_pi hKc, fun f hf i _ => htK i hf⟩ | no goals | 2c4ce7cb112d1885 |
Algebra.FiniteType.iff_quotient_freeAlgebra | Mathlib/RingTheory/FiniteType.lean | theorem iff_quotient_freeAlgebra :
FiniteType R A ↔
∃ (s : Finset A) (f : FreeAlgebra R s →ₐ[R] A), Surjective f | R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ FiniteType R A ↔ ∃ s f, Surjective ⇑f | constructor | case mp
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ FiniteType R A → ∃ s f, Surjective ⇑f
case mpr
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ (∃ s f, Surjective ⇑f) → FiniteType R A | 173fcd34459e6b97 |
FixedPoints.isIntegral | Mathlib/FieldTheory/Fixed.lean | theorem isIntegral [Finite G] (x : F) : IsIntegral (FixedPoints.subfield G F) x | G : Type u
inst✝³ : Group G
F : Type v
inst✝² : Field F
inst✝¹ : MulSemiringAction G F
inst✝ : Finite G
x : F
⊢ IsIntegral (↥(subfield G F)) x | cases nonempty_fintype G | case intro
G : Type u
inst✝³ : Group G
F : Type v
inst✝² : Field F
inst✝¹ : MulSemiringAction G F
inst✝ : Finite G
x : F
val✝ : Fintype G
⊢ IsIntegral (↥(subfield G F)) x | db4be496b7cabbf3 |
CategoryTheory.Limits.biprod.associator_inv_natural | Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.lean | theorem biprod.associator_inv_natural {U V W X Y Z : C} (f : U ⟶ X) (g : V ⟶ Y) (h : W ⟶ Z) :
biprod.map f (biprod.map g h) ≫ (biprod.associator _ _ _).inv
= (biprod.associator _ _ _).inv ≫ biprod.map (biprod.map f g) h | C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
U V W X Y Z : C
f : U ⟶ X
g : V ⟶ Y
h : W ⟶ Z
⊢ map f (map g h) ≫ (associator X Y Z).inv = (associator U V W).inv ≫ map (map f g) h | aesop_cat | no goals | 53a178e62bb2837e |
jacobiSum_mul_jacobiSum_inv | Mathlib/NumberTheory/JacobiSum/Basic.lean | /-- If `χ` and `φ` are multiplicative characters on a finite field `F` with values in another
field `F'` such that `χ`, `φ` and `χφ` are all nontrivial and `char F' ≠ char F`, then
`J(χ,φ) * J(χ⁻¹,φ⁻¹) = #F` (in `F'`). -/
lemma jacobiSum_mul_jacobiSum_inv (h : ringChar F' ≠ ringChar F) {χ φ : MulChar F F'} (hχ : χ ≠ 1)... | case intro.intro.a
F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
... | rw [map_mul, ← jacobiSum_ringHomComp, ← jacobiSum_ringHomComp] | case intro.intro.a
F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
... | 5b48f17367fbeedf |
Finsupp.mapDomain_injective | Mathlib/Data/Finsupp/Basic.lean | theorem mapDomain_injective {f : α → β} (hf : Function.Injective f) :
Function.Injective (mapDomain f : (α →₀ M) → β →₀ M) | α : Type u_1
β : Type u_2
M : Type u_5
inst✝ : AddCommMonoid M
f : α → β
hf : Injective f
v₁ v₂ : α →₀ M
eq : mapDomain f v₁ = mapDomain f v₂
a : α
⊢ (mapDomain f v₁) (f a) = (mapDomain f v₂) (f a) | rw [eq] | no goals | 7c9da2f7a7413ad4 |
Std.DHashMap.Internal.List.getValueCast_insertEntry | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast_insertEntry [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α}
{v : β k} {h} : getValueCast a (insertEntry k v l) h =
if h' : k == a then cast (congrArg β (eq_of_beq h')) v
else getValueCast a l (containsKey_of_containsKey_insertEntry h (Bool.eq_false_iff.2 h')) | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List ((a : α) × β a)
k a : α
v : β k
h : containsKey a (insertEntry k v l) = true
⊢ getValueCast a (insertEntry k v l) h = if h' : (k == a) = true then cast ⋯ v else getValueCast a l ⋯ | rw [← Option.some_inj, ← getValueCast?_eq_some_getValueCast, apply_dite Option.some,
getValueCast?_insertEntry] | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List ((a : α) × β a)
k a : α
v : β k
h : containsKey a (insertEntry k v l) = true
⊢ (if h : (k == a) = true then some (cast ⋯ v) else getValueCast? a l) =
if h : (k == a) = true then some (cast ⋯ v) else some (getValueCast a l ⋯) | 39ea179860319449 |
WeierstrassCurve.Affine.CoordinateRing.degree_norm_ne_one | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | lemma degree_norm_ne_one [IsDomain R] (x : W.CoordinateRing) :
(Algebra.norm R[X] x).degree ≠ 1 | case intro.intro
R : Type u
inst✝¹ : CommRing R
W : Affine R
inst✝ : IsDomain R
p q : R[X]
⊢ 2 • p.degree ⊔ (2 • q.degree + 3) ≠ 1 | rcases p.degree with (_ | _ | _ | _) <;> cases q.degree | case intro.intro.none.bot
R : Type u
inst✝¹ : CommRing R
W : Affine R
inst✝ : IsDomain R
p q : R[X]
⊢ 2 • none ⊔ (2 • ⊥ + 3) ≠ 1
case intro.intro.none.coe
R : Type u
inst✝¹ : CommRing R
W : Affine R
inst✝ : IsDomain R
p q : R[X]
a✝ : ℕ
⊢ 2 • none ⊔ (2 • ↑a✝ + 3) ≠ 1
case intro.intro.some.zero.bot
R : Type u
inst✝¹ : ... | 0deae90da8556036 |
MeasureTheory.Measure.integral_comp_smul | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | theorem integral_comp_smul (f : E → F) (R : ℝ) :
∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ | case pos
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
hF : CompleteSpace F
⊢ ∫ (x : E), f (R... | rcases eq_or_ne R 0 with (rfl | hR) | case pos.inl
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
hF : CompleteSpace F
⊢ ∫ (x : E), f (0 •... | 9139a34e8ce6538f |
CategoryTheory.Abelian.has_injective_coseparator | Mathlib/CategoryTheory/Generator/Abelian.lean | theorem has_injective_coseparator [HasLimits C] [EnoughInjectives C] (G : C) (hG : IsSeparator G) :
∃ G : C, Injective G ∧ IsCoseparator G | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasLimits C
inst✝ : EnoughInjectives C
G : C
hG : IsSeparator G
this✝ : WellPowered.{v, v, u} C
this : HasProductsOfShape (Subobject (op G)) C
T : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))
X Y : C
f : X ⟶ Y
hf : ∀ (h : Y ⟶ T), ... | refine (Preadditive.isSeparator_iff _).1 hG _ fun h => ?_ | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasLimits C
inst✝ : EnoughInjectives C
G : C
hG : IsSeparator G
this✝ : WellPowered.{v, v, u} C
this : HasProductsOfShape (Subobject (op G)) C
T : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))
X Y : C
f : X ⟶ Y
hf : ∀ (h : Y ⟶ T), ... | 6b085b71b2f6e5e0 |
Order.succ_le_succ | Mathlib/Order/SuccPred/Basic.lean | theorem succ_le_succ (h : a ≤ b) : succ a ≤ succ b | case pos
α : Type u_1
inst✝¹ : Preorder α
inst✝ : SuccOrder α
a b : α
h : a ≤ b
hb : IsMax b
hba : b ≤ a
⊢ succ a ≤ succ b | exact (hb <| hba.trans <| le_succ _).trans (le_succ _) | no goals | 72f7740b504b5cec |
List.erase_eq_eraseP' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem erase_eq_eraseP' (a : α) (l : List α) : l.erase a = l.eraseP (· == a) | case nil
α : Type u_1
inst✝ : BEq α
a : α
⊢ [].erase a = eraseP (fun x => x == a) [] | simp | no goals | a7d8cd443cc25e52 |
