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 |
|---|---|---|---|---|---|---|
AddCircle.closedBall_ae_eq_ball | Mathlib/MeasureTheory/Group/AddCircle.lean | theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =ᵐ[volume] ball x ε | case inl
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hε : ε ≤ 0
⊢ closedBall x ε =ᶠ[ae volume] ball x ε | rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closedBall,
min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ENNReal.ofReal_eq_zero] | case inl
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hε : ε ≤ 0
⊢ 2 * ε ≤ 0 | d4d4a4368219224a |
PiNat.iUnion_cylinder_update | Mathlib/Topology/MetricSpace/PiNat.lean | theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
⋃ k, cylinder (update x n k) (n + 1) = cylinder x n | case h.mpr
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
H : ∀ i < n, y i = x i
i : ℕ
hi : i < n + 1
⊢ y i = update x n (y n) i | rcases Nat.lt_succ_iff_lt_or_eq.1 hi with (h'i | rfl) | case h.mpr.inl
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
H : ∀ i < n, y i = x i
i : ℕ
hi : i < n + 1
h'i : i < n
⊢ y i = update x n (y n) i
case h.mpr.inr
E : ℕ → Type u_1
x y : (n : ℕ) → E n
i : ℕ
H : ∀ i_1 < i, y i_1 = x i_1
hi : i < i + 1
⊢ y i = update x i (y i) i | 822a9c224bc12ada |
CategoryTheory.FinitaryExtensive.isVanKampen_finiteCoproducts_Fin | Mathlib/CategoryTheory/Extensive.lean | theorem FinitaryExtensive.isVanKampen_finiteCoproducts_Fin [FinitaryExtensive C] {n : ℕ}
{F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsVanKampenColimit c | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryExtensive C
n : ℕ
F : Discrete (Fin n) ⥤ C
c : Cocone F
hc : IsColimit c
f : Fin n → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor f
⊢ IsVanKampenColimit c | clear_value f | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryExtensive C
n : ℕ
F : Discrete (Fin n) ⥤ C
c : Cocone F
hc : IsColimit c
f : Fin n → C
this : F = Discrete.functor f
⊢ IsVanKampenColimit c | 1a3b75887fa9e8f3 |
Num.castNum_shiftRight | Mathlib/Data/Num/Lemmas.lean | theorem castNum_shiftRight (m : Num) (n : Nat) : ↑(m >>> n) = (m : ℕ) >>> (n : ℕ) | n : ℕ
IH : ∀ (m : PosNum), ↑(m >>> n) = ↑(pos m) >>> n
m : PosNum
m✝ : ℕ
⊢ (m✝ + m✝) / 2 = m✝ | omega | no goals | f09e639cb27e0ca3 |
Submodule.inf_comap_le_comap_add | Mathlib/Algebra/Module/Submodule/Map.lean | theorem inf_comap_le_comap_add (f₁ f₂ : M →ₛₗ[τ₁₂] M₂) :
comap f₁ q ⊓ comap f₂ q ≤ comap (f₁ + f₂) q | R : Type u_1
R₂ : Type u_3
M : Type u_5
M₂ : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : Semiring R₂
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
q : Submodule R₂ M₂
f₁ f₂ : M →ₛₗ[τ₁₂] M₂
m : M
h : m ∈ comap f₁ q ⊓ comap f₂ q
⊢ f₁ m + f₂ m ∈ q | change f₁ m ∈ q ∧ f₂ m ∈ q at h | R : Type u_1
R₂ : Type u_3
M : Type u_5
M₂ : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : Semiring R₂
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
q : Submodule R₂ M₂
f₁ f₂ : M →ₛₗ[τ₁₂] M₂
m : M
h : f₁ m ∈ q ∧ f₂ m ∈ q
⊢ f₁ m + f₂ m ∈ q | eee85a1daac4f5a6 |
Profinite.NobelingProof.GoodProducts.maxTail_isGood | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem maxTail_isGood (l : MaxProducts C ho)
(h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
l.val.Tail.isGood (C' C ho) | case intro.intro.intro.intro
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inh... | apply Submodule.add_mem | case intro.intro.intro.intro.h₁
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : ... | 71fdaf74452759e3 |
ENNReal.ofReal_rpow_of_nonneg | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | theorem ofReal_rpow_of_nonneg {x p : ℝ} (hx_nonneg : 0 ≤ x) (hp_nonneg : 0 ≤ p) :
ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) | case pos
x p : ℝ
hx_nonneg : 0 ≤ x
hp_nonneg : 0 ≤ p
hp0 : p = 0
⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) | simp [hp0] | no goals | e5b13e84a28af913 |
List.mem_permutationsAux2' | Mathlib/Data/List/Permutation.lean | theorem mem_permutationsAux2' {t : α} {ts : List α} {ys : List α} {l : List α} :
l ∈ (permutationsAux2 t ts [] ys id).2 ↔
∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l = l₁ ++ t :: l₂ ++ ts | case h
α : Type u_1
t : α
ts ys l x✝ : List α
⊢ id x✝ = [] ++ x✝ | rfl | no goals | 8ad876353a13ec25 |
LieSubmodule.lie_baseChange | Mathlib/Algebra/Lie/BaseChange.lean | lemma lie_baseChange {I : LieIdeal R L} {N : LieSubmodule R L M} :
⁅I, N⁆.baseChange A = ⁅I.baseChange A, N.baseChange A⁆ | case refine_2.intro.intro.intro.intro.zero
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
I : LieIdeal R L
N : LieSu... | simp | no goals | c0e93f202362c69d |
Profinite.exists_locallyConstant_finite_aux | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom | case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyCon... | let fs : ∀ a : α, j0 ⟶ j a := fun a => (hj0 (hj a)).some | case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyCon... | ad55d9835f0fad2e |
FirstOrder.Language.Theory.bot_imp | Mathlib/ModelTheory/Equivalence.lean | lemma bot_imp (φ : L.BoundedFormula α n) : ⊥ ⟹[T] φ := fun M v xs => by
simp only [BoundedFormula.realize_imp, BoundedFormula.realize_bot, false_implies]
| L : Language
T : L.Theory
α : Type w
n : ℕ
φ : L.BoundedFormula α n
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
⊢ (⊥ ⟹ φ).Realize v xs | simp only [BoundedFormula.realize_imp, BoundedFormula.realize_bot, false_implies] | no goals | 19d2fa9fa4e18463 |
RingOfIntegers.isUnit_norm | Mathlib/NumberTheory/NumberField/Norm.lean | theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x | K : Type u_2
inst✝⁵ : Field K
F : Type u_3
inst✝⁴ : Field F
inst✝³ : Algebra K F
inst✝² : Algebra.IsSeparable K F
inst✝¹ : FiniteDimensional K F
inst✝ : CharZero K
x : 𝓞 F
⊢ IsUnit ((norm K) x) ↔ IsUnit x | letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K | K : Type u_2
inst✝⁵ : Field K
F : Type u_3
inst✝⁴ : Field F
inst✝³ : Algebra K F
inst✝² : Algebra.IsSeparable K F
inst✝¹ : FiniteDimensional K F
inst✝ : CharZero K
x : 𝓞 F
this : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
⊢ IsUnit ((norm K) x) ↔ IsUnit x | 81c4016c5a7e9693 |
Ideal.Filtration.submodule_eq_span_le_iff_stable_ge | Mathlib/RingTheory/Filtration.lean | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | case neg.intro.intro.refine_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I •... | rw [add_comm, ← monomial_smul_single] | case neg.intro.intro.refine_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I •... | fba02d553ffa8877 |
CategoryTheory.Localization.Monoidal.tensorHom_id | Mathlib/CategoryTheory/Localization/Monoidal.lean | lemma tensorHom_id {X₁ X₂ : LocalizedMonoidal L W ε} (f : X₁ ⟶ X₂) (Y : LocalizedMonoidal L W ε) :
