ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_iff_eq_sup_sdiff
theorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y)
Goal: x ≤ z ↔ y = z ⊔ y \ x
-- sdiff_sup
theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: y \ (x ⊔ z) = y \ x ⊓ y \ z
-- sdiff_eq_sdiff_iff_inf_eq_inf
theorem sdiff_eq_sdiff_iff_inf_eq_inf : y \ x = y \ z ↔ y ⊓ x = y ⊓ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: y \ x = y \ z ↔ y ⊓ x = y ⊓ z
-- sdiff_eq_self_iff_disjoint
theorem sdiff_eq_self_iff_disjoint : x \ y = x ↔ Disjoint y x 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y = x ↔ Disjoint y x
-- sdiff_eq_self_iff_disjoint'
theorem sdiff_eq_self_iff_disjoint' : x \ y = x ↔ Disjoint x y 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y = x ↔ Disjoint x y
-- sdiff_lt
theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α] (hx : y ≤ x) (hy : y ≠ ⊥)
Goal: x \ y < x
-- le_sdiff_iff
@[simp]
theorem le_sdiff_iff : x ≤ y \ x ↔ x = ⊥ 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x ≤ y \ x ↔ x = ⊥
-- sdiff_eq_right
@[simp] lemma sdiff_eq_right : x \ y = y ↔ x = ⊥ ∧ y = ⊥ 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y = y ↔ x = ⊥ ∧ y = ⊥
-- sdiff_ne_right
lemma sdiff_ne_right : x \ y ≠ y ↔ x ≠ ⊥ ∨ y ≠ ⊥ 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ y ≠ y ↔ x ≠ ⊥ ∨ y ≠ ⊥
-- sdiff_lt_sdiff_right
theorem sdiff_lt_sdiff_right (h : x < y) (hz : z ≤ x) : x \ z < y \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (h : x < y) (hz : z ≤ x)
Goal: x \ z < y \ z
-- sup_inf_inf_sdiff
theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z
-- sdiff_sdiff_right
theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z
-- sdiff_sdiff_right'
theorem sdiff_sdiff_right' : x \ (y \ z) = x \ y ⊔ x ⊓ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: x \ (y \ z) = x \ y ⊔ x ⊓ z
-- sdiff_sdiff_eq_sdiff_sup
theorem sdiff_sdiff_eq_sdiff_sup (h : z ≤ x) : x \ (y \ z) = x \ y ⊔ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (h : z ≤ x)
Goal: x \ (y \ z) = x \ y ⊔ z
-- sdiff_sdiff_right_self
@[simp]
theorem sdiff_sdiff_right_self : x \ (x \ y) = x ⊓ y 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x \ (x \ y) = x ⊓ y
-- sdiff_sdiff_eq_self
theorem sdiff_sdiff_eq_self (h : y ≤ x) : x \ (x \ y) = y 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α] (h : y ≤ x)
Goal: x \ (x \ y) = y
-- sdiff_eq_symm
theorem sdiff_eq_symm (hy : y ≤ x) (h : x \ y = z) : x \ z = y 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hy : y ≤ x) (h : x \ y = z)
Goal: x \ z = y
-- sdiff_eq_comm
theorem sdiff_eq_comm (hy : y ≤ x) (hz : z ≤ x) : x \ y = z ↔ x \ z = y 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hy : y ≤ x) (hz : z ≤ x)
Goal: x \ y = z ↔ x \ z = y
-- eq_of_sdiff_eq_sdiff
theorem eq_of_sdiff_eq_sdiff (hxz : x ≤ z) (hyz : y ≤ z) (h : z \ x = z \ y) : x = y 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hxz : x ≤ z) (hyz : y ≤ z) (h : z \ x = z \ y)