Set.Countable.isPathConnected_compl_of_one_lt_rank | Mathlib/Analysis/NormedSpace/Connected.lean | theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ | E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib ... | have N : {c - x} ∩ s = ∅ := by
simpa only [singleton_inter_eq_empty, mem_compl_iff, Ia] using ha | E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib ... | a2405a5e761a2c4d |
Complex.continuousAt_arg_coe_angle | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x | case neg
x : ℂ
h : x ≠ 0
hs : x ∉ slitPlane
⊢ ContinuousAt (((Real.Angle.coe ∘ arg) ∘ Neg.neg) ∘ Neg.neg) x | refine ContinuousAt.comp ?_ continuous_neg.continuousAt | case neg
x : ℂ
h : x ≠ 0
hs : x ∉ slitPlane
⊢ ContinuousAt ((Real.Angle.coe ∘ arg) ∘ Neg.neg) (-x) | ff54db4c8efcf6dc |
LightCondMod.isDiscrete_tfae | Mathlib/Condensed/Discrete/Characterization.lean | theorem isDiscrete_tfae (M : LightCondMod.{u} R) :
TFAE
[ M.IsDiscrete
, IsIso ((LightCondensed.discreteUnderlyingAdj _).counit.app M)
, (LightCondensed.discrete _).essImage M
, (LightCondMod.LocallyConstant.functor R).essImage M
, IsIso ((LightCondMod.LocallyConstant.adjunction R).counit.app M... | R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
tfae_1_iff_4 : LightCondensed.IsDiscrete M ↔ (functor R).ess... | exact ⟨isColimitOfPreserves (forget (ModuleCat R)) (h S).some⟩ | no goals | 76d6b28951472ebd |
GenContFract.contsAux_eq_contsAux_squashGCF_of_le | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | theorem contsAux_eq_contsAux_squashGCF_of_le {m : ℕ} :
m ≤ n → contsAux g m = (squashGCF g n).contsAux m :=
Nat.strong_induction_on m
(by
clear m
intro m IH m_le_n
rcases m with - | m'
· rfl
· rcases n with - | n'
· exact (m'.not_succ_le_zero m_le_n).elim
-- 1 ≰ 0... | case succ.succ.zero
K : Type u_1
g : GenContFract K
inst✝ : DivisionRing K
n' : ℕ
IH : ∀ m < 0 + 1, m ≤ n' + 1 → g.contsAux m = (g.squashGCF (n' + 1)).contsAux m
m_le_n : 0 + 1 ≤ n' + 1
⊢ g.contsAux (0 + 1) = (g.squashGCF (n' + 1)).contsAux (0 + 1) | rfl | no goals | cf127cafa12b15bc |
Module.End.map_add_of_iInf_iSup_genEigenspace_ne_bot_of_commute | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | @[deprecated map_add_of_iInf_genEigenspace_ne_bot_of_commute (since := "2024-10-23")]
lemma map_add_of_iInf_iSup_genEigenspace_ne_bot_of_commute [NoZeroSMulDivisors R M]
{L F : Type*} [Add L] [FunLike F L (End R M)] [AddHomClass F L (End R M)] (f : F)
(μ : L → R) (h_ne : ⨅ x, ⨆ k : ℕ, (f x).genEigenspace (μ x) ... | R : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : NoZeroSMulDivisors R M
L : Type u_1
F : Type u_2
inst✝² : Add L
inst✝¹ : FunLike F L (End R M)
inst✝ : AddHomClass F L (End R M)
f : F
μ : L → R
h_ne : ⨅ x, ⨆ k, ((f x).genEigenspace (μ x)) ↑k ≠ ⊥
h : ∀ (x y : L), Commute (f ... | simp_rw [iSup_genEigenspace_eq] at h_ne | R : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : NoZeroSMulDivisors R M
L : Type u_1
F : Type u_2
inst✝² : Add L
inst✝¹ : FunLike F L (End R M)
inst✝ : AddHomClass F L (End R M)
f : F
μ : L → R
h : ∀ (x y : L), Commute (f x) (f y)
x y : L
h_ne : ⨅ x, (f x).maxGenEigenspace ... | d1ddaabe6646eb77 |
CategoryTheory.Subgroupoid.mem_full_objs_iff | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | theorem mem_full_objs_iff {c : C} : c ∈ (full D).objs ↔ c ∈ D | C : Type u
inst✝ : Groupoid C
D : Set C
c : C
⊢ c ∈ (full D).objs ↔ c ∈ D | rw [full_objs] | no goals | 08dce4792f1e3b3a |
Matrix.mul_eq_one_comm | Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean | theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1 | n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A✝ B✝ A B : Matrix n n R
hAB : A * B = 1
h0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) = detp 1 A * detp 1 B + detp (-1) A * detp (-1) B
⊢ B * A = 1 | replace h := congr(B * $(detp_smul_adjp hAB)) | n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A✝ B✝ A B : Matrix n n R
hAB : A * B = 1
h0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) = detp 1 A * detp 1 B + detp (-1) A * detp (-1) B
h : B * (A + (detp 1 A • adjp (-1) B + detp (-1) A • adjp 1 B)) = B * (detp 1 A... | 7b3f481849b56cef |
Fintype.mem_span_image_iff_exists_fun | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem Fintype.mem_span_image_iff_exists_fun {s : Set α} [Fintype s] :
x ∈ span R (v '' s) ↔ ∃ c : s → R, ∑ i, c i • v i = x | α : Type u_1
M : Type u_2
R : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
v : α → M
x : M
s : Set α
inst✝ : Fintype ↑s
⊢ x ∈ span R (v '' s) ↔ ∃ c, ∑ i : ↑s, c i • v ↑i = x | rw [← mem_span_range_iff_exists_fun, image_eq_range] | no goals | 9b1bb32beed91183 |
one_lt_mabs | Mathlib/Algebra/Order/Group/Unbundled/Abs.lean | @[to_additive (attr := simp) abs_pos] lemma one_lt_mabs : 1 < |a|ₘ ↔ a ≠ 1 | case inl
α : Type u_1
inst✝² : Group α
inst✝¹ : LinearOrder α
inst✝ : MulLeftMono α
a : α
ha : a < 1
⊢ 1 < mabs a ↔ a ≠ 1 | simp [mabs_of_lt_one ha, neg_pos, ha.ne, ha] | no goals | 246ff238fd37060c |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRup | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem sat_of_insertRup {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f) (c : DefaultClause n)
(p : PosFin n → Bool) (pf : p ⊨ f) :
(insertRupUnits f (negate c)).2 = true → p ⊨ c | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false = true → ∃ i, f.assignments[i.val] = both
i : PosFin n
hboth : (List.foldl inse... | intro i | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false = true → ∃ i, f.assignments[i.val] = both
i✝ : PosFin n
hboth : (List.foldl ins... | 6b3be4f74b21096f |
ContinuousGroupoid.mem_of_source_eq_empty | Mathlib/Geometry/Manifold/IsManifold/Basic.lean | /-- Any change of coordinates with empty source belongs to `continuousGroupoid`. -/
lemma ContinuousGroupoid.mem_of_source_eq_empty (f : PartialHomeomorph H H)
(hf : f.source = ∅) : f ∈ continuousGroupoid H | 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
f : PartialHomeomorph H H
hf : f.source = ∅
⊢ f ∈ contDiffGroupoid 0 I | exact ContDiffGroupoid.mem_of_source_eq_empty f hf | no goals | dbba7f8ea2b6cf15 |
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepFun.indepFun_finset (S T : Finset ι) (hST : Disjoint S T)
(hf_Indep : iIndepFun m f κ μ) (hf_meas : ∀ i, Measurable (f i)) :
IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) κ μ | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᶠ[ae μ] ⇑η
hη : IsMa... | rwa [dif_pos hiT] at h | no goals | 0da746b38698ee90 |
HolderOnWith.hausdorffMeasure_image_le | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem hausdorffMeasure_image_le (h : HolderOnWith C r f s) (hr : 0 < r) {d : ℝ} (hd : 0 ≤ d) :
μH[d] (f '' s) ≤ (C : ℝ≥0∞) ^ d * μH[r * d] s | case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ⊤
