ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isCoatomic_dual_iff_isAtomic
@[simp]
theorem isCoatomic_dual_iff_isAtomic [OrderBot α] : IsCoatomic αᵒᵈ ↔ IsAtomic α 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α]
Goal: IsCoatomic αᵒᵈ ↔ IsAtomic α
-- isAtomic_dual_iff_isCoatomic
@[simp]
theorem isAtomic_dual_iff_isCoatomic [OrderTop α] : IsAtomic αᵒᵈ ↔ IsCoatomic α 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α]
Goal: IsAtomic αᵒᵈ ↔ IsCoatomic α
-- isAtomic_iff_forall_isAtomic_Iic
theorem isAtomic_iff_forall_isAtomic_Iic [OrderBot α] :
    IsAtomic α ↔ ∀ x : α, IsAtomic (Set.Iic x) 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α]
Goal: IsAtomic α ↔ ∀ (x : α), IsAtomic ↑(Set.Iic x)
-- isCoatomic_iff_forall_isCoatomic_Ici
theorem isCoatomic_iff_forall_isCoatomic_Ici [OrderTop α] :
    IsCoatomic α ↔ ∀ x : α, IsCoatomic (Set.Ici x) 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α]
Goal: IsCoatomic α ↔ ∀ (x : α), IsCoatomic ↑(Set.Ici x)
-- IsStronglyAtomic.exists_covBy_le_of_lt
@[mk_iff]
class IsStronglyAtomic (α : Type*) [Preorder α] : Prop where
  exists_covBy_le_of_lt : ∀ (a b : α), a < b → ∃ x, a ⋖ x ∧ x ≤ b
Arguments: {α : Type u_5} [Preorder α] [self : IsStronglyAtomic α] (a b : α)
Goal: a < b → ∃ x, a ⋖ x ∧ x ≤ b
-- isStronglyAtomic_iff
@[mk_iff]
class IsStronglyAtomic (α : Type*) [Preorder α] : Prop where
  exists_covBy_le_of_lt : ∀ (a b : α), a < b → ∃ x, a ⋖ x ∧ x ≤ b
Arguments: (α : Type u_5) [Preorder α]
Goal: IsStronglyAtomic α ↔ ∀ (a b : α), a < b → ∃ x, a ⋖ x ∧ x ≤ b
-- exists_covBy_le_of_lt
theorem exists_covBy_le_of_lt [IsStronglyAtomic α] (h : a < b) : ∃ x, a ⋖ x ∧ x ≤ b 
Arguments: {α : Type u_4} {a b : α} [Preorder α] [IsStronglyAtomic α] (h : a < b)
Goal: ∃ x, a ⋖ x ∧ x ≤ b
-- isStronglyCoatomic_iff
@[mk_iff]
class IsStronglyCoatomic (α : Type*) [Preorder α] : Prop where
  (exists_le_covBy_of_lt : ∀ (a b : α), a < b → ∃ x, a ≤ x ∧ x ⋖ b)
Arguments: (α : Type u_5) [Preorder α]
Goal: IsStronglyCoatomic α ↔ ∀ (a b : α), a < b → ∃ x, a ≤ x ∧ x ⋖ b
-- IsStronglyCoatomic.exists_le_covBy_of_lt
@[mk_iff]
class IsStronglyCoatomic (α : Type*) [Preorder α] : Prop where
  (exists_le_covBy_of_lt : ∀ (a b : α), a < b → ∃ x, a ≤ x ∧ x ⋖ b)
Arguments: {α : Type u_5} [Preorder α] [self : IsStronglyCoatomic α] (a b : α)
Goal: a < b → ∃ x, a ≤ x ∧ x ⋖ b
-- exists_le_covBy_of_lt
theorem exists_le_covBy_of_lt [IsStronglyCoatomic α] (h : a < b) : ∃ x, a ≤ x ∧ x ⋖ b 
Arguments: {α : Type u_4} {a b : α} [Preorder α] [IsStronglyCoatomic α] (h : a < b)
Goal: ∃ x, a ≤ x ∧ x ⋖ b
-- isStronglyAtomic_dual_iff_is_stronglyCoatomic
theorem isStronglyAtomic_dual_iff_is_stronglyCoatomic :
    IsStronglyAtomic αᵒᵈ ↔ IsStronglyCoatomic α 
Arguments: {α : Type u_4} [Preorder α]
Goal: IsStronglyAtomic αᵒᵈ ↔ IsStronglyCoatomic α
-- isStronglyCoatomic_dual_iff_is_stronglyAtomic
@[simp] theorem isStronglyCoatomic_dual_iff_is_stronglyAtomic :
    IsStronglyCoatomic αᵒᵈ ↔ IsStronglyAtomic α 