Goal: x = y
-- sdiff_sdiff_left'
theorem sdiff_sdiff_left' : (x \ y) \ z = x \ y ⊓ x \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: (x \ y) \ z = x \ y ⊓ x \ z
-- sdiff_sdiff_sup_sdiff
theorem sdiff_sdiff_sup_sdiff : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x) 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x)
-- sdiff_sdiff_sup_sdiff'
theorem sdiff_sdiff_sup_sdiff' : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y
-- sdiff_sdiff_sdiff_cancel_left
lemma sdiff_sdiff_sdiff_cancel_left (hca : z ≤ x) : (x \ y) \ (x \ z) = z \ y 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hca : z ≤ x)
Goal: (x \ y) \ (x \ z) = z \ y
-- sdiff_sdiff_sdiff_cancel_right
lemma sdiff_sdiff_sdiff_cancel_right (hcb : z ≤ y) : (x \ z) \ (y \ z) = x \ y 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (hcb : z ≤ y)
Goal: (x \ z) \ (y \ z) = x \ y
-- inf_sdiff
theorem inf_sdiff : (x ⊓ y) \ z = x \ z ⊓ y \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: (x ⊓ y) \ z = x \ z ⊓ y \ z
-- inf_sdiff_assoc
theorem inf_sdiff_assoc : (x ⊓ y) \ z = x ⊓ y \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: (x ⊓ y) \ z = x ⊓ y \ z
-- inf_sdiff_right_comm
theorem inf_sdiff_right_comm : x \ z ⊓ y = (x ⊓ y) \ z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: x \ z ⊓ y = (x ⊓ y) \ z
-- inf_sdiff_distrib_left
theorem inf_sdiff_distrib_left (a b c : α) : a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c) 
Arguments: {α : Type u} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c)
-- inf_sdiff_distrib_right
theorem inf_sdiff_distrib_right (a b c : α) : a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c) 
Arguments: {α : Type u} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c)
-- disjoint_sdiff_comm
theorem disjoint_sdiff_comm : Disjoint (x \ z) y ↔ Disjoint x (y \ z) 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α]
Goal: Disjoint (x \ z) y ↔ Disjoint x (y \ z)
-- sup_eq_sdiff_sup_sdiff_sup_inf
theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y 
Arguments: {α : Type u} {x y : α} [GeneralizedBooleanAlgebra α]
Goal: x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y
-- sup_lt_of_lt_sdiff_left
theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (h : y < z \ x) (hxz : x ≤ z)
Goal: x ⊔ y < z
-- sup_lt_of_lt_sdiff_right
theorem sup_lt_of_lt_sdiff_right (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z 
Arguments: {α : Type u} {x y z : α} [GeneralizedBooleanAlgebra α] (h : x < z \ y) (hyz : y ≤ z)
Goal: x ⊔ y < z
-- BooleanAlgebra.inf_compl_le_bot
class BooleanAlgebra (α : Type u) extends
    DistribLattice α, HasCompl α, SDiff α, HImp α, Top α, Bot α where
    inf_compl_le_bot : ∀ x : α, x ⊓ xᶜ ≤ ⊥
    top_le_sup_compl : ∀ x : α, ⊤ ≤ x ⊔ xᶜ
    le_top : ∀ a : α, a ≤ ⊤
    bot_le : ∀ a : α, ⊥ ≤ a
    sdiff := fun x y => x ⊓ yᶜ