f ⊗ 𝟙 Y = f ▷ Y | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : MonoidalCategory C
inst✝¹ : W.IsMonoidal
inst✝ : L.IsLocalization W
unit : D
ε : L.obj (𝟙_ C) ≅ unit
X₁ X₂ : LocalizedMonoidal L W ε
f : X₁ ⟶ X₂
Y : LocalizedMonoidal L W ε
⊢ f ⊗ 𝟙 Y = f ▷... | simp [monoidalCategoryStruct] | no goals | 9a2ec4ffc79b6300 |
MeasureTheory.integral_Iic_deriv_mul_eq_sub | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem integral_Iic_deriv_mul_eq_sub
(hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
(huv : IntegrableOn (u' * v + u * v') (Iic a))
(h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
∫ (x : ℝ) in Iic a, u' x * v x + u x * v' x = ... | A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[Iic a \ {a}] a) (𝓝 a')
h_infty : Tends... | apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq | A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Iic a) volume
h_zero : Tendsto (u * v) (𝓝[Iic a \ {a}] a) (𝓝 a')
h_infty : Tends... | 12a95a68fda4b736 |
CategoryTheory.Functor.IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite | Mathlib/CategoryTheory/Sites/DenseSubsite/SheafEquiv.lean | lemma isIso_ranCounit_app_of_isDenseSubsite (Y : Sheaf J A) (U X) :
IsIso ((yoneda.map ((G.op.ranCounit.app Y.val).app (op U))).app (op X)) | case refine_2.mk.mk.unit.op.mk.mk.unit.op.mk.up.up.refl.refl.intro.intro.h
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (Structu... | let I' : GrothendieckTopology.Cover.Arrow ⟨_, IsDenseSubsite.imageSieve_mem J K G g⟩ :=
⟨_, I.f ≫ i, ⟨l _ _ _ _ _ I.hf, by simp [hl]⟩⟩ | case refine_2.mk.mk.unit.op.mk.mk.unit.op.mk.up.up.refl.refl.intro.intro.h
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (Structu... | b6d9495444a0fb1a |
Array.unattach_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem unattach_filter {p : α → Prop} {l : Array { x // p x }}
{f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
(l.filter f).unattach = l.unattach.filter g | α : Type u_1
p : α → Prop
l : Array { x // p x }
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
⊢ (filter f l).unattach = filter g l.unattach | cases l | case mk
α : Type u_1
p : α → Prop
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
toList✝ : List { x // p x }
⊢ (filter f { toList := toList✝ }).unattach = filter g { toList := toList✝ }.unattach | fa65ebe8d1be407c |
CategoryTheory.GrothendieckTopology.W_inverseImage_whiskeringLeft | Mathlib/CategoryTheory/Sites/Equivalence.lean | lemma W_inverseImage_whiskeringLeft :
K.W.inverseImage ((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op) = J.W | case h
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q... | constructor | case h.mp
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
... | 5d43f5abfc41ecac |
CategoryTheory.SmallObject.prop_iterationFunctor_map_succ | Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.lean | lemma prop_iterationFunctor_map_succ (j : κ.ord.toType) :
(succStruct I κ).prop ((iterationFunctor I κ).map (homOfLE (Order.le_succ j))) | C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.toType
inst✝ : I.IsCardinalForSmallObjectArgument κ
j : κ.ord.toType
⊢ (succStruct I κ).prop ((iterationFunctor I κ).map (homOfLE ⋯)) | have := hasIterationOfShape I κ | C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.toType
inst✝ : I.IsCardinalForSmallObjectArgument κ
j : κ.ord.toType
this : HasIterationOfShape κ.ord.toType C
⊢ (succStruct I κ).prop ((iterationFunctor I κ).map (homOfLE ⋯)) | f761277f814afc30 |
MeasureTheory.integral_prod_smul | Mathlib/MeasureTheory/Integral/Prod.lean | theorem integral_prod_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : α → 𝕜) (g : β → E) :
∫ z, f z.1 • g z.2 ∂μ.prod ν = (∫ x, f x ∂μ) • ∫ y, g y ∂ν | α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
⊢ ∫ (z : α × β), f z.1 • g ... | by_cases hE : CompleteSpace E | case pos
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SFinite ν
inst✝³ : NormedSpace ℝ E
inst✝² : SFinite μ
𝕜 : Type u_5
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : CompleteSpace... | 07ecf6c0f674e95c |
Array.getElem?_zero_filterMap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean | theorem getElem?_zero_filterMap (f : α → Option β) (l : Array α) : (l.filterMap f)[0]? = l.findSome? f | case mk
α : Type u_1
β : Type u_2
f : α → Option β
toList✝ : List α
⊢ (filterMap f { toList := toList✝ })[0]? = findSome? f { toList := toList✝ } | simp [← List.head?_eq_getElem?] | no goals | 877722bfc128ac0c |
DFA.pumping_lemma | Mathlib/Computability/DFA.lean | theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
(hlen : Fintype.card σ ≤ List.length x) :
∃ a b c,
x = a ++ b ++ c ∧
a.length + b.length ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts | case right.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
hx✝ : x ∈ M.accepts
hlen✝ : Fintype.card σ ≤ x.length
b : List α
hnil : b ≠ []
c' a' b' : List α
hb' : b' ∈ {b}∗
hlen : a'.length + b.length ≤ Fintype.card σ
hb : M.evalFrom (M.evalFrom M.start a') ... | have h := M.evalFrom_of_pow hb hb' | case right.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
hx✝ : x ∈ M.accepts
hlen✝ : Fintype.card σ ≤ x.length
b : List α
hnil : b ≠ []
c' a' b' : List α
hb' : b' ∈ {b}∗
hlen : a'.length + b.length ≤ Fintype.card σ
hb : M.evalFrom (M.evalFrom M.start a') ... | d5e61b96b7f98969 |
ENNReal.inner_le_weight_mul_Lp_of_nonneg | Mathlib/Analysis/MeanInequalities.lean | /-- **Weighted Hölder inequality**. -/
lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0∞) :
∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ | case h.e'_3.a
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ⊤) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ⊤
this :
∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ a ∈ s, ↑(w a).toNNReal) ^ (1 -... | obtain hw | hw := eq_or_ne (w i) 0 | case h.e'_3.a.inl
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ⊤) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ⊤
this :
∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ a ∈ s, ↑(w a).toNNReal) ^ ... | bb7d779f184c526a |
ENNReal.sub_add_eq_add_sub | Mathlib/Data/ENNReal/Order.lean | theorem sub_add_eq_add_sub (hab : b ≤ a) (b_ne_top : b ≠ ∞) :
a - b + c = a + c - b | a b c : ℝ≥0∞
hab : b ≤ a
b_ne_top : b ≠ ⊤
⊢ a - b + c = a + c - b | by_cases c_top : c = ∞ | case pos
a b c : ℝ≥0∞
hab : b ≤ a
b_ne_top : b ≠ ⊤
c_top : c = ⊤
⊢ a - b + c = a + c - b
case neg
a b c : ℝ≥0∞
hab : b ≤ a
b_ne_top : b ≠ ⊤
c_top : ¬c = ⊤
⊢ a - b + c = a + c - b | a99c742e77a4a6ce |