Arguments: {α : Type u_4} [Preorder α]
Goal: IsStronglyCoatomic αᵒᵈ ↔ IsStronglyAtomic α
-- Set.OrdConnected.isStronglyAtomic
theorem Set.OrdConnected.isStronglyAtomic [IsStronglyAtomic α] {s : Set α}
    (h : Set.OrdConnected s) : IsStronglyAtomic s 
Arguments: {α : Type u_4} [Preorder α] [IsStronglyAtomic α] {s : Set α} (h : s.OrdConnected)
Goal: IsStronglyAtomic ↑s
-- Set.OrdConnected.isStronglyCoatomic
theorem Set.OrdConnected.isStronglyCoatomic [IsStronglyCoatomic α] {s : Set α}
    (h : Set.OrdConnected s) : IsStronglyCoatomic s 
Arguments: {α : Type u_4} [Preorder α] [IsStronglyCoatomic α] {s : Set α} (h : s.OrdConnected)
Goal: IsStronglyCoatomic ↑s
-- IsStronglyAtomic.of_wellFounded_lt
theorem IsStronglyAtomic.of_wellFounded_lt (h : WellFounded ((· < ·) : α → α → Prop)) :
    IsStronglyAtomic α 
Arguments: {α : Type u_2} [PartialOrder α] (h : WellFounded fun x x_1 => x < x_1)
Goal: IsStronglyAtomic α
-- IsStronglyCoatomic.of_wellFounded_gt
theorem IsStronglyCoatomic.of_wellFounded_gt (h : WellFounded ((· > ·) : α → α → Prop)) :
    IsStronglyCoatomic α 
Arguments: {α : Type u_2} [PartialOrder α] (h : WellFounded fun x x_1 => x > x_1)
Goal: IsStronglyCoatomic α
-- isAtomic_of_orderBot_wellFounded_lt
theorem isAtomic_of_orderBot_wellFounded_lt [OrderBot α]
    (h : WellFounded ((· < ·) : α → α → Prop)) : IsAtomic α 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] (h : WellFounded fun x x_1 => x < x_1)
Goal: IsAtomic α
-- isCoatomic_of_orderTop_gt_wellFounded
theorem isCoatomic_of_orderTop_gt_wellFounded [OrderTop α]
    (h : WellFounded ((· > ·) : α → α → Prop)) : IsCoatomic α 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] (h : WellFounded fun x x_1 => x > x_1)
Goal: IsCoatomic α
-- BooleanAlgebra.le_iff_atom_le_imp
theorem le_iff_atom_le_imp {α} [BooleanAlgebra α] [IsAtomic α] {x y : α} :
    x ≤ y ↔ ∀ a, IsAtom a → a ≤ x → a ≤ y 
Arguments: {α : Type u_4} [BooleanAlgebra α] [IsAtomic α] {x y : α}
Goal: x ≤ y ↔ ∀ (a : α), IsAtom a → a ≤ x → a ≤ y
-- BooleanAlgebra.eq_iff_atom_le_iff
theorem eq_iff_atom_le_iff {α} [BooleanAlgebra α] [IsAtomic α] {x y : α} :
    x = y ↔ ∀ a, IsAtom a → (a ≤ x ↔ a ≤ y) 
Arguments: {α : Type u_4} [BooleanAlgebra α] [IsAtomic α] {x y : α}
Goal: x = y ↔ ∀ (a : α), IsAtom a → (a ≤ x ↔ a ≤ y)
-- IsAtomistic.eq_sSup_atoms
class IsAtomistic : Prop where
    eq_sSup_atoms : ∀ b : α, ∃ s : Set α, b = sSup s ∧ ∀ a, a ∈ s → IsAtom a
Arguments: {α : Type u_2} [CompleteLattice α] [self : IsAtomistic α] (b : α)
Goal: ∃ s, b = sSup s ∧ ∀ a ∈ s, IsAtom a
-- IsCoatomistic.eq_sInf_coatoms
class IsCoatomistic : Prop where
    eq_sInf_coatoms : ∀ b : α, ∃ s : Set α, b = sInf s ∧ ∀ a, a ∈ s → IsCoatom a
Arguments: {α : Type u_2} [CompleteLattice α] [self : IsCoatomistic α] (b : α)
Goal: ∃ s, b = sInf s ∧ ∀ a ∈ s, IsCoatom a
-- isCoatomistic_dual_iff_isAtomistic
@[simp]
theorem isCoatomistic_dual_iff_isAtomistic : IsCoatomistic αᵒᵈ ↔ IsAtomistic α 