    himp := fun x y => y ⊔ xᶜ
    sdiff_eq : ∀ x y : α, x \ y = x ⊓ yᶜ := by aesop
    himp_eq : ∀ x y : α, x ⇨ y = y ⊔ xᶜ := by aesop
Arguments: {α : Type u} [self : BooleanAlgebra α] (x : α)
Goal: x ⊓ xᶜ ≤ ⊥
-- BooleanAlgebra.himp_eq
class BooleanAlgebra (α : Type u) extends
    DistribLattice α, HasCompl α, SDiff α, HImp α, Top α, Bot α where
    inf_compl_le_bot : ∀ x : α, x ⊓ xᶜ ≤ ⊥
    top_le_sup_compl : ∀ x : α, ⊤ ≤ x ⊔ xᶜ
    le_top : ∀ a : α, a ≤ ⊤
    bot_le : ∀ a : α, ⊥ ≤ a
    sdiff := fun x y => x ⊓ yᶜ
    himp := fun x y => y ⊔ xᶜ
    sdiff_eq : ∀ x y : α, x \ y = x ⊓ yᶜ := by aesop
    himp_eq : ∀ x y : α, x ⇨ y = y ⊔ xᶜ := by aesop
Arguments: {α : Type u} [self : BooleanAlgebra α] (x y : α)
Goal: x ⇨ y = y ⊔ xᶜ
-- BooleanAlgebra.sdiff_eq
class BooleanAlgebra (α : Type u) extends
    DistribLattice α, HasCompl α, SDiff α, HImp α, Top α, Bot α where
    inf_compl_le_bot : ∀ x : α, x ⊓ xᶜ ≤ ⊥
    top_le_sup_compl : ∀ x : α, ⊤ ≤ x ⊔ xᶜ
    le_top : ∀ a : α, a ≤ ⊤
    bot_le : ∀ a : α, ⊥ ≤ a
    sdiff := fun x y => x ⊓ yᶜ
    himp := fun x y => y ⊔ xᶜ
    sdiff_eq : ∀ x y : α, x \ y = x ⊓ yᶜ := by aesop
    himp_eq : ∀ x y : α, x ⇨ y = y ⊔ xᶜ := by aesop
Arguments: {α : Type u} [self : BooleanAlgebra α] (x y : α)
Goal: x \ y = x ⊓ yᶜ
-- BooleanAlgebra.le_top
class BooleanAlgebra (α : Type u) extends
    DistribLattice α, HasCompl α, SDiff α, HImp α, Top α, Bot α where
    inf_compl_le_bot : ∀ x : α, x ⊓ xᶜ ≤ ⊥
    top_le_sup_compl : ∀ x : α, ⊤ ≤ x ⊔ xᶜ
    le_top : ∀ a : α, a ≤ ⊤
    bot_le : ∀ a : α, ⊥ ≤ a
    sdiff := fun x y => x ⊓ yᶜ
    himp := fun x y => y ⊔ xᶜ
    sdiff_eq : ∀ x y : α, x \ y = x ⊓ yᶜ := by aesop
    himp_eq : ∀ x y : α, x ⇨ y = y ⊔ xᶜ := by aesop
Arguments: {α : Type u} [self : BooleanAlgebra α] (a : α)
Goal: a ≤ ⊤
-- BooleanAlgebra.bot_le
class BooleanAlgebra (α : Type u) extends
    DistribLattice α, HasCompl α, SDiff α, HImp α, Top α, Bot α where
    inf_compl_le_bot : ∀ x : α, x ⊓ xᶜ ≤ ⊥
    top_le_sup_compl : ∀ x : α, ⊤ ≤ x ⊔ xᶜ
    le_top : ∀ a : α, a ≤ ⊤
    bot_le : ∀ a : α, ⊥ ≤ a
    sdiff := fun x y => x ⊓ yᶜ
    himp := fun x y => y ⊔ xᶜ
    sdiff_eq : ∀ x y : α, x \ y = x ⊓ yᶜ := by aesop
    himp_eq : ∀ x y : α, x ⇨ y = y ⊔ xᶜ := by aesop
Arguments: {α : Type u} [self : BooleanAlgebra α] (a : α)
Goal: ⊥ ≤ a
-- BooleanAlgebra.top_le_sup_compl
class BooleanAlgebra (α : Type u) extends
    DistribLattice α, HasCompl α, SDiff α, HImp α, Top α, Bot α where
    inf_compl_le_bot : ∀ x : α, x ⊓ xᶜ ≤ ⊥
    top_le_sup_compl : ∀ x : α, ⊤ ≤ x ⊔ xᶜ
    le_top : ∀ a : α, a ≤ ⊤
    bot_le : ∀ a : α, ⊥ ≤ a
    sdiff := fun x y => x ⊓ yᶜ
    himp := fun x y => y ⊔ xᶜ
    sdiff_eq : ∀ x y : α, x \ y = x ⊓ yᶜ := by aesop
    himp_eq : ∀ x y : α, x ⇨ y = y ⊔ xᶜ := by aesop
Arguments: {α : Type u} [self : BooleanAlgebra α] (x : α)
Goal: ⊤ ≤ x ⊔ xᶜ
-- inf_compl_eq_bot'