Projectivization.finrank_submodule | Mathlib/LinearAlgebra/Projectivization/Basic.lean | theorem finrank_submodule (v : ℙ K V) : finrank K v.submodule = 1 | K : Type u_1
V : Type u_2
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v : ℙ K V
⊢ finrank K ↥(Submodule.span K {v.rep}) = 1 | exact finrank_span_singleton v.rep_nonzero | no goals | 26b80ba46c4da72a |
IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_HaarMeasure | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_HaarMeasure {𝓕 : Set G}
(h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) [IsMulLeftInvariant μ] [SigmaFinite μ]
{V : Set (G ⧸ Γ)} (hV : (interior V).Nonempty) (meas_V : MeasurableSet V)
(hμK : μ V = ν ((π ⁻¹' V) ∩ 𝓕)) (neTopV : μ V ≠ ⊤) :
QuotientMeasu... | case neZeroV.a
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ... | simp | no goals | f605250f87a9b920 |
MeasureTheory.setLIntegral_const | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem setLIntegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s | α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
c : ℝ≥0∞
⊢ ∫⁻ (x : α) in s, c ∂μ = c * μ s | rw [lintegral_const, Measure.restrict_apply_univ] | no goals | b3521c67413cc4cc |
Cardinal.preAleph_isNormal | Mathlib/SetTheory/Cardinal/Aleph.lean | theorem preAleph_isNormal : IsNormal (ord ∘ preAleph) | ⊢ IsNormal (ord ∘ ⇑preAleph) | convert isNormal_preOmega | case h.e'_1
⊢ ord ∘ ⇑preAleph = ⇑preOmega | 97ce12621c6780c7 |
Set.Ioi_mul_Ici_subset' | Mathlib/Algebra/Order/Group/Pointwise/Interval.lean | theorem Ioi_mul_Ici_subset' (a b : α) : Ioi a * Ici b ⊆ Ioi (a * b) | case intro.intro.intro.intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b : α
this : MulLeftMono α
y : α
hya : y ∈ Ioi a
z : α
hzb : z ∈ Ici b
⊢ (fun x1 x2 => x1 * x2) y z ∈ Ioi (a * b) | exact mul_lt_mul_of_lt_of_le hya hzb | no goals | 4d5d88efdc9d7508 |
starConvex_iff_div | Mathlib/Analysis/Convex/Star.lean | theorem starConvex_iff_div : StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s →
∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s :=
⟨fun h y hy a b ha hb hab => by
apply h hy
· positivity
· positivity
· rw [← add_div]
exact div_self hab.ne',
fun h y hy a b ha hb hab => b... | case a
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x : E
s : Set E
h : StarConvex 𝕜 x s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : 0 < a + b
⊢ a / (a + b) + b / (a + b) = 1 | rw [← add_div] | case a
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x : E
s : Set E
h : StarConvex 𝕜 x s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : 0 < a + b
⊢ (a + b) / (a + b) = 1 | cfbb609cf4b8dfa0 |
mellin_comp_mul_left | Mathlib/Analysis/MellinTransform.lean | theorem mellin_comp_mul_left (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (fun t => f (a * t)) s = (a : ℂ) ^ (-s) • mellin f s | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f (a * t) = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t | have : EqOn (fun t : ℝ => (t : ℂ) ^ (s - 1) • f (a * t))
(fun t : ℝ => (a : ℂ) ^ (1 - s) • (fun u : ℝ => (u : ℂ) ^ (s - 1) • f u) (a * t))
(Ioi 0) := fun t ht ↦ by
dsimp only
rw [ofReal_mul, mul_cpow_ofReal_nonneg ha.le (le_of_lt ht), ← mul_smul,
(by ring : 1 - s = -(s - 1)), cpow_neg, inv_mul_cancel_... | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this : EqOn (fun t => ↑t ^ (s - 1) • f (a * t)) (fun t => ↑a ^ (1 - s) • (fun u => ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f (a * t) = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t | eba6520b29ea7f1f |
Set.pi_nonempty_iff | Mathlib/Data/Set/Prod.lean | theorem pi_nonempty_iff : (s.pi t).Nonempty ↔ ∀ i, ∃ x, i ∈ s → x ∈ t i | ι : Type u_1
α : ι → Type u_2
s : Set ι
t : (i : ι) → Set (α i)
⊢ (s.pi t).Nonempty ↔ ∀ (i : ι), ∃ x, i ∈ s → x ∈ t i | simp [Classical.skolem, Set.Nonempty] | no goals | 29679f3fb73380bd |
CategoryTheory.braiding_rightUnitor_aux₂ | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | theorem braiding_rightUnitor_aux₂ (X : C) :
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) = 𝟙_ C ◁ (λ_ X).hom :=
calc
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) =
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X : C
⊢ (α_ (𝟙_ C) (𝟙_ C) X).inv ≫ (ρ_ (𝟙_ C)).hom ▷ X = 𝟙_ C ◁ (λ_ X).hom | rw [triangle_assoc_comp_right] | no goals | 32413f0ef3e1a702 |
CategoryTheory.ShortComplex.quasiIso_iff_isIso_rightHomologyMap' | Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | lemma quasiIso_iff_isIso_rightHomologyMap' (φ : S₁ ⟶ S₂)
(h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :
QuasiIso φ ↔ IsIso (rightHomologyMap' φ h₁ h₂) | C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ QuasiIso φ ↔ IsIso (rightHomologyMap' φ h₁ h₂) | rw [γ.quasiIso_iff, γ.rightHomologyMap'_eq] | no goals | 78d183e4735284d3 |
iUnion_Icc_intCast | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem iUnion_Icc_intCast : ⋃ n : ℤ, Icc (n : α) (n + 1) = Set.univ | α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Archimedean α
⊢ ⋃ n, Icc (↑n) (↑n + 1) = univ | simpa only [zero_add] using iUnion_Icc_add_intCast (0 : α) | no goals | b9c8899dcf9d08cc |
TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | theorem iSup_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendieckTopology X U) :
iSup (coveringOfPresieve U R) = U | case a
X : TopCat
U : Opens ↑X
R : Presieve U
hR : Sieve.generate R ∈ (Opens.grothendieckTopology ↑X) U
⊢ iSup (coveringOfPresieve U R) ≤ U | refine iSup_le ?_ | case a
X : TopCat
U : Opens ↑X
R : Presieve U
hR : Sieve.generate R ∈ (Opens.grothendieckTopology ↑X) U
⊢ ∀ (i : (V : Opens ↑X) × { f // R f }), coveringOfPresieve U R i ≤ U | c64af7441d1c23c0 |
WittVector.map_frobeniusPoly | Mathlib/RingTheory/WittVector/Frobenius.lean | theorem map_frobeniusPoly (n : ℕ) :
MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n | case h.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
⊢ C (↑p ^ (j + 1) * (↑((p ^ (n - i)).choose (j + 1)) * (↑p ^ i * ⅟↑p ^ n))) =
C ((Int.castRingHom ℚ) ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1))) * ↑p) | rw [C_inj] | case h.h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
⊢ ↑p ^ (j + 1) * (↑((p ^ (n - i)).choose (j + 1)) * (↑p ^ i * ⅟↑p ^ n)) =
(Int.castRingHom ℚ) ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1))) * ↑p | 1eb072545457e9c3 |
IsCyclotomicExtension.Rat.cyclotomicRing_isIntegralClosure_of_prime | Mathlib/NumberTheory/Cyclotomic/Rat.lean | theorem cyclotomicRing_isIntegralClosure_of_prime :
IsIntegralClosure (CyclotomicRing p ℤ ℚ) ℤ (CyclotomicField p ℚ) | p : ℕ+
hp : Fact (Nat.Prime ↑p)