Arguments: {α : Type u_2} [CompleteLattice α]
Goal: IsCoatomistic αᵒᵈ ↔ IsAtomistic α
-- isAtomistic_dual_iff_isCoatomistic
@[simp]
theorem isAtomistic_dual_iff_isCoatomistic : IsAtomistic αᵒᵈ ↔ IsCoatomistic α 
Arguments: {α : Type u_2} [CompleteLattice α]
Goal: IsAtomistic αᵒᵈ ↔ IsCoatomistic α
-- sSup_atoms_le_eq
@[simp]
theorem sSup_atoms_le_eq (b : α) : sSup { a : α | IsAtom a ∧ a ≤ b } = b 
Arguments: {α : Type u_2} [CompleteLattice α] [IsAtomistic α] (b : α)
Goal: sSup {a | IsAtom a ∧ a ≤ b} = b
-- sSup_atoms_eq_top
@[simp]
theorem sSup_atoms_eq_top : sSup { a : α | IsAtom a } = ⊤ 
Arguments: {α : Type u_2} [CompleteLattice α] [IsAtomistic α]
Goal: sSup {a | IsAtom a} = ⊤
-- le_iff_atom_le_imp
theorem le_iff_atom_le_imp {a b : α} : a ≤ b ↔ ∀ c : α, IsAtom c → c ≤ a → c ≤ b 
Arguments: {α : Type u_2} [CompleteLattice α] [IsAtomistic α] {a b : α}
Goal: a ≤ b ↔ ∀ (c : α), IsAtom c → c ≤ a → c ≤ b
-- eq_iff_atom_le_iff
theorem eq_iff_atom_le_iff {a b : α} : a = b ↔ ∀ c, IsAtom c → (c ≤ a ↔ c ≤ b) 
Arguments: {α : Type u_2} [CompleteLattice α] [IsAtomistic α] {a b : α}
Goal: a = b ↔ ∀ (c : α), IsAtom c → (c ≤ a ↔ c ≤ b)
-- IsSimpleOrder.eq_bot_or_eq_top
class IsSimpleOrder (α : Type*) [LE α] [BoundedOrder α] extends Nontrivial α : Prop where
    eq_bot_or_eq_top : ∀ a : α, a = ⊥ ∨ a = ⊤
Arguments: {α : Type u_4} [LE α] [BoundedOrder α] [self : IsSimpleOrder α] (a : α)
Goal: a = ⊥ ∨ a = ⊤
-- isSimpleOrder_iff_isSimpleOrder_orderDual
theorem isSimpleOrder_iff_isSimpleOrder_orderDual [LE α] [BoundedOrder α] :
    IsSimpleOrder α ↔ IsSimpleOrder αᵒᵈ 
Arguments: {α : Type u_2} [LE α] [BoundedOrder α]
Goal: IsSimpleOrder α ↔ IsSimpleOrder αᵒᵈ
-- IsSimpleOrder.bot_ne_top
theorem IsSimpleOrder.bot_ne_top [LE α] [BoundedOrder α] [IsSimpleOrder α] : (⊥ : α) ≠ (⊤ : α) 
Arguments: {α : Type u_2} [LE α] [BoundedOrder α] [IsSimpleOrder α]
Goal: ⊥ ≠ ⊤
-- isAtom_top
@[simp]
theorem isAtom_top : IsAtom (⊤ : α) 
Arguments: {α : Type u_2} [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α]
Goal: IsAtom ⊤
-- isCoatom_bot
@[simp]
theorem isCoatom_bot : IsCoatom (⊥ : α) 
Arguments: {α : Type u_2} [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α]
Goal: IsCoatom ⊥
-- bot_covBy_top
theorem bot_covBy_top : (⊥ : α) ⋖ ⊤ 
Arguments: {α : Type u_2} [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α]
Goal: ⊥ ⋖ ⊤
-- IsSimpleOrder.eq_bot_of_lt
theorem eq_bot_of_lt : a = ⊥ 
Arguments: {α : Type u_2} [Preorder α] [BoundedOrder α] [IsSimpleOrder α] {a b : α} (h : a < b)
Goal: a = ⊥
-- IsSimpleOrder.eq_top_of_lt
theorem eq_top_of_lt : b = ⊤ 
Arguments: {α : Type u_2} [Preorder α] [BoundedOrder α] [IsSimpleOrder α] {a b : α} (h : a < b)
Goal: b = ⊤
-- IsSimpleOrder.equivBool_symm_apply
@[simps]
def equivBool {α} [DecidableEq α] [LE α] [BoundedOrder α] [IsSimpleOrder α] : α ≃ Bool 
Arguments: {α : Type u_4} [DecidableEq α] [LE α] [BoundedOrder α] [IsSimpleOrder α] (x : Bool)
Goal: IsSimpleOrder.equivBool.symm x = Bool.casesOn x ⊥ ⊤