⊢ μH[d] (f... | simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hCd0 hCd,
← ENNReal.tsum_mul_left] | case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ⊤
⊢ ⨆ r,
... | cf145f55a786d8cb |
Turing.TM2to1.tr_respects_aux₂ | Mathlib/Computability/TuringMachine.lean | theorem tr_respects_aux₂ [DecidableEq K] {k : K} {q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ} {v : σ}
{S : ∀ k, List (Γ k)} {L : ListBlank (∀ k, Option (Γ k))}
(hL : ∀ k, L.map (proj k) = ListBlank.mk ((S k).map some).reverse) (o : StAct K Γ σ k) :
let v' := stVar v (S k) o
let Sk' := stWrite v (S k) o
le... | case push
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Γ k
this :
∀ (L R : Lis... | refine ListBlank.ext fun i ↦ ?_ | case push
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Γ k
this :
∀ (L R : Lis... | 77a7f2ed036af218 |
MvPolynomial.aevalTower_id | Mathlib/Algebra/MvPolynomial/Eval.lean | theorem aevalTower_id :
aevalTower (AlgHom.id S S) = (aeval : (σ → S) → MvPolynomial σ S →ₐ[S] S) | case h.hf
σ : Type u_1
S : Type u_2
inst✝ : CommSemiring S
x✝ : σ → S
i✝ : σ
⊢ (aevalTower (AlgHom.id S S) x✝) (X i✝) = (aeval x✝) (X i✝) | simp only [aevalTower_X, aeval_X] | no goals | 3985ba8a1a02a809 |
MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α}
{μ : Measure α} [SigmaFinite μ] {f : α →ₛ ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ | case pos.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (support ⇑f₁) (support ⇑f₂)
h₁ :
∀ {L : ℝ≥0∞},
L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ
h₂ :
∀ {L : ℝ≥0∞},
... | simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_right, zero_le'] | no goals | 63c32f2aa18a237b |
SetTheory.PGame.Domineering.card_of_mem_left | Mathlib/SetTheory/Game/Domineering.lean | theorem card_of_mem_left {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) : 2 ≤ Finset.card b | b : Board
m : ℤ × ℤ
h : m ∈ left b
w₁ : m ∈ b
w₂ : (m.1, m.2 - 1) ∈ Finset.erase b m
i₁ : (Finset.erase b m).card < Finset.card b
i₂ : 0 < (Finset.erase b m).card
⊢ 2 ≤ Finset.card b | exact Nat.lt_of_le_of_lt i₂ i₁ | no goals | 6e4aa3d2a839c75a |
Array.getElem_modify | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_modify {as : Array α} {x i} (h : i < (as.modify x f).size) :
(as.modify x f)[i] = if x = i then f (as[i]'(by simpa using h)) else as[i]'(by simpa using h) | α : Type u_1
f : α → α
as : Array α
x : Nat
h✝ : ¬x < as.size
h : x < (as.modify x f).size
⊢ ?m.511823 h | exact h | no goals | da4e64796b761406 |
VectorFourier.hasFDerivAt_fourierChar_smul | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | /-- The `w`-derivative of the Fourier transform integrand. -/
lemma hasFDerivAt_fourierChar_smul (v : V) (w : W) :
HasFDerivAt (fun w' ↦ 𝐞 (-L v w') • f v) (𝐞 (-L v w) • fourierSMulRight L f v) w | case h.e'_12.h.h.h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
v : V
w : W
ha : HasFDerivAt (fun w' => (L v) w') (L v) w
e_8✝ : S... | rw [ContinuousLinearMap.comp_apply, ContinuousLinearMap.neg_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, ← smul_assoc, smul_comm,
← smul_assoc, real_smul, real_smul, Submonoid.smul_def, smul_eq_mul] | case h.e'_12.h.h.h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
v : V
w : W
ha : HasFDerivAt (fun w' => (L v) w') (L v) w
e_8✝ : S... | 487c2b0a207b024e |
Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | lemma nhdsWithin_stolzCone_le_nhdsWithin_stolzSet {s : ℝ} (hs : 0 < s) :
∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1 | case h
s : ℝ
hs : 0 < s
M ε : ℝ
left✝ : 0 < M
hε : 0 < ε
H : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M
⊢ ∃ u, IsOpen u ∧ 1 ∈ u ∧ u ∩ stolzCone s ⊆ stolzSet M | refine ⟨{w | 1 - ε < w.re}, isOpen_lt continuous_const continuous_re, ?_, H⟩ | case h
s : ℝ
hs : 0 < s
M ε : ℝ
left✝ : 0 < M
hε : 0 < ε
H : {z | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M
⊢ 1 ∈ {w | 1 - ε < w.re} | eb9728e2d56d226d |
Algebra.Generators.Hom.toAlgHom_comp_apply | Mathlib/RingTheory/Generators.lean | @[simp]
lemma Hom.toAlgHom_comp_apply
[Algebra R R''] [IsScalarTower R R' R''] [IsScalarTower R' R'' S'']
[IsScalarTower R' S' S''] [IsScalarTower S S' S'']
(f : Hom P P') (g : Hom P' P'') (x) :
(g.comp f).toAlgHom x = g.toAlgHom (f.toAlgHom x) | case h_C
R : Type u
S : Type v
inst✝¹⁹ : CommRing R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
P : Generators R S
R' : Type u_2
S' : Type u_4
inst✝¹⁶ : CommRing R'
inst✝¹⁵ : CommRing S'
inst✝¹⁴ : Algebra R' S'
P' : Generators R' S'
R'' : Type u_1
S'' : Type u_3
inst✝¹³ : CommRing R''
inst✝¹² : CommRing S''
inst✝¹¹ : Al... | simp only [← MvPolynomial.algebraMap_eq, AlgHom.map_algebraMap] | no goals | 4ee937b263fb4ba0 |
SimplicialObject.Splitting.cofan_inj_πSummand_eq_id | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | theorem cofan_inj_πSummand_eq_id [HasZeroMorphisms C] {Δ : SimplexCategoryᵒᵖ} (A : IndexSet Δ) :
(s.cofan Δ).inj A ≫ s.πSummand A = 𝟙 _ | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X : SimplicialObject C
s : Splitting X
inst✝ : HasZeroMorphisms C
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ (s.cofan Δ).inj A ≫ s.πSummand A = 𝟙 (summand s.N Δ A) | simp [πSummand] | no goals | ca1ca7cd0248e505 |
MeasureTheory.UnifIntegrable.add | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem add (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)
(hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
UnifIntegrable (f + g) p μ | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ... | simp_rw [Pi.add_apply, Set.indicator_add'] | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ... | f4675bc9f77774c8 |
WittVector.map_frobeniusPoly | Mathlib/RingTheory/WittVector/Frobenius.lean | theorem map_frobeniusPoly (n : ℕ) :
MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n | p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
∀ m < n,
X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
(bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
⊢ C (↑p ^ i) * (bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTer... | rw [← IH i hi] | p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
∀ m < n,
X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
(bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
⊢ C (↑p ^ i) * (X i ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ... | 1eb072545457e9c3 |
Ideal.finite_quotient_pow | Mathlib/RingTheory/Ideal/Quotient/Index.lean | lemma Ideal.finite_quotient_pow (hI : I.FG) [Finite (R ⧸ I)] (n) : Finite (R ⧸ I ^ n) | R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
hI : I.FG
inst✝ : Finite (R ⧸ I)
n : ℕ