⊢ IsIntegralClosure (CyclotomicRing (p ^ 1) ℤ ℚ) ℤ (CyclotomicField (p ^ 1) ℚ) | exact cyclotomicRing_isIntegralClosure_of_prime_pow | no goals | 6fa714ef2d882b6c |
AkraBazziRecurrence.GrowsPolynomially.add_isLittleO | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x | f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ c * ‖f x‖
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
⊢ 0 < 1... | norm_num | no goals | dd533a6f58fe3819 |
Complex.norm_sub_mem_Icc_angle | Mathlib/Analysis/Complex/Angle.lean | /-- Chord-length is a multiple of arc-length up to constants. -/
lemma norm_sub_mem_Icc_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y) | x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ 0 ≤ |θ| | positivity | no goals | 5f1a2d709d27832f |
Rel.card_interedges_finpartition_left | Mathlib/Combinatorics/SimpleGraph/Density.lean | theorem card_interedges_finpartition_left [DecidableEq α] (P : Finpartition s) (t : Finset β) :
#(interedges r s t) = ∑ a ∈ P.parts, #(interedges r a t) | α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s : Finset α
inst✝ : DecidableEq α
P : Finpartition s
t : Finset β
⊢ #(interedges r s t) = ∑ a ∈ P.parts, #(interedges r a t) | classical
simp_rw [← P.biUnion_parts, interedges_biUnion_left, id]
rw [card_biUnion]
exact fun x hx y hy h ↦ interedges_disjoint_left r (P.disjoint hx hy h) _ | no goals | 1de26d6f738e9233 |
Finset.prod_finset_product | Mathlib/Algebra/BigOperators/Group/Finset/Sigma.lean | theorem prod_finset_product (r : Finset (γ × α)) (s : Finset γ) (t : γ → Finset α)
(h : ∀ p : γ × α, p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1) {f : γ × α → β} :
∏ p ∈ r, f p = ∏ c ∈ s, ∏ a ∈ t c, f (c, a) | α : Type u_3
β : Type u_4
γ : Type u_5
inst✝ : CommMonoid β
r : Finset (γ × α)
s : Finset γ
t : γ → Finset α
h : ∀ (p : γ × α), p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1
f : γ × α → β
⊢ ∏ p ∈ r, f p = ∏ c ∈ s, ∏ a ∈ t c, f (c, a) | refine Eq.trans ?_ (prod_sigma s t fun p => f (p.1, p.2)) | α : Type u_3
β : Type u_4
γ : Type u_5
inst✝ : CommMonoid β
r : Finset (γ × α)
s : Finset γ
t : γ → Finset α
h : ∀ (p : γ × α), p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1
f : γ × α → β
⊢ ∏ p ∈ r, f p = ∏ x ∈ s.sigma t, f (x.fst, x.snd) | d218649e57b2dbb6 |
Subgroup.closure_mul_image_eq | Mathlib/GroupTheory/Schreier.lean | theorem closure_mul_image_eq (hR : IsComplement H R) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g => g * (hR.toRightFun g : G)⁻¹) = H | G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
⊢ closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) = H | refine le_antisymm hU fun h hh => ?_ | G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
h : G
hh : h ∈ H
⊢ h ∈ closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) | 608d4904b9787c07 |
CategoryTheory.Coverage.toGrothendieck_eq_sInf | Mathlib/CategoryTheory/Sites/Coverage.lean | theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S : Presieve X
hS : S ∈ K.covering X
⊢ S ∈ (ofGrothendieck C (toGrothendieck C K)).covering X | apply Saturate.of _ _ hS | no goals | d86a2e08211f498a |
Dynamics.coverMincard_univ | Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean | lemma coverMincard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
coverMincard T F univ n = 1 | case intro
X : Type u_1
T : X → X
F : Set X
n : ℕ
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F univ n {x}
⊢ coverMincard T F univ n ≤ 1 | rw [← Finset.coe_singleton] at this | case intro
X : Type u_1
T : X → X
F : Set X
n : ℕ
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F univ n ↑{x}
⊢ coverMincard T F univ n ≤ 1 | 1832f6ac09cf7d05 |
List.sublist_mergeSort | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean | theorem sublist_mergeSort
(trans : ∀ (a b c : α), le a b → le b c → le a c)
(total : ∀ (a b : α), le a b || le b a) :
∀ {c : List α} (_ : c.Pairwise le) (_ : c <+ l),
c <+ mergeSort l le
| _, _, .slnil => nil_sublist _
| c, hc, @Sublist.cons _ _ l a h => by
obtain ⟨l₁, l₂, h₁, h₂, -⟩ := mergeSor... | case intro.intro.intro.intro
α : Type u_1
le : α → α → Bool
l✝ : List α
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
c : List α
hc : Pairwise (fun a b => le a b = true) c
l : List α
a : α
h : c <+ l
l₁ l₂ : List α
h₁ : (a :: l).mergeSort le = l₁ ++ ... | exact h'.middle a | no goals | c9ef9462c7f97fe4 |
hasFDerivAt_integral_of_dominated_of_fderiv_le | Mathlib/Analysis/Calculus/ParametricIntegral.lean | theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F' : H → α → H →L[𝕜] E} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable (F' x₀) μ)
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a)
(bound_integrable :... | α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
ε : ℝ
F' : H → α → H →L[𝕜] E
ε_pos ... | letI : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H | α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
ε : ℝ
F' : H → α → H →L[𝕜] E
ε_pos ... | 86f496bd7a3bfae7 |
contDiffGroupoid_zero_eq | Mathlib/Geometry/Manifold/IsManifold/Basic.lean | theorem contDiffGroupoid_zero_eq : contDiffGroupoid 0 I = 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
u : PartialHomeomorph H H
a✝ : u ∈ ⊤.members
⊢ { property := fun f s => ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I)... | simp only [contDiffOn_zero] | 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
u : PartialHomeomorph H H
a✝ : u ∈ ⊤.members
⊢ ContinuousOn (↑I ∘ ↑u ∘ ↑I.symm) (↑I.symm ⁻¹' u.source ∩ range ↑I) ∧
ContinuousOn ... | 1ef8c45760ff3aa4 |
WeierstrassCurve.Affine.Point.toClass_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | lemma toClass_eq_zero (P : W.Point) : toClass P = 0 ↔ P = 0 | F : Type u
inst✝ : Field F
W : Affine F
x✝ y✝ : F
h : W.Equation x✝ y✝
right✝ : evalEval x✝ y✝ W.polynomialX ≠ 0 ∨ evalEval x✝ y✝ W.polynomialY ≠ 0
hP : toClass (some ⋯) = 0
⊢ ↑(CoordinateRing.XYIdeal' ⋯) = ↑?m.692865 | rfl | no goals | 110279bc1e8b3f89 |
IsAlgClosed.ringEquiv_of_equiv_of_char_eq | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | theorem ringEquiv_of_equiv_of_char_eq (p : ℕ) [CharP K p] [CharP L p] (hK : ℵ₀ < #K)
(hKL : Nonempty (K ≃ L)) : Nonempty (K ≃+* L) | case inr
K : Type u
L : Type v
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : IsAlgClosed K
inst✝² : IsAlgClosed L
p : ℕ
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
inst✝¹ : CharP K 0
inst✝ : CharP L 0
hp : True
⊢ Nonempty (K ≃+* L) | letI : CharZero K := CharP.charP_to_charZero K | case inr
K : Type u
L : Type v
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : IsAlgClosed K
inst✝² : IsAlgClosed L
p : ℕ
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
inst✝¹ : CharP K 0
inst✝ : CharP L 0
hp : True