theorem inf_compl_eq_bot' : x ⊓ xᶜ = ⊥ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: x ⊓ xᶜ = ⊥
-- sup_compl_eq_top
@[simp]
theorem sup_compl_eq_top : x ⊔ xᶜ = ⊤ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: x ⊔ xᶜ = ⊤
-- compl_sup_eq_top
@[simp]
theorem compl_sup_eq_top : xᶜ ⊔ x = ⊤ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: xᶜ ⊔ x = ⊤
-- isCompl_compl
theorem isCompl_compl : IsCompl x xᶜ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: IsCompl x xᶜ
-- sdiff_eq
theorem sdiff_eq : x \ y = x ⊓ yᶜ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x \ y = x ⊓ yᶜ
-- himp_eq
theorem himp_eq : x ⇨ y = y ⊔ xᶜ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x ⇨ y = y ⊔ xᶜ
-- hnot_eq_compl
@[simp]
theorem hnot_eq_compl : ￢x = xᶜ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: ￢x = xᶜ
-- top_sdiff
theorem top_sdiff : ⊤ \ x = xᶜ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: ⊤ \ x = xᶜ
-- eq_compl_iff_isCompl
theorem eq_compl_iff_isCompl : x = yᶜ ↔ IsCompl x y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x = yᶜ ↔ IsCompl x y
-- compl_eq_iff_isCompl
theorem compl_eq_iff_isCompl : xᶜ = y ↔ IsCompl x y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: xᶜ = y ↔ IsCompl x y
-- compl_eq_comm
theorem compl_eq_comm : xᶜ = y ↔ yᶜ = x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: xᶜ = y ↔ yᶜ = x
-- eq_compl_comm
theorem eq_compl_comm : x = yᶜ ↔ y = xᶜ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x = yᶜ ↔ y = xᶜ
-- compl_compl
@[simp]
theorem compl_compl (x : α) : xᶜᶜ = x 
Arguments: {α : Type u} [BooleanAlgebra α] (x : α)
Goal: xᶜᶜ = x
-- compl_comp_compl
theorem compl_comp_compl : compl ∘ compl = @id α 
Arguments: {α : Type u} [BooleanAlgebra α]
Goal: compl ∘ compl = id
-- compl_involutive
@[simp]
theorem compl_involutive : Function.Involutive (compl : α → α) 
Arguments: {α : Type u} [BooleanAlgebra α]
Goal: Function.Involutive compl
-- compl_bijective
theorem compl_bijective : Function.Bijective (compl : α → α) 
Arguments: {α : Type u} [BooleanAlgebra α]
Goal: Function.Bijective compl
-- compl_surjective
theorem compl_surjective : Function.Surjective (compl : α → α) 
Arguments: {α : Type u} [BooleanAlgebra α]
Goal: Function.Surjective compl
-- compl_injective
theorem compl_injective : Function.Injective (compl : α → α) 
Arguments: {α : Type u} [BooleanAlgebra α]
Goal: Function.Injective compl
-- compl_inj_iff
@[simp]
theorem compl_inj_iff : xᶜ = yᶜ ↔ x = y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: xᶜ = yᶜ ↔ x = y
-- IsCompl.compl_eq_iff
theorem IsCompl.compl_eq_iff (h : IsCompl x y) : zᶜ = y ↔ z = x 
Arguments: {α : Type u} {x y z : α} [BooleanAlgebra α] (h : IsCompl x y)
Goal: zᶜ = y ↔ z = x
-- compl_eq_top
@[simp]
theorem compl_eq_top : xᶜ = ⊤ ↔ x = ⊥ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: xᶜ = ⊤ ↔ x = ⊥
-- compl_eq_bot
@[simp]