-- IsSimpleOrder.equivBool_apply
@[simps]
def equivBool {α} [DecidableEq α] [LE α] [BoundedOrder α] [IsSimpleOrder α] : α ≃ Bool 
Arguments: {α : Type u_4} [DecidableEq α] [LE α] [BoundedOrder α] [IsSimpleOrder α] (x : α)
Goal: IsSimpleOrder.equivBool x = decide (x = ⊤)
-- isSimpleOrder_iff_isAtom_top
theorem isSimpleOrder_iff_isAtom_top [PartialOrder α] [BoundedOrder α] :
    IsSimpleOrder α ↔ IsAtom (⊤ : α) 
Arguments: {α : Type u_2} [PartialOrder α] [BoundedOrder α]
Goal: IsSimpleOrder α ↔ IsAtom ⊤
-- isSimpleOrder_iff_isCoatom_bot
theorem isSimpleOrder_iff_isCoatom_bot [PartialOrder α] [BoundedOrder α] :
    IsSimpleOrder α ↔ IsCoatom (⊥ : α) 
Arguments: {α : Type u_2} [PartialOrder α] [BoundedOrder α]
Goal: IsSimpleOrder α ↔ IsCoatom ⊥
-- Set.isSimpleOrder_Iic_iff_isAtom
theorem isSimpleOrder_Iic_iff_isAtom [PartialOrder α] [OrderBot α] {a : α} :
    IsSimpleOrder (Iic a) ↔ IsAtom a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderBot α] {a : α}
Goal: IsSimpleOrder ↑(Set.Iic a) ↔ IsAtom a
-- Set.isSimpleOrder_Ici_iff_isCoatom
theorem isSimpleOrder_Ici_iff_isCoatom [PartialOrder α] [OrderTop α] {a : α} :
    IsSimpleOrder (Ici a) ↔ IsCoatom a 
Arguments: {α : Type u_2} [PartialOrder α] [OrderTop α] {a : α}
Goal: IsSimpleOrder ↑(Set.Ici a) ↔ IsCoatom a
-- OrderEmbedding.isAtom_of_map_bot_of_image
theorem isAtom_of_map_bot_of_image [OrderBot α] [OrderBot β] (f : β ↪o α) (hbot : f ⊥ = ⊥) {b : β}
    (hb : IsAtom (f b)) : IsAtom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] (f : β ↪o α) (hbot : f ⊥ = ⊥) {b : β} (hb : IsAtom (f b))
Goal: IsAtom b
-- OrderEmbedding.isCoatom_of_map_top_of_image
theorem isCoatom_of_map_top_of_image [OrderTop α] [OrderTop β] (f : β ↪o α) (htop : f ⊤ = ⊤)
    {b : β} (hb : IsCoatom (f b)) : IsCoatom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] (f : β ↪o α) (htop : f ⊤ = ⊤) {b : β} (hb : IsCoatom (f b))
Goal: IsCoatom b
-- GaloisInsertion.isAtom_of_u_bot
theorem isAtom_of_u_bot [OrderBot α] [OrderBot β] {l : α → β} {u : β → α}
    (gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) {b : β} (hb : IsAtom (u b)) : IsAtom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] {l : α → β} {u : β → α} (gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) {b : β} (hb : IsAtom (u b))
Goal: IsAtom b
-- GaloisInsertion.isAtom_iff
theorem isAtom_iff [OrderBot α] [IsAtomic α] [OrderBot β] {l : α → β} {u : β → α}
    (gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) (h_atom : ∀ a, IsAtom a → u (l a) = a) (a : α) :
    IsAtom (l a) ↔ IsAtom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [IsAtomic α] [OrderBot β] {l : α → β} {u : β → α} (gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) (h_atom : ∀ (a : α), IsAtom a → u (l a) = a) (a : α)
Goal: IsAtom (l a) ↔ IsAtom a
-- GaloisInsertion.isAtom_iff'
theorem isAtom_iff' [OrderBot α] [IsAtomic α] [OrderBot β] {l : α → β} {u : β → α}