this : CharZero K := CharP.charP_to_charZero K
⊢ Nonempty (K ≃+* L) | de2eb87d94ec75b8 |
Matrix.circulant_single_one | Mathlib/LinearAlgebra/Matrix/Circulant.lean | theorem circulant_single_one (α n) [Zero α] [One α] [DecidableEq n] [AddGroup n] :
circulant (Pi.single 0 1 : n → α) = (1 : Matrix n n α) | case a
α : Type u_5
n : Type u_6
inst✝³ : Zero α
inst✝² : One α
inst✝¹ : DecidableEq n
inst✝ : AddGroup n
i j : n
⊢ circulant (Pi.single 0 1) i j = 1 i j | simp [one_apply, Pi.single_apply, sub_eq_zero] | no goals | cecb6ab7fa24e964 |
Std.DHashMap.Internal.Raw₀.contains_eq_isSome_getKey? | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem contains_eq_isSome_getKey? [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {a : α} :
m.contains a = (m.getKey? a).isSome | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
a : α
⊢ m.contains a = (m.getKey? a).isSome | simp_to_model using List.containsKey_eq_isSome_getKey? | no goals | 0108159507c5736e |
emultiplicity_factor_dvd_iso_eq_emultiplicity_of_mem_normalizedFactors | Mathlib/RingTheory/ChainOfDivisors.lean | theorem emultiplicity_factor_dvd_iso_eq_emultiplicity_of_mem_normalizedFactors {m p : M} {n : N}
(hm : m ≠ 0) (hn : n ≠ 0) (hp : p ∈ normalizedFactors m)
{d : { l : M // l ∣ m } ≃ { l : N // l ∣ n }} (hd : ∀ l l', (d l : N) ∣ d l' ↔ (l : M) ∣ l') :
emultiplicity (d ⟨p, dvd_of_mem_normalizedFactors hp⟩ : N) ... | M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n ... | rw [associatesEquivOfUniqueUnits_symm_apply] | M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n ... | ab3d99bfba24fa92 |
LinearMap.ker_ne_bot_of_finrank_lt | Mathlib/LinearAlgebra/FiniteDimensional.lean | lemma ker_ne_bot_of_finrank_lt [FiniteDimensional K V] [FiniteDimensional K V₂] {f : V →ₗ[K] V₂}
(h : finrank K V₂ < finrank K V) :
LinearMap.ker f ≠ ⊥ | K : Type u
V : Type v
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module K V
V₂ : Type v'
inst✝³ : AddCommGroup V₂
inst✝² : Module K V₂
inst✝¹ : FiniteDimensional K V
inst✝ : FiniteDimensional K V₂
f : V →ₗ[K] V₂
h : finrank K V₂ < finrank K V
h₁ : finrank K ↥(range f) + finrank K ↥(ker f) = finrank K V
⊢ ... | have h₂ : finrank K (LinearMap.range f) ≤ finrank K V₂ := (LinearMap.range f).finrank_le | K : Type u
V : Type v
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module K V
V₂ : Type v'
inst✝³ : AddCommGroup V₂
inst✝² : Module K V₂
inst✝¹ : FiniteDimensional K V
inst✝ : FiniteDimensional K V₂
f : V →ₗ[K] V₂
h : finrank K V₂ < finrank K V
h₁ : finrank K ↥(range f) + finrank K ↥(ker f) = finrank K V
h₂... | 518682793e112d46 |
Nimber.mem_invSet_of_lt_invAux | Mathlib/SetTheory/Nimber/Field.lean | theorem mem_invSet_of_lt_invAux (h : b < invAux a) : b ∈ invSet a | a b : Nimber
h : b < sInf a.invSetᶜ
⊢ b ∈ a.invSet | have := not_mem_of_lt_csInf h ⟨_, bot_mem_lowerBounds _⟩ | a b : Nimber
h : b < sInf a.invSetᶜ
this : b ∉ a.invSetᶜ
⊢ b ∈ a.invSet | eec3b95274cfa652 |
Monotone.measure_iUnion | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | theorem _root_.Monotone.measure_iUnion [Preorder ι] [IsDirected ι (· ≤ ·)]
[(atTop : Filter ι).IsCountablyGenerated] {s : ι → Set α} (hs : Monotone s) :
μ (⋃ i, s i) = ⨆ i, μ (s i) | case inr.intro.intro
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
h✝ : Nonempty ι
x : ℕ → ι
hxm : Monotone x
hx : Tendsto x atTop atTop
⊢ μ (⋃ i, s i) = ⨆ i, μ (s i) | rw [← hs.iUnion_comp_tendsto_atTop hx, ← Monotone.iSup_comp_tendsto_atTop _ hx] | case inr.intro.intro
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hs : Monotone s
h✝ : Nonempty ι
x : ℕ → ι
hxm : Monotone x
hx : Tendsto x atTop atTop
⊢ μ (⋃ a, s (x a)) = ⨆ a, μ (s (x a))
... | 1b5eb1867e5189d1 |
Ordinal.nmul_nadd | Mathlib/SetTheory/Ordinal/NaturalOps.lean | theorem nmul_nadd (a b c : Ordinal) : a ⨳ (b ♯ c) = a ⨳ b ♯ a ⨳ c | case refine_2.intro.intro.intro.intro
a b c d : Ordinal.{u_1}
hd✝ : d < a ⨳ b
a' : Ordinal.{u_1}
ha : a' < a
b' : Ordinal.{u_1}
hb : b' < b
hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b'
⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) | have := nadd_lt_nadd_of_le_of_lt hd (nmul_nadd_lt ha (nadd_lt_nadd_right hb c)) | case refine_2.intro.intro.intro.intro
a b c d : Ordinal.{u_1}
hd✝ : d < a ⨳ b
a' : Ordinal.{u_1}
ha : a' < a
b' : Ordinal.{u_1}
hb : b' < b
hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b'
this : d ♯ a' ⨳ b' ♯ (a' ⨳ (b ♯ c) ♯ a ⨳ (b' ♯ c)) < a' ⨳ b ♯ a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ a' ⨳ (b' ♯ c))
⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) | aa8dbf4b11503401 |
Ideal.mem_leadingCoeff | Mathlib/RingTheory/Polynomial/Basic.lean | theorem mem_leadingCoeff (x) : x ∈ I.leadingCoeff ↔ ∃ p ∈ I, Polynomial.leadingCoeff p = x | case mp.intro.intro.intro.intro
R : Type u
inst✝ : CommSemiring R
I : Ideal R[X]
i : ℕ
p : R[X]
hpI : p ∈ I
left✝ : p.degree ≤ ↑i
⊢ ∃ p_1 ∈ I, p_1.leadingCoeff = p.leadingCoeff | exact ⟨p, hpI, rfl⟩ | no goals | 53d1bb7e4a02e31f |
Order.krullDim_eq_bot_iff | Mathlib/Order/KrullDimension.lean | @[simp]
lemma krullDim_eq_bot_iff : krullDim α = ⊥ ↔ IsEmpty α | α : Type u_1
inst✝ : Preorder α
⊢ (∀ (i : LTSeries α), ↑i.length ≤ ⊥) ↔ IsEmpty α | simp only [le_bot_iff, WithBot.natCast_ne_bot, isEmpty_iff] | α : Type u_1
inst✝ : Preorder α
⊢ LTSeries α → False ↔ α → False | 4f19822603a40f0d |
Array.fst_eq_of_mem_zipIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Range.lean | theorem fst_eq_of_mem_zipIdx {x : α × Nat} {l : Array α} {k : Nat} (h : x ∈ zipIdx l k) :
x.1 = l[x.2 - k]'(by have := le_snd_of_mem_zipIdx h; have := snd_lt_add_of_mem_zipIdx h; omega) | α : Type u_1
x : α × Nat
k : Nat
toList✝ : List α
h : x ∈ { toList := toList✝ }.zipIdx k
⊢ x ∈ { toList := toList✝ }.toList.zipIdx k | simpa using h | no goals | 7c90ecc68df3967d |
exists_mem_Ico_zpow | Mathlib/Algebra/Order/Archimedean/Basic.lean | theorem exists_mem_Ico_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ico (y ^ n) (y ^ (n + 1)) | α : Type u_1
inst✝² : LinearOrderedSemifield α
inst✝¹ : Archimedean α
x y : α
inst✝ : ExistsAddOfLE α
hx : 0 < x
hy : 1 < y
⊢ ∃ n, x ∈ Ico (y ^ n) (y ^ (n + 1)) | classical exact
let ⟨N, hN⟩ := pow_unbounded_of_one_lt x⁻¹ hy
have he : ∃ m : ℤ, y ^ m ≤ x :=
⟨-N,
le_of_lt
(by
rw [zpow_neg y ↑N, zpow_natCast]
exact (inv_lt_comm₀ hx (lt_trans (inv_pos.2 hx) hN)).1 hN)⟩
let ⟨M, hM⟩ := pow_unbounded_of_one_lt x hy
have hb... | no goals | 1b09f629ccad7679 |
CauSeq.lim_inv | Mathlib/Algebra/Order/CauSeq/Completion.lean | theorem lim_inv {f : CauSeq β abv} (hf : ¬LimZero f) : lim (inv f hf) = (lim f)⁻¹ :=
have hl : lim f ≠ 0 | α : Type u_1
inst✝³ : LinearOrderedField α
β : Type u_2