theorem compl_eq_bot : xᶜ = ⊥ ↔ x = ⊤ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: xᶜ = ⊥ ↔ x = ⊤
-- compl_inf
@[simp]
theorem compl_inf : (x ⊓ y)ᶜ = xᶜ ⊔ yᶜ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: (x ⊓ y)ᶜ = xᶜ ⊔ yᶜ
-- compl_le_compl_iff_le
@[simp]
theorem compl_le_compl_iff_le : yᶜ ≤ xᶜ ↔ x ≤ y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: yᶜ ≤ xᶜ ↔ x ≤ y
-- compl_lt_compl_iff_lt
@[simp] lemma compl_lt_compl_iff_lt : yᶜ < xᶜ ↔ x < y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: yᶜ < xᶜ ↔ x < y
-- compl_le_of_compl_le
theorem compl_le_of_compl_le (h : yᶜ ≤ x) : xᶜ ≤ y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α] (h : yᶜ ≤ x)
Goal: xᶜ ≤ y
-- compl_le_iff_compl_le
theorem compl_le_iff_compl_le : xᶜ ≤ y ↔ yᶜ ≤ x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: xᶜ ≤ y ↔ yᶜ ≤ x
-- compl_le_self
@[simp] theorem compl_le_self : xᶜ ≤ x ↔ x = ⊤ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α]
Goal: xᶜ ≤ x ↔ x = ⊤
-- compl_lt_self
@[simp] theorem compl_lt_self [Nontrivial α] : xᶜ < x ↔ x = ⊤ 
Arguments: {α : Type u} {x : α} [BooleanAlgebra α] [Nontrivial α]
Goal: xᶜ < x ↔ x = ⊤
-- sdiff_compl
@[simp]
theorem sdiff_compl : x \ yᶜ = x ⊓ y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x \ yᶜ = x ⊓ y
-- sup_inf_inf_compl
@[simp]
theorem sup_inf_inf_compl : x ⊓ y ⊔ x ⊓ yᶜ = x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x ⊓ y ⊔ x ⊓ yᶜ = x
-- compl_sdiff
theorem compl_sdiff : (x \ y)ᶜ = x ⇨ y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: (x \ y)ᶜ = x ⇨ y
-- compl_himp
@[simp]
theorem compl_himp : (x ⇨ y)ᶜ = x \ y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: (x ⇨ y)ᶜ = x \ y
-- compl_sdiff_compl
theorem compl_sdiff_compl : xᶜ \ yᶜ = y \ x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: xᶜ \ yᶜ = y \ x
-- compl_himp_compl
@[simp]
theorem compl_himp_compl : xᶜ ⇨ yᶜ = y ⇨ x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: xᶜ ⇨ yᶜ = y ⇨ x
-- disjoint_compl_left_iff
theorem disjoint_compl_left_iff : Disjoint xᶜ y ↔ y ≤ x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: Disjoint xᶜ y ↔ y ≤ x
-- disjoint_compl_right_iff
theorem disjoint_compl_right_iff : Disjoint x yᶜ ↔ x ≤ y 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: Disjoint x yᶜ ↔ x ≤ y
-- codisjoint_himp_self_left
theorem codisjoint_himp_self_left : Codisjoint (x ⇨ y) x 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: Codisjoint (x ⇨ y) x
-- codisjoint_himp_self_right
theorem codisjoint_himp_self_right : Codisjoint x (x ⇨ y) 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: Codisjoint x (x ⇨ y)
-- himp_le
theorem himp_le : x ⇨ y ≤ z ↔ y ≤ z ∧ Codisjoint x z 
Arguments: {α : Type u} {x y z : α} [BooleanAlgebra α]
Goal: x ⇨ y ≤ z ↔ y ≤ z ∧ Codisjoint x z
-- himp_le_iff
@[simp] lemma himp_le_iff : x ⇨ y ≤ x ↔ x = ⊤ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x ⇨ y ≤ x ↔ x = ⊤
-- himp_eq_left