⊢ Finite (R ⧸ I ^ n) | induction n with
| zero =>
simp only [pow_zero, Ideal.one_eq_top]
infer_instance
| succ n _ =>
exact Submodule.finite_quotient_smul (I ^ n) hI | no goals | a872582db39414d1 |
NNReal.sqrt_one | Mathlib/Data/Real/Sqrt.lean | @[simp] lemma sqrt_one : sqrt 1 = 1 | ⊢ sqrt 1 = 1 | simp | no goals | 292282fe2c272554 |
uniqueDiffOn_Ici | Mathlib/Analysis/Calculus/TangentCone.lean | theorem uniqueDiffOn_Ici (a : ℝ) : UniqueDiffOn ℝ (Ici a) :=
uniqueDiffOn_convex (convex_Ici a) <| by simp only [interior_Ici, nonempty_Ioi]
| a : ℝ
⊢ (interior (Ici a)).Nonempty | simp only [interior_Ici, nonempty_Ioi] | no goals | 04ac02507d87d628 |
Nat.preimage_Iio | Mathlib/Algebra/Order/Floor.lean | theorem preimage_Iio {a : α} : (Nat.cast : ℕ → α) ⁻¹' Set.Iio a = Set.Iio ⌈a⌉₊ | α : Type u_2
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a : α
⊢ Nat.cast ⁻¹' Iio a = Iio ⌈a⌉₊ | ext | case h
α : Type u_2
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a : α
x✝ : ℕ
⊢ x✝ ∈ Nat.cast ⁻¹' Iio a ↔ x✝ ∈ Iio ⌈a⌉₊ | 3b9a3727a0146f09 |
Orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.tan (o.oangle x (x + y)) = ‖y‖ / ‖x‖ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ (o.oangle x (x + y)).tan = ‖y‖ / ‖x‖ | rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.tan_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)] | no goals | f251633759d9e6d8 |
Profinite.NobelingProof.GoodProducts.chain'_cons_of_lt | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem chain'_cons_of_lt (l : MaxProducts C ho)
(q : Products I) (hq : q < l.val.Tail) :
List.Chain' (fun x x_1 ↦ x > x_1) (term I ho :: q.val) | case pos
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
l : ↑(MaxProducts C ho)
q : Products I
hq : List.Lex (fun x1 x2 => x1 < x2) ↑q []
this : Inhabited I
a : I
ha : a ∈ ↑q
hM : ↑(↑l).Tail = []
⊢ ... | simp only [List.not_lex_nil] at hq | no goals | df9d4955518f2722 |
Std.DHashMap.Raw.Const.get_eq_get! | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem get_eq_get! [EquivBEq α] [LawfulHashable α] [Inhabited β] (h : m.WF) {a : α} {h} :
get m a h = get! m a | α : Type u
inst✝⁴ : BEq α
inst✝³ : Hashable α
β : Type v
m : Raw α fun x => β
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited β
h✝ : m.WF
a : α
h : a ∈ m
⊢ get m a h = get! m a | simp_to_raw using Raw₀.Const.get_eq_get! | no goals | 5c589da07b220b00 |
Function.update_eq_updateFinset | Mathlib/Data/Finset/Update.lean | theorem update_eq_updateFinset {i y} :
Function.update x i y = updateFinset x {i} (uniqueElim y) | case pos
ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
i : ι
y : π i
j : ι
hj : j = i
⊢ update x i y j = updateFinset x {i} (uniqueElim y) j | cases hj | case pos.refl
ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝ : DecidableEq ι
i : ι
y : π i
⊢ update x i y i = updateFinset x {i} (uniqueElim y) i | f513edede4d792eb |
AlgebraicGeometry.isOpenImmersion_sigmaDesc | Mathlib/AlgebraicGeometry/Limits.lean | lemma isOpenImmersion_sigmaDesc
{X : Scheme} (α : ∀ i, f i ⟶ X) [∀ i, IsOpenImmersion (α i)]
(hα : Pairwise (Disjoint on (Set.range <| α · |>.base))) :
IsOpenImmersion (Sigma.desc α) | case left
ι : Type u
f : ι → Scheme
X : Scheme
α : (i : ι) → f i ⟶ X
inst✝ : ∀ (i : ι), IsOpenImmersion (α i)
hα : _root_.Pairwise (Disjoint on fun x => Set.range ⇑(ConcreteCategory.hom (α x).base))
⊢ Topology.IsOpenEmbedding (⇑(ConcreteCategory.hom (Sigma.desc α).base) ∘ ⇑(sigmaMk f)) | refine .of_continuous_injective_isOpenMap ?_ ?_ ?_ | case left.refine_1
ι : Type u
f : ι → Scheme
X : Scheme
α : (i : ι) → f i ⟶ X
inst✝ : ∀ (i : ι), IsOpenImmersion (α i)
hα : _root_.Pairwise (Disjoint on fun x => Set.range ⇑(ConcreteCategory.hom (α x).base))
⊢ Continuous (⇑(ConcreteCategory.hom (Sigma.desc α).base) ∘ ⇑(sigmaMk f))
case left.refine_2
ι : Type u
f : ι →... | a3962f9239181cbe |
essSup_map_measure | Mathlib/MeasureTheory/Function/EssSup.lean | theorem essSup_map_measure (hg : AEMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
essSup g (Measure.map f μ) = essSup (g ∘ f) μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : AEMeasurable g (Me... | exact h_eq.symm | no goals | 53d5e3fac9824186 |
Nat.mul_left_comm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean | theorem mul_left_comm (n m k : Nat) : n * (m * k) = m * (n * k) | n m k : Nat
⊢ n * (m * k) = m * (n * k) | rw [← Nat.mul_assoc, Nat.mul_comm n m, Nat.mul_assoc] | no goals | 074bc37179b83a35 |
segment_eq_Icc' | Mathlib/Analysis/Convex/Segment.lean | theorem segment_eq_Icc' (x y : 𝕜) : [x -[𝕜] y] = Icc (min x y) (max x y) | case inr
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
x y : 𝕜
h : y ≤ x
⊢ [x-[𝕜]y] = Icc (x ⊓ y) (x ⊔ y) | rw [segment_symm, segment_eq_Icc h, max_eq_left h, min_eq_right h] | no goals | 0e171dde18437a6d |
TrivSqZeroExt.fst_exp | Mathlib/Analysis/Normed/Algebra/TrivSqZeroExt.lean | theorem fst_exp (x : tsze R M) : fst (exp 𝕜 x) = exp 𝕜 x.fst | 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁷ : Field 𝕜
inst✝¹⁶ : CharZero 𝕜
inst✝¹⁵ : CommRing R
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Algebra 𝕜 R
inst✝¹² : Module 𝕜 M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module Rᵐᵒᵖ M
inst✝⁹ : IsCentralScalar R M
inst✝⁸ : IsScalarTower 𝕜 R M
inst✝⁷ : TopologicalSpace R
inst✝⁶ : Topolog... | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | no goals | 08caab94c1dc8dcb |
Finset.SupIndep.product | Mathlib/Order/SupIndep.lean | theorem SupIndep.product {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α}
(hs : s.SupIndep fun i => t.sup fun i' => f (i, i'))
(ht : t.SupIndep fun i' => s.sup fun i => f (i, i')) : (s ×ˢ t).SupIndep f | case mk
α : Type u_1
ι : Type u_3
ι' : Type u_4
inst✝¹ : DistribLattice α
inst✝ : OrderBot α
s : Finset ι
t : Finset ι'
f : ι × ι' → α
hs : s.SupIndep fun i => t.sup fun i' => f (i, i')
ht : t.SupIndep fun i' => s.sup fun i => f (i, i')
u : Finset (ι × ι')
hu : u ⊆ s ×ˢ t
i : ι
i' : ι'
hi : (i, i') ∈ s ×ˢ t
hiu : (i, i... | rintro ⟨j, j'⟩ hj | case mk.mk
α : Type u_1
ι : Type u_3
ι' : Type u_4
inst✝¹ : DistribLattice α
inst✝ : OrderBot α
s : Finset ι
t : Finset ι'
f : ι × ι' → α
hs : s.SupIndep fun i => t.sup fun i' => f (i, i')
ht : t.SupIndep fun i' => s.sup fun i => f (i, i')
u : Finset (ι × ι')
hu : u ⊆ s ×ˢ t
i : ι
i' : ι'
hi : (i, i') ∈ s ×ˢ t
hiu : (i... | 2d54695f6a47e602 |
CompHausLike.precoherent | Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean | theorem precoherent [HasExplicitPullbacks P] [HasExplicitFiniteCoproducts.{0} P]
(hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective f) :
Precoherent (CompHausLike P) | P : TopCat → Prop
inst✝¹ : HasExplicitPullbacks P