inst✝² : Field β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : IsComplete β abv
f✝ : CauSeq β abv
hf✝ : ¬f✝.LimZero
hl : f✝.lim ≠ 0
g f : CauSeq β abv
hf : ¬f.LimZero
h₂ : g - f * f.inv hf * g = 1 * g - f * f.inv hf * g
h₃ : f * f.inv hf * g = f * f.inv hf * g
... | rw [h₂, h₃, ← sub_mul] | no goals | 24d312013bd260a5 |
Set.image_subtype_val_Ioo | Mathlib/Order/Interval/Set/OrdConnected.lean | @[simp]
lemma image_subtype_val_Ioo {s : Set α} [OrdConnected s] (x y : s) :
Subtype.val '' Ioo x y = Ioo x.1 y :=
(OrderEmbedding.subtype (· ∈ s)).image_Ioo (by simpa) x y
| α : Type u_1
inst✝¹ : Preorder α
s : Set α
inst✝ : s.OrdConnected
x y : ↑s
⊢ (range ⇑(OrderEmbedding.subtype fun x => x ∈ s)).OrdConnected | simpa | no goals | 662224c3b98ac8ee |
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
= κ a (s ×ˢ Iic x) | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRa... | refine Measurable.ennreal_ofReal ?_ | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRa... | 2b5f63b20150553c |
BitVec.lt_of_getMsb?_isSome | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem lt_of_getMsb?_isSome (x : BitVec w) (i : Nat) : (getMsb? x i).isSome → i < w | w : Nat
x : BitVec w
i : Nat
⊢ (x.getMsb? i).isSome = true → i < w | if h : i < w then
simp [h]
else
simp [Nat.ge_of_not_lt h] | no goals | 1862cdbde333a541 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRatUnits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem assignmentsInvariant_insertRatUnits {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRatUnits f units).1 | case inr.inr
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitInvariant f.assignments ⋯ ratUnits assignme... | simp [hf.1] at cf | no goals | b6d274376b14ee0b |
hallMatchingsOn.nonempty | Mathlib/Combinatorics/Hall/Basic.lean | theorem hallMatchingsOn.nonempty {ι : Type u} {α : Type v} [DecidableEq α] (t : ι → Finset α)
(h : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) (ι' : Finset ι) :
Nonempty (hallMatchingsOn t ι') | case h.e'_3
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ι' : Finset ι
s' : Finset { x // x ∈ ι' }
⊢ #s' = #(image Subtype.val s') | simp only [card_image_of_injective s' Subtype.coe_injective] | no goals | cab7f21d2266f995 |
Rat.num_den_mk | Mathlib/Data/Rat/Lemmas.lean | theorem num_den_mk {q : ℚ} {n d : ℤ} (hd : d ≠ 0) (qdf : q = n /. d) :
∃ c : ℤ, n = c * q.num ∧ d = c * q.den | q : ℚ
n d : ℤ
hd : d ≠ 0
qdf : q = n /. d
hn : n ≠ 0
this : q.num * d = n * ↑q.den
⊢ q.num ∣ n | rw [qdf] | q : ℚ
n d : ℤ
hd : d ≠ 0
qdf : q = n /. d
hn : n ≠ 0
this : q.num * d = n * ↑q.den
⊢ (n /. d).num ∣ n | 6651e65d3ba1324c |
AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen | Mathlib/AlgebraicGeometry/AffineScheme.lean | theorem basicOpen_basicOpen_is_basicOpen (g : Γ(X, X.basicOpen f)) :
∃ f' : Γ(X, U), X.basicOpen f' = X.basicOpen g | case h
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
x : ↑Γ(X, U)
n : ℕ
⊢ X.basicOpen (f * x) =
X.basicOpen
(x • ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑Γ(X, X.basicOpen f)) ⟨(fun x => f ^ x) n, ⋯⟩)) | rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra,
Scheme.basicOpen_res] | case h
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
x : ↑Γ(X, U)
n : ℕ
⊢ X.basicOpen f ⊓ X.basicOpen x =
X.basicOpen f ⊓ X.basicOpen x ⊓
X.basicOpen ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑Γ(X, X.basicOpen f)) ⟨(fun x => f ^ x) n, ⋯⟩) | 32b1bf2d9457c577 |
Besicovitch.exists_closedBall_covering_tsum_measure_le | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ]
[Measure.OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧
(s ⊆ ⋃ x ∈ t, closedBa... | α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set ... | rw [measure_iUnion] | case hn
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
... | 181682c2ed74f98b |
MultilinearMap.norm_image_sub_le_of_bound' | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | theorem norm_image_sub_le_of_bound' [DecidableEq ι] (f : MultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : ∀ i, E i) :
‖f m₁ - f m₂‖ ≤ C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ | case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC ... | simp | no goals | b7626dd64eddfe02 |
CochainComplex.HomComplex.δ_map | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | @[simp]
lemma δ_map : δ n m (z.map Φ) = (δ n m z).map Φ | case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : ¬n + 1 = m
⊢ δ n m (z.map Φ) = (δ n m z).map Φ | simp only [δ_shape _ _ hnm, Cochain.map_zero] | no goals | 3a413234451ae7e7 |
Array.size_eraseP | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Erase.lean | theorem size_eraseP {l : Array α} : (l.eraseP p).size = if l.any p then l.size - 1 else l.size | case isFalse
α : Type u_1
p : α → Bool
l : Array α
h : ¬∃ i x, p l[i] = true
⊢ ∀ (i : Nat) (h : i < l.size), ¬p l[i] = true | simp_all | no goals | caa85a89730c5300 |
Std.Tactic.BVDecide.BVPred.denote_getD_eq_getLsbD | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/GetLsbD.lean | theorem denote_getD_eq_getLsbD (aig : AIG α) (assign : α → Bool) (x : BitVec w)
(xv : AIG.RefVec aig w) (falseRef : AIG.Ref aig)
(hx : ∀ idx hidx, ⟦aig, xv.get idx hidx, assign⟧ = x.getLsbD idx)
(hfalse : ⟦aig, falseRef, assign⟧ = false) :
∀ idx, ⟦aig, xv.getD idx falseRef, assign⟧ = x.getLsbD idx | case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
x : BitVec w
xv : aig.RefVec w
falseRef : aig.Ref
hx : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := xv.get idx hidx }⟧ = x.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
idx : Na... | rw [hx] | no goals | bc1180b785603c7f |
hasFDerivAt_update | Mathlib/Analysis/Calculus/FDeriv/Pi.lean | theorem hasFDerivAt_update (x : ∀ i, E i) {i : ι} (y : E i) :
HasFDerivAt (Function.update x i) (.pi (Pi.single i (.id 𝕜 (E i)))) y | case pos
𝕜 : Type u_1
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : (i : ι) → E i
i : ι
y : E i
l : E i →L[𝕜] (i : ι) → E i := ContinuousLinearMap.pi (Pi.single i (Conti... | subst hji | case pos
𝕜 : Type u_1
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : (i : ι) → E i
j : ι
y : E j
l : E j →L[𝕜] (i : ι) → E i := ContinuousLinearMap.pi (Pi.single j (Conti... | 1a0c7205e23fe373 |
zero_mem_tangentCone | Mathlib/Analysis/Calculus/TangentCone.lean | theorem zero_mem_tangentCone {s : Set E} {x : E} (hx : (𝓝[s \ {x}] x).NeBot) :
0 ∈ tangentConeAt 𝕜 s x | case intro.intro.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : (𝓝[s \ {x}] x).NeBot
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hv : ∀ (n : ℕ), v n ∈ s \ {x} ∩ Metric.ball x (u n * u n)