@[simp] lemma himp_eq_left : x ⇨ y = x ↔ x = ⊤ ∧ y = ⊤ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x ⇨ y = x ↔ x = ⊤ ∧ y = ⊤
-- himp_ne_right
lemma himp_ne_right : x ⇨ y ≠ x ↔ x ≠ ⊤ ∨ y ≠ ⊤ 
Arguments: {α : Type u} {x y : α} [BooleanAlgebra α]
Goal: x ⇨ y ≠ x ↔ x ≠ ⊤ ∨ y ≠ ⊤
-- Bool.sup_eq_bor
@[simp]
theorem Bool.sup_eq_bor : (· ⊔ ·) = or 
Goal: (fun x x_1 => x ⊔ x_1) = or
-- Bool.inf_eq_band
@[simp]
theorem Bool.inf_eq_band : (· ⊓ ·) = and 
Goal: (fun x x_1 => x ⊓ x_1) = and
-- Bool.compl_eq_bnot
@[simp]
theorem Bool.compl_eq_bnot : HasCompl.compl = not 
Goal: compl = not
-- CountableInterFilter.countable_sInter_mem
class CountableInterFilter (l : Filter α) : Prop where
    countable_sInter_mem : ∀ S : Set (Set α), S.Countable → (∀ s ∈ S, s ∈ l) → ⋂₀ S ∈ l
Arguments: {α : Type u_2} {l : Filter α} [self : CountableInterFilter l] (S : Set (Set α))
Goal: S.Countable → (∀ s ∈ S, s ∈ l) → ⋂₀ S ∈ l
-- countable_sInter_mem
theorem countable_sInter_mem {S : Set (Set α)} (hSc : S.Countable) : ⋂₀ S ∈ l ↔ ∀ s ∈ S, s ∈ l 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {S : Set (Set α)} (hSc : S.Countable)
Goal: ⋂₀ S ∈ l ↔ ∀ s ∈ S, s ∈ l
-- countable_iInter_mem
theorem countable_iInter_mem [Countable ι] {s : ι → Set α} : (⋂ i, s i) ∈ l ↔ ∀ i, s i ∈ l 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} [CountableInterFilter l] [Countable ι] {s : ι → Set α}
Goal: ⋂ i, s i ∈ l ↔ ∀ (i : ι), s i ∈ l
-- countable_bInter_mem
theorem countable_bInter_mem {ι : Type*} {S : Set ι} (hS : S.Countable) {s : ∀ i ∈ S, Set α} :
    (⋂ i, ⋂ hi : i ∈ S, s i ‹_›) ∈ l ↔ ∀ i, ∀ hi : i ∈ S, s i ‹_› ∈ l 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {ι : Type u_4} {S : Set ι} (hS : S.Countable) {s : (i : ι) → i ∈ S → Set α}
Goal: ⋂ i, ⋂ (hi : i ∈ S), s i hi ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), s i hi ∈ l
-- eventually_countable_forall
theorem eventually_countable_forall [Countable ι] {p : α → ι → Prop} :
    (∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} [CountableInterFilter l] [Countable ι] {p : α → ι → Prop}
Goal: (∀ᶠ (x : α) in l, ∀ (i : ι), p x i) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p x i
-- eventually_countable_ball
theorem eventually_countable_ball {ι : Type*} {S : Set ι} (hS : S.Countable)
    {p : α → ∀ i ∈ S, Prop} :
    (∀ᶠ x in l, ∀ i hi, p x i hi) ↔ ∀ i hi, ∀ᶠ x in l, p x i hi 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {ι : Type u_4} {S : Set ι} (hS : S.Countable) {p : α → (i : ι) → i ∈ S → Prop}
Goal: (∀ᶠ (x : α) in l, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᶠ (x : α) in l, p x i hi
-- EventuallyLE.countable_iUnion
theorem EventuallyLE.countable_iUnion [Countable ι] {s t : ι → Set α} (h : ∀ i, s i ≤ᶠ[l] t i) :
    ⋃ i, s i ≤ᶠ[l] ⋃ i, t i 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} [CountableInterFilter l] [Countable ι] {s t : ι → Set α} (h : ∀ (i : ι), s i ≤ᶠ[l] t i)