inst✝ : HasExplicitFiniteCoproducts P
hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective ⇑(ConcreteCategory.hom f)
this : Preregular (CompHausLike P)
⊢ Precoherent (CompHausLike P) | infer_instance | no goals | 8f34ea5175cf990c |
MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfUnion_of_mem | Mathlib/MeasureTheory/SetSemiring.lean | lemma pairwiseDisjoint_disjointOfUnion_of_mem (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) :
PairwiseDisjoint (hC.disjointOfUnion hJ j : Set (Set α)) id | α : Type u_1
C : Set (Set α)
j : Set α
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
hj : j ∈ J
⊢ ↑(hC.disjointOfUnion hJ j) ⊆ ⋃ x ∈ J, ↑(hC.disjointOfUnion hJ x) | exact subset_iUnion₂_of_subset j hj fun ⦃a⦄ a ↦ a | no goals | ff86e76868b74bd9 |
CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_map | Mathlib/CategoryTheory/Sites/CoverLifting.lean | lemma liftAux_map {Y : C} (f : G.obj Y ⟶ X) {W : C} (g : W ⟶ Y) (i : S.Arrow)
(h : G.obj W ⟶ i.Y) (w : h ≫ i.f = G.map g ≫ f) :
liftAux hF α s f ≫ F.map g.op = s.ι i ≫ R.map h.op ≫ α.app _ :=
(Multifork.IsLimit.fac
(hF.isLimitMultifork ⟨_, G.cover_lift J K (K.pullback_stable f S.2)⟩) _ _
⟨W, g, by s... | C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} D
G : C ⥤ D
A : Type w
inst✝¹ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : G.IsCocontinuous J K
F : Cᵒᵖ ⥤ A
hF : Presheaf.IsSheaf J F
R : Dᵒᵖ ⥤ A
α : G.op ⋙ R ⟶ F
X : D
S : K.Cover X
s : Multifork... | simpa only [← w] using S.1.downward_closed i.hf h | no goals | 2d92bf4845379c1b |
EReal.measurable_const_mul | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | private lemma measurable_const_mul (c : EReal) : Measurable fun (x : EReal) ↦ c * x | c : EReal
⊢ (fun p => ⊥ * ↑p) = fun p => if p = 0 then 0 else if p < 0 then ⊤ else ⊥ | ext p | case h
c : EReal
p : ℝ
⊢ ⊥ * ↑p = if p = 0 then 0 else if p < 0 then ⊤ else ⊥ | d167c2c55a5c43ff |
Polynomial.monomial_mem_lifts | Mathlib/Algebra/Polynomial/Lifts.lean | theorem monomial_mem_lifts {s : S} (n : ℕ) (h : s ∈ Set.range f) : monomial n s ∈ lifts f | case intro
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
r : R
h : f r ∈ Set.range ⇑f
⊢ (monomial n) (f r) ∈ lifts f | use monomial n r | case h
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
r : R
h : f r ∈ Set.range ⇑f
⊢ (mapRingHom f) ((monomial n) r) = (monomial n) (f r) | 2f89a13c06272ab6 |
MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι}
[l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞)
(hF_meas : ∀ᶠ n in l, Measurable (F n)) (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, F n a ≤ bound a)
(h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n =... | case intro.refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_5
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤
h_lim : ∀ᵐ (a : α) ∂μ... | exact Nat.le_add_left _ _ | no goals | 879da8b2b29c96c4 |
EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical | Mathlib/Geometry/Euclidean/Sphere/Power.lean | theorem mul_dist_eq_mul_dist_of_cospherical {a b c d p : P} (h : Cospherical ({a, b, c, d} : Set P))
(hapb : ∃ k₁ : ℝ, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p))
(hcpd : ∃ k₂ : ℝ, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)) :
dist a p * dist b p = dist c p * dist d p | V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)
hcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)
q : P
r : ℝ
h' : ∀ p ∈ {a, b, c, d}, dist p q = r... | simp | no goals | 1215002a48fa6426 |
extract_gcd | Mathlib/Algebra/GCDMonoid/Basic.lean | theorem extract_gcd {α : Type*} [CancelCommMonoidWithZero α] [GCDMonoid α] (x y : α) :
∃ x' y', x = gcd x y * x' ∧ y = gcd x y * y' ∧ IsUnit (gcd x' y') | case pos.intro
α : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
h : gcd 0 0 = 0
⊢ ∃ x' y', 0 = gcd 0 0 * x' ∧ 0 = gcd 0 0 * y' ∧ Associated (gcd x' y') 1 | exact ⟨1, 1, by rw [h, zero_mul], by rw [h, zero_mul], gcd_one_left' 1⟩ | no goals | d8751c012f27fe65 |
Polynomial.isIntegral_isLocalization_polynomial_quotient | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Idea... | case intro.mk.mk
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
... | refine @IsIntegral.of_mem_closure'' Rₘ _ Sₘ _ φ'
((algebraMap (R[X] ⧸ P) Sₘ).comp (Ideal.Quotient.mk P) '' insert X { p | p.degree ≤ 0 }) ?_
((algebraMap (R[X] ⧸ P) Sₘ) p') ?_ | case intro.mk.mk.refine_1
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
... | 4f255f198e9b3f92 |
Finsupp.span_image_eq_map_linearCombination | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem span_image_eq_map_linearCombination (s : Set α) :
span R (v '' s) = Submodule.map (linearCombination R v) (supported R R s) | case h₁
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
s : Set α
x : M
hx : x ∈ v '' s
⊢ x ∈ ↑(map (linearCombination R v) (supported R R s)) | rw [Set.mem_image] at hx | case h₁
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
s : Set α
x : M
hx : ∃ x_1 ∈ s, v x_1 = x
⊢ x ∈ ↑(map (linearCombination R v) (supported R R s)) | 4d7a89362346e5b3 |
MvPolynomial.quotient_mk_comp_C_injective | Mathlib/RingTheory/MvPolynomial.lean | theorem quotient_mk_comp_C_injective [Field K] (I : Ideal (MvPolynomial σ K)) (hI : I ≠ ⊤) :
Function.Injective ((Ideal.Quotient.mk I).comp MvPolynomial.C) | σ : Type u
K : Type v
inst✝ : Field K
I : Ideal (MvPolynomial σ K)
hI : I ≠ ⊤
x : K
hx : ((Ideal.Quotient.mk I).comp C) x = 0
⊢ x = 0 | rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem] at hx | σ : Type u
K : Type v
inst✝ : Field K
I : Ideal (MvPolynomial σ K)
hI : I ≠ ⊤
x : K
hx : C x ∈ I
⊢ x = 0 | eb34f5768c30a3a4 |
Affine.Simplex.eq_mongePoint_of_forall_mem_mongePlane | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem eq_mongePoint_of_forall_mem_mongePlane {n : ℕ} {s : Simplex ℝ P (n + 2)} {i₁ : Fin (n + 3)}
{p : P} (h : ∀ i₂, i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂) : p = s.mongePoint | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
h' :
∀ (i₂ : Fin (n + 3)),
i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈ (Submod... | have hv : p -ᵥ s.mongePoint ∈ vectorSpan ℝ (Set.range s.points) := by
let s₁ : Finset (Fin (n + 3)) := univ.erase i₁
obtain ⟨i₂, h₂⟩ := card_pos.1 (show 0 < #s₁ by simp [s₁, card_erase_of_mem])
have h₁₂ : i₁ ≠ i₂ := (ne_of_mem_erase h₂).symm
exact (Submodule.mem_inf.1 (h' i₂ h₁₂)).2 | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
h' :
∀ (i₂ : Fin (n + 3)),
i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈ (Submod... | 41c76fab4d39bac3 |
AlgebraicGeometry.IsLocalAtTarget.of_openCover | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | lemma of_openCover (H : ∀ i, P (𝒰.pullbackHom f i)) : P f | P : MorphismProperty Scheme