... | let d n := v n - x | case intro.intro.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : (𝓝[s \ {x}] x).NeBot
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hv : ∀ (n : ℕ), v n ∈ s \ {x} ∩ Metric.ball x (u n * u n)
... | 15a846dd0ca352f4 |
CircleDeg1Lift.translationNumber_mul_of_commute | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | theorem translationNumber_mul_of_commute {f g : CircleDeg1Lift} (h : Commute f g) :
τ (f * g) = τ f + τ g | f g : CircleDeg1Lift
h : Commute f g
⊢ Tendsto (fun x => f.transnumAuxSeq x + g.transnumAuxSeq x) atTop (𝓝 (τ (f * g))) | simp only [transnumAuxSeq, ← add_div] | f g : CircleDeg1Lift
h : Commute f g
⊢ Tendsto (fun x => ((f ^ 2 ^ x) 0 + (g ^ 2 ^ x) 0) / 2 ^ x) atTop (𝓝 (τ (f * g))) | 37a2cff7ea8a6696 |
MonoidAlgebra.support_single_mul_eq_image | Mathlib/Algebra/MonoidAlgebra/Support.lean | theorem support_single_mul_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
(hr : ∀ y, r * y = 0 ↔ y = 0) {x : G} (lx : IsLeftRegular x) :
(single x r * f : MonoidAlgebra k G).support = Finset.image (x * ·) f.support | k : Type u₁
G : Type u₂
inst✝² : Semiring k
inst✝¹ : DecidableEq G
inst✝ : Mul G
f : MonoidAlgebra k G
r : k
hr : ∀ (y : k), r * y = 0 ↔ y = 0
x : G
lx : IsLeftRegular x
⊢ (single x r * f).support = image (fun x_1 => x * x_1) f.support | refine subset_antisymm (support_single_mul_subset f _ _) fun y hy => ?_ | k : Type u₁
G : Type u₂
inst✝² : Semiring k
inst✝¹ : DecidableEq G
inst✝ : Mul G
f : MonoidAlgebra k G
r : k
hr : ∀ (y : k), r * y = 0 ↔ y = 0
x : G
lx : IsLeftRegular x
y : G
hy : y ∈ image (fun x_1 => x * x_1) f.support
⊢ y ∈ (single x r * f).support | 277ab9834610ad72 |
KaehlerDifferential.ideal_fg | Mathlib/RingTheory/Kaehler/Basic.lean | theorem KaehlerDifferential.ideal_fg [EssFiniteType R S] :
(KaehlerDifferential.ideal R S).FG | R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
x : S
I : Ideal (S ⊗[R] S) := Ideal.span ↑(Finset.image (fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))
⊢ (IsScalarTower.toAlgHom R (S ⊗[R] S) (S ⊗[R] S ⧸ I)).comp TensorProduct.includeRight =
(... | apply EssFiniteType.algHom_ext | case H
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
x : S
I : Ideal (S ⊗[R] S) := Ideal.span ↑(Finset.image (fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))
⊢ ∀ s ∈ EssFiniteType.finset R S,
((IsScalarTower.toAlgHom R (S ⊗[R] S) (S ⊗[R] S ⧸... | 960b040da609c2c7 |
Stream'.WSeq.exists_dropn_of_mem | Mathlib/Data/Seq/WSeq.lean | theorem exists_dropn_of_mem {s : WSeq α} {a} (h : a ∈ s) :
∃ n s', some (a, s') ∈ destruct (drop s n) :=
let ⟨n, h⟩ := exists_get?_of_mem h
⟨n, by
rcases (head_terminates_iff _).1 ⟨⟨_, h⟩⟩ with ⟨⟨o, om⟩⟩
have := Computation.mem_unique (Computation.mem_map _ om) h
rcases o with - | o
· injection ... | case mk.intro.some.mk
α : Type u
s : WSeq α
a : α
h✝ : a ∈ s
n : ℕ
h : some a ∈ s.get? n
a' : α
s' : WSeq α
om : some (a, s') ∈ (s.drop n).destruct
i : a' = a
⊢ ∃ s', some (a, s') ∈ (s.drop n).destruct | exact ⟨_, om⟩ | no goals | 9eebc16ac70760fb |
contDiffOn_succ_iff_fderivWithin | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem contDiffOn_succ_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 (n + 1) f s ↔
DifferentiableOn 𝕜 f s ∧ (n = ω → AnalyticOn 𝕜 f s) ∧
ContDiffOn 𝕜 n (fderivWithin 𝕜 f s) s | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
H : ContDiffOn 𝕜 (n + 1) ... | rw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
H : ContDiffOn 𝕜 (n + 1) ... | 84290533f50e9cf3 |
AlgebraicGeometry.genericPoint_eq_bot_of_affine | Mathlib/AlgebraicGeometry/FunctionField.lean | theorem genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain R] :
genericPoint (Spec R) = (⊥ : PrimeSpectrum R) | R : CommRingCat
inst✝ : IsDomain ↑R
⊢ PrimeSpectrum.zeroLocus ↑⊥.asIdeal = Set.univ | rw [← PrimeSpectrum.zeroLocus_singleton_zero] | R : CommRingCat
inst✝ : IsDomain ↑R
⊢ PrimeSpectrum.zeroLocus ↑⊥.asIdeal = PrimeSpectrum.zeroLocus {0} | a2652ebb426e503b |
PFunctor.liftp_iff | Mathlib/Data/PFunctor/Univariate/Basic.lean | theorem liftp_iff {α : Type u} (p : α → Prop) (x : P α) :
Liftp p x ↔ ∃ a f, x = ⟨a, f⟩ ∧ ∀ i, p (f i) | case mp
P : PFunctor.{u}
α : Type u
p : α → Prop
x : ↑P α
⊢ Liftp p x → ∃ a f, x = ⟨a, f⟩ ∧ ∀ (i : P.B a), p (f i) | rintro ⟨y, hy⟩ | case mp.intro
P : PFunctor.{u}
α : Type u
p : α → Prop
x : ↑P α
y : ↑P (Subtype p)
hy : Subtype.val <$> y = x
⊢ ∃ a f, x = ⟨a, f⟩ ∧ ∀ (i : P.B a), p (f i) | c4aadc94c9da47b9 |
Finset.prod_dite_irrel | Mathlib/Algebra/BigOperators/Group/Finset/Defs.lean | theorem prod_dite_irrel (p : Prop) [Decidable p] (s : Finset α) (f : p → α → β) (g : ¬p → α → β) :
∏ x ∈ s, (if h : p then f h x else g h x) =
if h : p then ∏ x ∈ s, f h x else ∏ x ∈ s, g h x | α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
p : Prop
inst✝ : Decidable p
s : Finset α
f : p → α → β
g : ¬p → α → β
⊢ (∏ x ∈ s, if h : p then f h x else g h x) = if h : p then ∏ x ∈ s, f h x else ∏ x ∈ s, g h x | split_ifs with h <;> rfl | no goals | 99275d01a286f9df |
Int.nneg_mul_add_sq_of_abs_le_one | Mathlib/Algebra/Order/Ring/Cast.lean | lemma nneg_mul_add_sq_of_abs_le_one (n : ℤ) (hx : |x| ≤ 1) : (0 : R) ≤ n * x + n * n | R : Type u_1
inst✝ : LinearOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hn : n < 0
this : x + ↑n ≤ 1 + -1
⊢ x + ↑n ≤ 0 | rwa [add_neg_cancel] at this | no goals | bd5028d11b4586e9 |
Prime.dvd_of_dvd_pow | Mathlib/Algebra/Prime/Defs.lean | theorem dvd_of_dvd_pow {a : M} {n : ℕ} (h : p ∣ a ^ n) : p ∣ a | case zero
M : Type u_1
inst✝ : CommMonoidWithZero M
p : M
hp : Prime p
a : M
h : p ∣ 1
⊢ p ∣ a | have := isUnit_of_dvd_one h | case zero
M : Type u_1
inst✝ : CommMonoidWithZero M
p : M
hp : Prime p
a : M
h : p ∣ 1
this : IsUnit p
⊢ p ∣ a | d27fe6ad3c2106b8 |
LipschitzWith.completion_extension | Mathlib/Topology/MetricSpace/Completion.lean | theorem LipschitzWith.completion_extension [MetricSpace β] [CompleteSpace β] {f : α → β}
{K : ℝ≥0} (h : LipschitzWith K f) : LipschitzWith K (Completion.extension f) :=
LipschitzWith.of_dist_le_mul fun x y => induction_on₂ x y
(isClosed_le (by fun_prop) (by fun_prop)) <| by
simpa only [extension_coe h.u... | α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : MetricSpace β
inst✝ : CompleteSpace β
f : α → β
K : ℝ≥0
h : LipschitzWith K f
x y : Completion α
⊢ Continuous fun x => dist (Completion.extension f x.1) (Completion.extension f x.2) | fun_prop | no goals | 5a616850749db2c3 |
Set.einfsep_pos_of_finite | Mathlib/Topology/MetricSpace/Infsep.lean | theorem einfsep_pos_of_finite [Finite s] : 0 < s.einfsep | case neg
α : Type u_1
inst✝¹ : EMetricSpace α
s : Set α
inst✝ : Finite ↑s
val✝ : Fintype ↑s
hs : s.Subsingleton