Goal: ⋃ i, s i ≤ᶠ[l] ⋃ i, t i
-- EventuallyEq.countable_iUnion
theorem EventuallyEq.countable_iUnion [Countable ι] {s t : ι → Set α} (h : ∀ i, s i =ᶠ[l] t i) :
    ⋃ i, s i =ᶠ[l] ⋃ i, t i 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} [CountableInterFilter l] [Countable ι] {s t : ι → Set α} (h : ∀ (i : ι), s i =ᶠ[l] t i)
Goal: ⋃ i, s i =ᶠ[l] ⋃ i, t i
-- EventuallyLE.countable_bUnion
theorem EventuallyLE.countable_bUnion {ι : Type*} {S : Set ι} (hS : S.Countable)
    {s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
    ⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_› 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {ι : Type u_4} {S : Set ι} (hS : S.Countable) {s t : (i : ι) → i ∈ S → Set α} (h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi)
Goal: ⋃ i, ⋃ (h : i ∈ S), s i h ≤ᶠ[l] ⋃ i, ⋃ (h : i ∈ S), t i h
-- EventuallyEq.countable_bUnion
theorem EventuallyEq.countable_bUnion {ι : Type*} {S : Set ι} (hS : S.Countable)
    {s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi =ᶠ[l] t i hi) :
    ⋃ i ∈ S, s i ‹_› =ᶠ[l] ⋃ i ∈ S, t i ‹_› 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {ι : Type u_4} {S : Set ι} (hS : S.Countable) {s t : (i : ι) → i ∈ S → Set α} (h : ∀ (i : ι) (hi : i ∈ S), s i hi =ᶠ[l] t i hi)
Goal: ⋃ i, ⋃ (h : i ∈ S), s i h =ᶠ[l] ⋃ i, ⋃ (h : i ∈ S), t i h
-- EventuallyLE.countable_iInter
theorem EventuallyLE.countable_iInter [Countable ι] {s t : ι → Set α} (h : ∀ i, s i ≤ᶠ[l] t i) :
    ⋂ i, s i ≤ᶠ[l] ⋂ i, t i 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} [CountableInterFilter l] [Countable ι] {s t : ι → Set α} (h : ∀ (i : ι), s i ≤ᶠ[l] t i)
Goal: ⋂ i, s i ≤ᶠ[l] ⋂ i, t i
-- EventuallyEq.countable_iInter
theorem EventuallyEq.countable_iInter [Countable ι] {s t : ι → Set α} (h : ∀ i, s i =ᶠ[l] t i) :
    ⋂ i, s i =ᶠ[l] ⋂ i, t i 
Arguments: {ι : Sort u_1} {α : Type u_2} {l : Filter α} [CountableInterFilter l] [Countable ι] {s t : ι → Set α} (h : ∀ (i : ι), s i =ᶠ[l] t i)
Goal: ⋂ i, s i =ᶠ[l] ⋂ i, t i
-- EventuallyLE.countable_bInter
theorem EventuallyLE.countable_bInter {ι : Type*} {S : Set ι} (hS : S.Countable)
    {s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
    ⋂ i ∈ S, s i ‹_› ≤ᶠ[l] ⋂ i ∈ S, t i ‹_› 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {ι : Type u_4} {S : Set ι} (hS : S.Countable) {s t : (i : ι) → i ∈ S → Set α} (h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi)
Goal: ⋂ i, ⋂ (h : i ∈ S), s i h ≤ᶠ[l] ⋂ i, ⋂ (h : i ∈ S), t i h
-- EventuallyEq.countable_bInter
theorem EventuallyEq.countable_bInter {ι : Type*} {S : Set ι} (hS : S.Countable)
    {s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi =ᶠ[l] t i hi) :
    ⋂ i ∈ S, s i ‹_› =ᶠ[l] ⋂ i ∈ S, t i ‹_› 
Arguments: {α : Type u_2} {l : Filter α} [CountableInterFilter l] {ι : Type u_4} {S : Set ι} (hS : S.Countable) {s t : (i : ι) → i ∈ S → Set α} (h : ∀ (i : ι) (hi : i ∈ S), s i hi =ᶠ[l] t i hi)