    (gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) (h_atom : ∀ a, IsAtom a → u (l a) = a) (b : β) :
    IsAtom (u b) ↔ IsAtom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [IsAtomic α] [OrderBot β] {l : α → β} {u : β → α} (gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) (h_atom : ∀ (a : α), IsAtom a → u (l a) = a) (b : β)
Goal: IsAtom (u b) ↔ IsAtom b
-- GaloisInsertion.isCoatom_of_image
theorem isCoatom_of_image [OrderTop α] [OrderTop β] {l : α → β} {u : β → α}
    (gi : GaloisInsertion l u) {b : β} (hb : IsCoatom (u b)) : IsCoatom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] {l : α → β} {u : β → α} (gi : GaloisInsertion l u) {b : β} (hb : IsCoatom (u b))
Goal: IsCoatom b
-- GaloisInsertion.isCoatom_iff
theorem isCoatom_iff [OrderTop α] [IsCoatomic α] [OrderTop β] {l : α → β} {u : β → α}
    (gi : GaloisInsertion l u) (h_coatom : ∀ a : α, IsCoatom a → u (l a) = a) (b : β) :
    IsCoatom (u b) ↔ IsCoatom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [IsCoatomic α] [OrderTop β] {l : α → β} {u : β → α} (gi : GaloisInsertion l u) (h_coatom : ∀ (a : α), IsCoatom a → u (l a) = a) (b : β)
Goal: IsCoatom (u b) ↔ IsCoatom b
-- GaloisCoinsertion.isCoatom_of_l_top
theorem isCoatom_of_l_top [OrderTop α] [OrderTop β] {l : α → β} {u : β → α}
    (gi : GaloisCoinsertion l u) (hbot : l ⊤ = ⊤) {a : α} (hb : IsCoatom (l a)) : IsCoatom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) (hbot : l ⊤ = ⊤) {a : α} (hb : IsCoatom (l a))
Goal: IsCoatom a
-- GaloisCoinsertion.isCoatom_iff
theorem isCoatom_iff [OrderTop α] [OrderTop β] [IsCoatomic β] {l : α → β} {u : β → α}
    (gi : GaloisCoinsertion l u) (htop : l ⊤ = ⊤) (h_coatom : ∀ b, IsCoatom b → l (u b) = b)
    (b : β) : IsCoatom (u b) ↔ IsCoatom b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] [IsCoatomic β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) (htop : l ⊤ = ⊤) (h_coatom : ∀ (b : β), IsCoatom b → l (u b) = b) (b : β)
Goal: IsCoatom (u b) ↔ IsCoatom b
-- GaloisCoinsertion.isCoatom_iff'
theorem isCoatom_iff' [OrderTop α] [OrderTop β] [IsCoatomic β] {l : α → β} {u : β → α}
    (gi : GaloisCoinsertion l u) (htop : l ⊤ = ⊤) (h_coatom : ∀ b, IsCoatom b → l (u b) = b)
    (a : α) : IsCoatom (l a) ↔ IsCoatom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] [IsCoatomic β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) (htop : l ⊤ = ⊤) (h_coatom : ∀ (b : β), IsCoatom b → l (u b) = b) (a : α)
Goal: IsCoatom (l a) ↔ IsCoatom a
-- GaloisCoinsertion.isAtom_of_image
theorem isAtom_of_image [OrderBot α] [OrderBot β] {l : α → β} {u : β → α}
    (gi : GaloisCoinsertion l u) {a : α} (hb : IsAtom (l a)) : IsAtom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) {a : α} (hb : IsAtom (l a))
Goal: IsAtom a
-- GaloisCoinsertion.isAtom_iff
theorem isAtom_iff [OrderBot α] [OrderBot β] [IsAtomic β] {l : α → β} {u : β → α}
    (gi : GaloisCoinsertion l u) (h_atom : ∀ b, IsAtom b → l (u b) = b) (a : α) :