⊢ 0 < s.einfsep | exact hs.einfsep.symm ▸ WithTop.top_pos | no goals | a37f1bdc73519b23 |
Polynomial.scaleRoots_eval₂_mul_of_commute | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | theorem scaleRoots_eval₂_mul_of_commute {p : S[X]} (f : S →+* A) (a : A) (s : S)
(hsa : Commute (f s) a) (hf : ∀ s₁ s₂, Commute (f s₁) (f s₂)) :
eval₂ f (f s * a) (scaleRoots p s) = f s ^ p.natDegree * eval₂ f a p | S : Type u_2
A : Type u_3
inst✝¹ : Semiring S
inst✝ : Semiring A
p : S[X]
f : S →+* A
a : A
s : S
hsa : Commute (f s) a
hf : ∀ (s₁ s₂ : S), Commute (f s₁) (f s₂)
i : ℕ
_hi : i ∈ p.support
⊢ f (p.coeff i * s ^ (p.natDegree - i)) * (f s * a) ^ i = f (p.coeff i) * f s ^ (p.natDegree - i + i) * a ^ i | simp_rw [f.map_mul, f.map_pow, pow_add, hsa.mul_pow, mul_assoc] | no goals | 72127d2a3a809905 |
InnerProductSpace.Core.inner_self_of_eq_zero | Mathlib/Analysis/InnerProductSpace/Defs.lean | theorem inner_self_of_eq_zero {x : F} : x = 0 → ⟪x, x⟫ = 0 | 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
⊢ ⟪0, 0⟫_𝕜 = 0 | exact inner_zero_left _ | no goals | 676cfea59056a4fa |
IsCoprime.pow_left_iff | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem IsCoprime.pow_left_iff (hm : 0 < m) : IsCoprime (x ^ m) y ↔ IsCoprime x y | R : Type u
inst✝ : CommSemiring R
x y : R
m : ℕ
hm : 0 < m
h : IsCoprime (∏ _x ∈ range m, x) y
⊢ IsCoprime x y | exact h.of_prod_left 0 (Finset.mem_range.mpr hm) | no goals | ecd9cc21301320f6 |
List.range'_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem range'_eq_append_iff : range' s n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs = range' s k ∧ ys = range' (s + k) (n - k) | case succ.mpr.intro.intro.intro.zero
n : Nat
ih : ∀ {s : Nat} {xs ys : List Nat}, range' s n = xs ++ ys ↔ ∃ k, k ≤ n ∧ xs = range' s k ∧ ys = range' (s + k) (n - k)
s : Nat
h : 0 ≤ n + 1
⊢ range' s 0 = [] ∧ range' (s + 0) (n + 1 - 0) = s :: range' (s + 1) n ∨
∃ a', range' s 0 = s :: a' ∧ range' (s + 1) n = a' ++ ra... | simp [range'_succ] | no goals | 001f41225221ffd1 |
CategoryTheory.Limits.IsCofiltered.sequentialFunctor_initial_aux | Mathlib/CategoryTheory/Limits/Shapes/Countable.lean | theorem sequentialFunctor_initial_aux (j : J) : ∃ (n : ℕ), sequentialFunctor_obj J n ≤ j | case intro
J : Type u_2
inst✝² : Countable J
inst✝¹ : Preorder J
inst✝ : IsCofiltered J
j : J
m : ℕ
h : ⋯.choose m = j
⊢ ∃ n, sequentialFunctor_obj J n ≤ j | refine ⟨m + 1, ?_⟩ | case intro
J : Type u_2
inst✝² : Countable J
inst✝¹ : Preorder J
inst✝ : IsCofiltered J
j : J
m : ℕ
h : ⋯.choose m = j
⊢ sequentialFunctor_obj J (m + 1) ≤ j | 532687255dc32248 |
connectedComponent_eq_iInter_isClopen | Mathlib/Topology/Separation/Regular.lean | theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x : X) :
connectedComponent x = ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s | case intro.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed (⋂ s, ↑s)
a b : Set X
ha : IsClosed a
hb : IsClosed b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen u
hv : IsOpen v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
H1 : Disjoint (... | rwa [← disjoint_compl_left_iff_subset, disjoint_iff_inter_eq_empty,
← not_nonempty_iff_eq_empty] | no goals | cba2fad96fde2a59 |
StarOrderedRing.mul_le_mul_of_nonneg_left | Mathlib/Algebra/Order/Ring/Star.lean | private lemma mul_le_mul_of_nonneg_left {R : Type*} [CommSemiring R] [PartialOrder R]
[StarRing R] [StarOrderedRing R] {a b c : R} (hab : a ≤ b) (hc : 0 ≤ c) : c * a ≤ c * b | case mem.intro
R : Type u_1
inst✝³ : CommSemiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
a b c : R
hab : a ≤ b
x : R
⊢ star x * a * x ≤ star x * b * x | exact conjugate_le_conjugate hab x | no goals | 28f106ff69a62b17 |
List.drop_set | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean | theorem drop_set {l : List α} {n m : Nat} {a : α} :
(l.set m a).drop n = if m < n then l.drop n else (l.drop n).set (m - n) a | case succ.nil
α : Type u_1
a : α
n✝ : Nat
hn : ∀ {l : List α} {m : Nat}, drop n✝ (l.set m a) = if m < n✝ then drop n✝ l else (drop n✝ l).set (m - n✝) a
m : Nat
⊢ drop (n✝ + 1) ([].set m a) = if m < n✝ + 1 then drop (n✝ + 1) [] else (drop (n✝ + 1) []).set (m - (n✝ + 1)) a | simp | no goals | 0d47a5ed137632dc |
Dynamics.coverEntropyInf_eq_iSup_netEntropyInfEntourage | Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean | theorem coverEntropyInf_eq_iSup_netEntropyInfEntourage :
coverEntropyInf T F = ⨆ U ∈ 𝓤 X, netEntropyInfEntourage T F U | case a
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
⊢ netEntropyInfEntourage T F U ≤ coverEntropyInf T F | apply (netEntropyInfEntourage_antitone T F (symmetrizeRel_subset_self U)).trans | case a
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
⊢ (fun U => netEntropyInfEntourage T F U) (symmetrizeRel U) ≤ coverEntropyInf T F | e89a5a4b832783dc |
Set.Finite.exists_finset_coe | Mathlib/Data/Set/Finite/Basic.lean | theorem Finite.exists_finset_coe {s : Set α} (h : s.Finite) : ∃ s' : Finset α, ↑s' = s | α : Type u
s : Set α
h : s.Finite
⊢ ∃ s', ↑s' = s | cases h.nonempty_fintype | case intro
α : Type u
s : Set α
h : s.Finite
val✝ : Fintype ↑s
⊢ ∃ s', ↑s' = s | 89eb8218371bdc8d |
Polynomial.integralNormalization_coeff_mul_leadingCoeff_pow | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | theorem integralNormalization_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ natDegree p) :
(integralNormalization p).coeff i * p.leadingCoeff ^ i =
p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) | R : Type u
inst✝ : Semiring R
p : R[X]
i : ℕ
hp : 1 ≤ p.natDegree
⊢ p.integralNormalization.coeff i * p.leadingCoeff ^ i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) | rw [integralNormalization_coeff] | R : Type u
inst✝ : Semiring R
p : R[X]
i : ℕ
hp : 1 ≤ p.natDegree
⊢ (if p.degree = ↑i then 1 else p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i)) * p.leadingCoeff ^ i =
p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) | 435e215726619a62 |
sphere_pi | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | /-- A sphere in a product space is a union of spheres on each component restricted to the closed
ball. -/
lemma sphere_pi (x : ∀ b, π b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r | case inr.inl
β : Type u_2
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : dist x x✝ ≤ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0 | rw [dist_pi_le_iff le_rfl] at hx | case inr.inl
β : Type u_2
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : ∀ (b : β), dist (x b) (x✝ b) ≤ 0
⊢ x ∈ Function.eval default ⁻¹' sphere ... | f35020355db84c52 |
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