Goal: ⋂ i, ⋂ (h : i ∈ S), s i h =ᶠ[l] ⋂ i, ⋂ (h : i ∈ S), t i h
-- Filter.mem_ofCountableInter
@[simp]
theorem Filter.mem_ofCountableInter {l : Set (Set α)}
    (hl : ∀ S : Set (Set α), S.Countable → S ⊆ l → ⋂₀ S ∈ l) (h_mono : ∀ s t, s ∈ l → s ⊆ t → t ∈ l)
    {s : Set α} : s ∈ Filter.ofCountableInter l hl h_mono ↔ s ∈ l 
Arguments: {α : Type u_2} {l : Set (Set α)} (hl : ∀ (S : Set (Set α)), S.Countable → S ⊆ l → ⋂₀ S ∈ l) (h_mono : ∀ (s t : Set α), s ∈ l → s ⊆ t → t ∈ l) {s : Set α}
Goal: s ∈ Filter.ofCountableInter l hl h_mono ↔ s ∈ l
-- Filter.mem_ofCountableUnion
@[simp]
theorem Filter.mem_ofCountableUnion {l : Set (Set α)} {hunion hmono s} :
    s ∈ ofCountableUnion l hunion hmono ↔ l sᶜ 
Arguments: {α : Type u_2} {l : Set (Set α)} {hunion : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l} {hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l} {s : Set α}
Goal: s ∈ Filter.ofCountableUnion l hunion hmono ↔ l sᶜ
-- Filter.mem_countableGenerate_iff
theorem mem_countableGenerate_iff {s : Set α} :
    s ∈ countableGenerate g ↔ ∃ S : Set (Set α), S ⊆ g ∧ S.Countable ∧ ⋂₀ S ⊆ s 
Arguments: {α : Type u_2} {g : Set (Set α)} {s : Set α}
Goal: s ∈ Filter.countableGenerate g ↔ ∃ S ⊆ g, S.Countable ∧ ⋂₀ S ⊆ s
-- Filter.le_countableGenerate_iff_of_countableInterFilter
theorem le_countableGenerate_iff_of_countableInterFilter {f : Filter α} [CountableInterFilter f] :
    f ≤ countableGenerate g ↔ g ⊆ f.sets 
Arguments: {α : Type u_2} {g : Set (Set α)} {f : Filter α} [CountableInterFilter f]
Goal: f ≤ Filter.countableGenerate g ↔ g ⊆ f.sets
-- Filter.countableGenerate_isGreatest
theorem countableGenerate_isGreatest :
    IsGreatest { f : Filter α | CountableInterFilter f ∧ g ⊆ f.sets } (countableGenerate g) 
Arguments: {α : Type u_2} (g : Set (Set α))
Goal: IsGreatest {f | CountableInterFilter f ∧ g ⊆ f.sets} (Filter.countableGenerate g)
-- Filter.zero_zeroAtFilter
theorem zero_zeroAtFilter [Zero β] [TopologicalSpace β] (l : Filter α) :
    ZeroAtFilter l (0 : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [TopologicalSpace β] (l : Filter α)
Goal: l.ZeroAtFilter 0
-- Filter.ZeroAtFilter.add
nonrec theorem ZeroAtFilter.add [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β]
    {l : Filter α} {f g : α → β} (hf : ZeroAtFilter l f) (hg : ZeroAtFilter l g) :
    ZeroAtFilter l (f + g) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β] {l : Filter α} {f g : α → β} (hf : l.ZeroAtFilter f) (hg : l.ZeroAtFilter g)
Goal: l.ZeroAtFilter (f + g)
-- Filter.ZeroAtFilter.neg
nonrec theorem ZeroAtFilter.neg [TopologicalSpace β] [AddGroup β] [ContinuousNeg β] {l : Filter α}
    {f : α → β} (hf : ZeroAtFilter l f) : ZeroAtFilter l (-f) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace β] [AddGroup β] [ContinuousNeg β] {l : Filter α} {f : α → β} (hf : l.ZeroAtFilter f)
Goal: l.ZeroAtFilter (-f)