hP : IsLocalAtTarget P
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
H : ∀ (i : 𝒰.1), P (Scheme.Cover.pullbackHom 𝒰 f i)
⊢ P f | apply of_iSup_eq_top (fun i ↦ (𝒰.map i).opensRange) 𝒰.iSup_opensRange | P : MorphismProperty Scheme
hP : IsLocalAtTarget P
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
H : ∀ (i : 𝒰.1), P (Scheme.Cover.pullbackHom 𝒰 f i)
⊢ ∀ (i : 𝒰.J), P (f ∣_ Scheme.Hom.opensRange (𝒰.map i)) | 8adfa259eb8081d8 |
FinitePlace.norm_eq_one_iff_not_mem | Mathlib/NumberTheory/NumberField/FinitePlaces.lean | theorem FinitePlace.norm_eq_one_iff_not_mem (x : 𝓞 (WithVal (v.valuation K))) :
‖embedding v x‖ = 1 ↔ x ∉ v.asIdeal | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
v : HeightOneSpectrum (𝓞 K)
x : 𝓞 (WithVal (valuation K v))
⊢ v.intValuationDef x = 1 ↔ 1 ≤ v.intValuationDef x | exact (intValuation_le_one v x).ge_iff_eq.symm | no goals | e9451970be705207 |
IsDedekindDomain.selmerGroup.fromUnit_ker | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | theorem fromUnit_ker [hn : Fact <| 0 < n] :
(@fromUnit R _ _ K _ _ _ n).ker = (powMonoidHom n : Rˣ →* Rˣ).range | case h.mk.mp.intro.mk
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
n : ℕ
hn : Fact (0 < n)
val✝ inv✝ : R
val_inv✝ : val✝ * inv✝ = 1
inv_val✝ : inv✝ * val✝ = 1
hx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_v... | rw [Units.val_pow_eq_pow_val] at hv | case h.mk.mp.intro.mk
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
n : ℕ
hn : Fact (0 < n)
val✝ inv✝ : R
val_inv✝ : val✝ * inv✝ = 1
inv_val✝ : inv✝ * val✝ = 1
hx✝ : { val := val✝, inv := inv✝, val_inv := val_inv✝, inv_val := inv_v... | 252946ccdda732f7 |
Set.PartiallyWellOrderedOn.exists_min_bad_of_exists_bad | Mathlib/Order/WellFoundedSet.lean | theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
(∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f | case intro
α : Type u_2
r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
fun t =>
Nat.rec ⟨↑(minBadSeqOfBadSeq r rk s 0 f0 hf0), ⋯⟩ (fun n fn => ⟨↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn ⋯), ⋯⟩) t
h : ∀ (m n : ℕ), m ≤ n → ↑(fs ... | refine ⟨fun n => (fs n).1 n, ⟨fun n => (fs n).2.1.1 n, fun m n mn => ?_⟩, fun n g hg1 hg2 => ?_⟩ | case intro.refine_1
α : Type u_2
r : α → α → Prop
rk : α → ℕ
s : Set α
f0 : ℕ → α
hf0 : IsBadSeq r s f0
fs : (n : ℕ) → { f // IsBadSeq r s f ∧ IsMinBadSeq r rk s n f } :=
fun t =>
Nat.rec ⟨↑(minBadSeqOfBadSeq r rk s 0 f0 hf0), ⋯⟩ (fun n fn => ⟨↑(minBadSeqOfBadSeq r rk s (n + 1) ↑fn ⋯), ⋯⟩) t
h : ∀ (m n : ℕ), m ≤ ... | 30e737a048136643 |
MeasureTheory.AECover.aemeasurable | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem AECover.aemeasurable {β : Type*} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot]
{f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
(hfm : ∀ i, AEMeasurable f (μ.restrict <| φ i)) : AEMeasurable f μ | α : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i))
⊢ AEMeasurable f μ | obtain ⟨u, hu⟩ := l.exists_seq_tendsto | case intro
α : Type u_1
ι : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i))
u : ℕ → ι
hu : Tendsto u atTop l
⊢ AEMeasurable... | d6c0989e9b5280ae |
Finsupp.DegLex.wellFounded | Mathlib/Data/Finsupp/MonomialOrder/DegLex.lean | theorem wellFounded
{r : α → α → Prop} [IsTrichotomous α r] (hr : WellFounded (Function.swap r))
{s : ℕ → ℕ → Prop} (hs : WellFounded s) (hs0 : ∀ ⦃n⦄, ¬ s n 0) :
WellFounded (Finsupp.DegLex r s) | α : Type u_1
r : α → α → Prop
inst✝ : IsTrichotomous α r
hr : WellFounded (Function.swap r)
s : ℕ → ℕ → Prop
hs : WellFounded s
hs0 : ∀ ⦃n : ℕ⦄, ¬s n 0
⊢ WellFounded (Finsupp.DegLex r s) | have wft := WellFounded.prod_lex hs (Finsupp.Lex.wellFounded' hs0 hs hr) | α : Type u_1
r : α → α → Prop
inst✝ : IsTrichotomous α r
hr : WellFounded (Function.swap r)
s : ℕ → ℕ → Prop
hs : WellFounded s
hs0 : ∀ ⦃n : ℕ⦄, ¬s n 0
wft : WellFounded (Prod.Lex s (Finsupp.Lex r s))
⊢ WellFounded (Finsupp.DegLex r s) | 499bcc8d6aec7c66 |
AnalyticOnNhd.iteratedFDeriv | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem AnalyticOnNhd.iteratedFDeriv [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) (n : ℕ) :
AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 n f) s | case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
n : ℕ
IH : AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 n f) s
⊢ A... | convert ContinuousLinearMap.comp_analyticOnNhd ?g IH.fderiv | case h.e.h.e'_9.h.e'_4
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
n : ℕ
IH : AnalyticOnNhd 𝕜 (iteratedFDeriv ... | 30a2541ac18d8023 |
Submonoid.mem_center_iff | Mathlib/GroupTheory/Submonoid/Center.lean | theorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g | M : Type u_1
inst✝ : Monoid M
z : M
⊢ z ∈ center M ↔ ∀ (g : M), g * z = z * g | rw [← Semigroup.mem_center_iff] | M : Type u_1
inst✝ : Monoid M
z : M
⊢ z ∈ center M ↔ z ∈ Set.center M | 3674e4ad8ce3ab8b |
Multipliable.multipliable_of_eq_one_or_self | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | theorem Multipliable.multipliable_of_eq_one_or_self (hf : Multipliable f)
(h : ∀ b, g b = 1 ∨ g b = f b) : Multipliable g | α : Type u_1
β : Type u_2
inst✝³ : CommGroup α
inst✝² : UniformSpace α
inst✝¹ : UniformGroup α
f g : β → α
inst✝ : CompleteSpace α
hf : Multipliable f
h : ∀ (b : β), g b = 1 ∨ g b = f b
e : Set α
he : e ∈ 𝓝 1
s : Finset β
hs : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
t : Finset β
ht : Disjoint t s
b : β
hbt :... | simp only [Finset.mem_filter, and_iff_right hbt] at hb | α : Type u_1
β : Type u_2
inst✝³ : CommGroup α
inst✝² : UniformSpace α
inst✝¹ : UniformGroup α
f g : β → α
inst✝ : CompleteSpace α
hf : Multipliable f
h : ∀ (b : β), g b = 1 ∨ g b = f b
e : Set α
he : e ∈ 𝓝 1
s : Finset β
hs : ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e
t : Finset β
ht : Disjoint t s
b : β
hbt :... | 8323c581b099ab4f |
Polynomial.irreducible_comp | Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean | theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g.Monic)
(hf : Irreducible f)
(hg : ∀ (E : Type u) [Field E] [Algebra K E] (x : E) (_ : minpoly K x = f),
Irreducible (g.map (algebraMap _ _) - C (AdjoinSimple.gen K x))) :
Irreducible (f.comp g) | K : Type u
inst✝ : Field K
f g : K[X]
hfm : f.Monic
hgm : g.Monic
hf : Irreducible f
hg :
∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),
minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))
hf' : f.natDegree ≠ 0
hg' : g.natDegree ≠ 0
H₁ : f.comp g ≠ 0
H... | rw [adjoin.finrank, ← key₁] | K : Type u
inst✝ : Field K
f g : K[X]
hfm : f.Monic
hgm : g.Monic
hf : Irreducible f
hg :
∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),
minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))
hf' : f.natDegree ≠ 0
hg' : g.natDegree ≠ 0
H₁ : f.comp g ≠ 0
H... | 9c1ae09b4550aa94 |
RingHom.locally_isStableUnderBaseChange | Mathlib/RingTheory/RingHom/Locally.lean | /-- If `P` is stable under base change, then so is `Locally P`. -/
lemma locally_isStableUnderBaseChange (hPi : RespectsIso P) (hPb : IsStableUnderBaseChange P) :
IsStableUnderBaseChange (Locally P) | case intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
hPb : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] => P
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algeb... | rw [locally_iff_exists hPi] | case intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
hPb : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] => P
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algeb... | b116e3feacfa87ad |
Stream'.WSeq.Equiv.ext | Mathlib/Data/Seq/WSeq.lean | theorem Equiv.ext {s t : WSeq α} (h : ∀ n, get? s n ~ get? t n) : s ~ʷ t :=
⟨fun s t => ∀ n, get? s n ~ get? t n, h, fun {s t} h => by
refine liftRel_def.2 ⟨?_, ?_⟩
· rw [← head_terminates_iff, ← head_terminates_iff]
exact terminates_congr (h 0)
· intro a b ma mb
rcases a with - | a <;> rcases... | case refine_2
α : Type u
s✝ t✝ : WSeq α
h✝ : ∀ (n : ℕ), s✝.get? n ~ t✝.get? n
s t : WSeq α
h : (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) s t
⊢ ∀ {a b : Option (α × WSeq α)},
a ∈ s.destruct → b ∈ t.destruct → LiftRelO (fun x1 x2 => x1 = x2) (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) a b | intro a b ma mb | case refine_2
α : Type u
s✝ t✝ : WSeq α
h✝ : ∀ (n : ℕ), s✝.get? n ~ t✝.get? n
s t : WSeq α
h : (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) s t
a b : Option (α × WSeq α)
ma : a ∈ s.destruct
mb : b ∈ t.destruct
⊢ LiftRelO (fun x1 x2 => x1 = x2) (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) a b | ffd3964c3d88e7cd |
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 inr.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 ↑s = Rin... | rw [hh'] | case inr.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 ↑s = Rin... | 37c27acde7e67205 |
Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux | Mathlib/Analysis/Complex/CauchyIntegral.lean | theorem circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux {R : ℝ} {c w : ℂ}
{f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ ball c R \ s)
(hc : ContinuousOn f (closedBall c R)) (hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x) :
(∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w... | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : Co... | have hc' : ContinuousOn (fun z => (z - w)⁻¹) (sphere c R) :=
(continuousOn_id.sub continuousOn_const).inv₀ fun z hz => sub_ne_zero.2 <| hne z hz | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : (insert w s).Countable
hcF : Co... | 5415387dfb69ba4b |
AlgebraicGeometry.Scheme.Hom.iInf_ker_openCover_map_comp | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma Hom.iInf_ker_openCover_map_comp (f : X ⟶ Y) [QuasiCompact f] (𝒰 : X.OpenCover) :
⨅ i, (𝒰.map i ≫ f).ker = f.ker | X Y : Scheme
f : X ⟶ Y
inst✝ : QuasiCompact f
𝒰 : X.OpenCover
⊢ ⨅ i, ker (𝒰.map i ≫ f) ≤ ker f | refine iInf_le_iff.mpr fun I hI U ↦ ?_ | X Y : Scheme
f : X ⟶ Y
inst✝ : QuasiCompact f
𝒰 : X.OpenCover
I : Y.IdealSheafData
hI : ∀ (i : 𝒰.J), I ≤ ker (𝒰.map i ≫ f)
U : ↑Y.affineOpens
⊢ I.ideal U ≤ (ker f).ideal U | 6d6088be8ad0dbb8 |
Polynomial.eval₂_zero | Mathlib/Algebra/Polynomial/Eval/Defs.lean | theorem eval₂_zero : (0 : R[X]).eval₂ f x = 0 | R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
x : S
⊢ eval₂ f x 0 = 0 | simp [eval₂_eq_sum] | no goals | b4be376432b7ceb2 |
IsUltrametricDist.closedBall_subset_trichotomy | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | lemma closedBall_subset_trichotomy :
closedBall x r ⊆ closedBall y s ∨ closedBall y s ⊆ closedBall x r ∨
Disjoint (closedBall x r) (closedBall y s) | case intro
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x✝ y✝ : X
r✝ s✝ : ℝ
x y : X
r s : ℝ
hrs : r ≤ s
h : (closedBall x r ∩ closedBall y s).Nonempty
hxz : h.some ∈ closedBall x r
hyz : h.some ∈ closedBall y s
hx : closedBall x r ⊆ closedBall x s
⊢ closedBall x r ⊆ closedBall y s | rwa [closedBall_eq_of_mem hyz |>.trans (closedBall_eq_of_mem <| hx hxz).symm] | no goals | 36785ffe53e1b7f3 |
StrictConvexOn.secant_strict_mono | Mathlib/Analysis/Convex/Slope.lean | theorem StrictConvexOn.secant_strict_mono (hf : StrictConvexOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s)
(hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y) :
(f x - f a) / (x - a) < (f y - f a) / (y - a) | case inl.inl
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f x - f a) / (x - a) < (f y - f a) / (y - a) | convert hf.secant_strict_mono_aux3 hx ha hxy hya using 1 <;> rw [← neg_div_neg_eq] <;>
field_simp | no goals | 0b61a551659476be |
Traversable.foldl_toList | Mathlib/Control/Fold.lean | theorem foldl_toList (f : α → β → α) (xs : t β) (x : α) :
foldl f x xs = List.foldl f x (toList xs) | α β : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
f : α → β → α
xs : t β
x : α
⊢ foldl f x xs = unop ((Foldl.ofFreeMonoid f) (FreeMonoid.ofList (toList xs))) x | simp only [foldl, toList_spec, foldMap_hom_free, foldl.ofFreeMonoid_comp_of, Foldl.get,
FreeMonoid.ofList_toList] | no goals | b62cb3180753dd22 |
Lean.Order.Array.monotone_allM | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_allM
{m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
(f : γ → α → m Bool) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
monotone (fun x => xs.allM (f x) start stop) | case hmono₁.hmono.h
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
y : α
⊢ monotone fun x => do
let __do_lift ← f x y
pure !__do_lift | apply monotone_bind | case hmono₁.hmono.h.hmono₁
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
y : α
⊢ monotone fun x => f x y
case hmono₁.hmono.h.hmono₂
γ : Type w
inst✝³ : Part... | cf015d7f59a8c09d |
Metric.Sigma.dist_triangle | Mathlib/Topology/MetricSpace/Gluing.lean | theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z | case mk.mk
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
z : (i : ι) × E i
i : ι
x : E i
j : ι
y : E j
⊢ dist ⟨i, x⟩ z ≤ dist ⟨i, x⟩ ⟨j, y⟩ + dist ⟨j, y⟩ z | rcases z with ⟨k, z⟩ | case mk.mk.mk
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
⊢ dist ⟨i, x⟩ ⟨k, z⟩ ≤ dist ⟨i, x⟩ ⟨j, y⟩ + dist ⟨j, y⟩ ⟨k, z⟩ | 5708d5e59b0fdab2 |
MeasureTheory.OuterMeasure.mkMetric'.trim_pre | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞)
(hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r | X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : OpensMeasurableSpace X
m : Set X → ℝ≥0∞
hcl : ∀ (s : Set X), m (closure s) = m s
r : ℝ≥0∞
s : Set X
hs : diam s ≤ r
⊢ (pre m r).trim s ≤ m s | rw [trim_eq_iInf] | X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : OpensMeasurableSpace X
m : Set X → ℝ≥0∞
hcl : ∀ (s : Set X), m (closure s) = m s
r : ℝ≥0∞
s : Set X
hs : diam s ≤ r
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), (pre m r) t ≤ m s | 85b2e283625a23bd |
Monoid.Coprod.con_inv_mul_cancel | Mathlib/GroupTheory/Coprod/Basic.lean | theorem con_inv_mul_cancel (x : FreeMonoid (G ⊕ H)) :
coprodCon G H (ofList (x.toList.map (Sum.map Inv.inv Inv.inv)).reverse * x) 1 | G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
x : FreeMonoid (G ⊕ H)
⊢ mk (ofList (List.map (Sum.map Inv.inv Inv.inv) (toList x)).reverse) * mk x = 1 | induction x using FreeMonoid.inductionOn' with
| one => simp
| mul_of x xs ihx =>
simp only [toList_of_mul, map_cons, reverse_cons, ofList_append, map_mul, ihx, ofList_singleton]
rwa [mul_assoc, ← mul_assoc (mk (of _)), mk_of_inv_mul, one_mul] | no goals | 3d66c61021a89ea4 |
CategoryTheory.BinaryCofan.isVanKampen_mk | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem BinaryCofan.isVanKampen_mk {X Y : C} (c : BinaryCofan X Y)
(cofans : ∀ X Y : C, BinaryCofan X Y) (colimits : ∀ X Y, IsColimit (cofans X Y))
(cones : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), PullbackCone f g)
(limits : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), IsLimit (cones f g))
(h₁ : ∀ {X' Y' : C} ... | case mpr.intro
C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
cofans : (X Y : C) → BinaryCofan X Y
colimits : (X Y : C) → IsColimit (cofans X Y)
cones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g
limits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)
h₁ :
∀ {X' Y' : C... | have he₂ : c'.inr = e₂.hom ≫ (cones f c.inr).fst := by simp [e₂] | case mpr.intro
C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
cofans : (X Y : C) → BinaryCofan X Y
colimits : (X Y : C) → IsColimit (cofans X Y)
cones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g
limits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)
h₁ :
∀ {X' Y' : C... | 34db0fb5e69f31b2 |
Finset.gcd_insert | Mathlib/Algebra/GCDMonoid/Finset.lean | theorem gcd_insert [DecidableEq β] {b : β} :
(insert b s : Finset β).gcd f = GCDMonoid.gcd (f b) (s.gcd f) | α : Type u_2
β : Type u_3
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
s : Finset β
f : β → α
inst✝ : DecidableEq β
b : β
⊢ (insert b s).gcd f = GCDMonoid.gcd (f b) (s.gcd f) | by_cases h : b ∈ s | case pos
α : Type u_2
β : Type u_3
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
s : Finset β
f : β → α
inst✝ : DecidableEq β
b : β
h : b ∈ s
⊢ (insert b s).gcd f = GCDMonoid.gcd (f b) (s.gcd f)
case neg
α : Type u_2
β : Type u_3
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
s... | a46d4493db74038f |
MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact | Mathlib/MeasureTheory/Measure/Regular.lean | /-- See also `IsCompact.measure_closure` for a version
that assumes the `σ`-algebra to be the Borel `σ`-algebra but makes no assumptions on `μ`. -/
lemma measure_closure_eq_of_isCompact [R1Space α] [OuterRegular μ]
{k : Set α} (hk : IsCompact k) : μ (closure k) = μ k | α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : R1Space α
inst✝ : μ.OuterRegular
k : Set α
hk : IsCompact k
u : Set α
ku : k ⊆ u
u_open : IsOpen u
⊢ μ (closure k) ≤ μ u | exact measure_mono (hk.closure_subset_of_isOpen u_open ku) | no goals | 2328160c60e59dcc |
Ordinal.IsFundamentalSequence.monotone | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem monotone (hf : IsFundamentalSequence a o f) {i j : Ordinal} (hi : i < o)
(hj : j < o) (hij : i ≤ j) : f i hi ≤ f j hj | a o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hf : a.IsFundamentalSequence o f
i j : Ordinal.{u}
hi : i < o
hj : j < o
hij : i ≤ j
⊢ f i hi ≤ f j hj | rcases lt_or_eq_of_le hij with (hij | rfl) | case inl
a o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hf : a.IsFundamentalSequence o f
i j : Ordinal.{u}
hi : i < o
hj : j < o
hij✝ : i ≤ j
hij : i < j
⊢ f i hi ≤ f j hj
case inr
a o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hf : a.IsFundamentalSequence o f
i : Ordinal.{u}
hi hj : i < ... | 2fb252185c6eb164 |
ComplexShape.not_mem_range_embeddingUpIntLE_iff | Mathlib/Algebra/Homology/Embedding/Basic.lean | lemma not_mem_range_embeddingUpIntLE_iff (n : ℤ) :
(∀ (i : ℕ), (embeddingUpIntLE p).f i ≠ n) ↔ p < n | p n : ℤ
h : ∀ (i : ℕ), (embeddingUpIntLE p).f i ≠ n
this : ¬p < n
⊢ p - ↑(p - n).natAbs = n | omega | no goals | 356a2857080ff2c8 |
Std.Sat.AIG.RefVec.map.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean | theorem go_get_aux {aig : AIG α} (curr : Nat) (hcurr : curr ≤ len) (s : RefVec aig curr)
(input : RefVec aig len) (f : (aig : AIG α) → Ref aig → Entrypoint α)
[LawfulOperator α Ref f] [LawfulMapOperator α f] :
-- The hfoo here is a trick to make the dependent type gods happy.
∀ (idx : Nat) (hidx : idx <... | case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx : idx < curr
res : RefVecEntry α len
h✝ :... | simp | no goals | bc7701018f2eb538 |
mellin_convergent_top_of_isBigO | Mathlib/Analysis/MellinTransform.lean | theorem mellin_convergent_top_of_isBigO {f : ℝ → ℝ}
(hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0)) {a s : ℝ}
(hf : f =O[atTop] (· ^ (-a))) (hs : s < a) :
∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioi c) | f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x => x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ b ≥ e, ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < e ⊔ 1
⊢ ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi (e ⊔ 1)), ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d | refine (ae_restrict_mem measurableSet_Ioi).mono fun t ht => ?_ | f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x => x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ b ≥ e, ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < e ⊔ 1
t : ℝ
ht : t ∈ Ioi (e ⊔ 1)
⊢ ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d | b881ecb56c0e56ab |
countable_setOf_covBy_right | Mathlib/Topology/Order/Basic.lean | theorem countable_setOf_covBy_right [OrderTopology α] [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x}... | apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1) | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x}... | 7a450f28627ecdc7 |
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