    IsAtom (l a) ↔ IsAtom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] [IsAtomic β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) (h_atom : ∀ (b : β), IsAtom b → l (u b) = b) (a : α)
Goal: IsAtom (l a) ↔ IsAtom a
-- OrderIso.isAtom_iff
@[simp]
theorem isAtom_iff [OrderBot α] [OrderBot β] (f : α ≃o β) (a : α) : IsAtom (f a) ↔ IsAtom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] (f : α ≃o β) (a : α)
Goal: IsAtom (f a) ↔ IsAtom a
-- OrderIso.isCoatom_iff
@[simp]
theorem isCoatom_iff [OrderTop α] [OrderTop β] (f : α ≃o β) (a : α) :
    IsCoatom (f a) ↔ IsCoatom a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] (f : α ≃o β) (a : α)
Goal: IsCoatom (f a) ↔ IsCoatom a
-- OrderIso.isSimpleOrder_iff
theorem isSimpleOrder_iff [BoundedOrder α] [BoundedOrder β] (f : α ≃o β) :
    IsSimpleOrder α ↔ IsSimpleOrder β 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [BoundedOrder α] [BoundedOrder β] (f : α ≃o β)
Goal: IsSimpleOrder α ↔ IsSimpleOrder β
-- OrderIso.isSimpleOrder
theorem isSimpleOrder [BoundedOrder α] [BoundedOrder β] [h : IsSimpleOrder β] (f : α ≃o β) :
    IsSimpleOrder α 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [BoundedOrder α] [BoundedOrder β] [h : IsSimpleOrder β] (f : α ≃o β)
Goal: IsSimpleOrder α
-- OrderIso.isAtomic_iff
protected theorem isAtomic_iff [OrderBot α] [OrderBot β] (f : α ≃o β) :
    IsAtomic α ↔ IsAtomic β 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] (f : α ≃o β)
Goal: IsAtomic α ↔ IsAtomic β
-- OrderIso.isCoatomic_iff
protected theorem isCoatomic_iff [OrderTop α] [OrderTop β] (f : α ≃o β) :
    IsCoatomic α ↔ IsCoatomic β 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] (f : α ≃o β)
Goal: IsCoatomic α ↔ IsCoatomic β
-- CompleteLattice.isStronglyAtomic
theorem CompleteLattice.isStronglyAtomic [IsUpperModularLattice α] [IsAtomistic α] :
    IsStronglyAtomic α 
Arguments: {α : Type u_2} [CompleteLattice α] [IsUpperModularLattice α] [IsAtomistic α]
Goal: IsStronglyAtomic α
-- CompleteLattice.isStronglyCoatomic
theorem CompleteLattice.isStronglyCoatomic [IsLowerModularLattice α] [IsCoatomistic α] :
    IsStronglyCoatomic α 
Arguments: {α : Type u_2} [CompleteLattice α] [IsLowerModularLattice α] [IsCoatomistic α]
Goal: IsStronglyCoatomic α
-- IsCompl.isAtom_iff_isCoatom
theorem isAtom_iff_isCoatom : IsAtom a ↔ IsCoatom b 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] {a b : α} (hc : IsCompl a b)
Goal: IsAtom a ↔ IsCoatom b
-- IsCompl.isCoatom_iff_isAtom
theorem isCoatom_iff_isAtom : IsCoatom a ↔ IsAtom b 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] {a b : α} (hc : IsCompl a b)
Goal: IsCoatom a ↔ IsAtom b
-- isCoatomic_of_isAtomic_of_complementedLattice_of_isModular
theorem isCoatomic_of_isAtomic_of_complementedLattice_of_isModular [IsAtomic α] :
    IsCoatomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α] [IsAtomic α]
Goal: IsCoatomic α
-- isAtomic_of_isCoatomic_of_complementedLattice_of_isModular
theorem isAtomic_of_isCoatomic_of_complementedLattice_of_isModular [IsCoatomic α] :
    IsAtomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α] [IsCoatomic α]
Goal: IsAtomic α
-- isAtomic_iff_isCoatomic
theorem isAtomic_iff_isCoatomic : IsAtomic α ↔ IsCoatomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α]
Goal: IsAtomic α ↔ IsCoatomic α
-- ComplementedLattice.isStronglyAtomic
theorem ComplementedLattice.isStronglyAtomic [IsAtomic α] : IsStronglyAtomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α] [IsAtomic α]
Goal: IsStronglyAtomic α
-- ComplementedLattice.isStronglyCoatomic
theorem ComplementedLattice.isStronglyCoatomic [IsCoatomic α] : IsStronglyCoatomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α] [IsCoatomic α]
Goal: IsStronglyCoatomic α
-- ComplementedLattice.isStronglyAtomic'
theorem ComplementedLattice.isStronglyAtomic' [h : IsAtomic α] : IsStronglyCoatomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α] [h : IsAtomic α]
Goal: IsStronglyCoatomic α
-- ComplementedLattice.isStronglyCoatomic'
theorem ComplementedLattice.isStronglyCoatomic' [h : IsCoatomic α] : IsStronglyAtomic α 
Arguments: {α : Type u_2} [Lattice α] [BoundedOrder α] [IsModularLattice α] [ComplementedLattice α] [h : IsCoatomic α]
Goal: IsStronglyAtomic α
-- Prop.isAtom_iff
@[simp] theorem isAtom_iff {p : Prop} : IsAtom p ↔ p 
Arguments: {p : Prop}
Goal: IsAtom p ↔ p
-- Prop.isCoatom_iff
@[simp] theorem isCoatom_iff {p : Prop} : IsCoatom p ↔ ¬ p 
Arguments: {p : Prop}
Goal: IsCoatom p ↔ ¬p
-- Pi.eq_bot_iff
protected theorem eq_bot_iff [∀ i, Bot (π i)] {f : ∀ i, π i} : f = ⊥ ↔ ∀ i, f i = ⊥ 
Arguments: {ι : Type u_4} {π : ι → Type u} [(i : ι) → Bot (π i)] {f : (i : ι) → π i}
Goal: f = ⊥ ↔ ∀ (i : ι), f i = ⊥
-- Pi.isAtom_iff
theorem isAtom_iff {f : ∀ i, π i} [∀ i, PartialOrder (π i)] [∀ i, OrderBot (π i)] :
    IsAtom f ↔ ∃ i, IsAtom (f i) ∧ ∀ j, j ≠ i → f j = ⊥ 
Arguments: {ι : Type u_4} {π : ι → Type u} {f : (i : ι) → π i} [(i : ι) → PartialOrder (π i)] [(i : ι) → OrderBot (π i)]
Goal: IsAtom f ↔ ∃ i, IsAtom (f i) ∧ ∀ (j : ι), j ≠ i → f j = ⊥
-- Pi.isAtom_single
theorem isAtom_single {i : ι} [DecidableEq ι] [∀ i, PartialOrder (π i)] [∀ i, OrderBot (π i)]
    {a : π i} (h : IsAtom a) : IsAtom (Function.update (⊥ : ∀ i, π i) i a) 
Arguments: {ι : Type u_4} {π : ι → Type u} {i : ι} [DecidableEq ι] [(i : ι) → PartialOrder (π i)] [(i : ι) → OrderBot (π i)] {a : π i} (h : IsAtom a)
Goal: IsAtom (Function.update ⊥ i a)
-- Pi.isAtom_iff_eq_single
theorem isAtom_iff_eq_single [DecidableEq ι] [∀ i, PartialOrder (π i)]
    [∀ i, OrderBot (π i)] {f : ∀ i, π i} :
    IsAtom f ↔ ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a 
Arguments: {ι : Type u_4} {π : ι → Type u} [DecidableEq ι] [(i : ι) → PartialOrder (π i)] [(i : ι) → OrderBot (π i)] {f : (i : ι) → π i}
Goal: IsAtom f ↔ ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a
-- isAtom_compl
@[simp] lemma isAtom_compl : IsAtom aᶜ ↔ IsCoatom a 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a : α}
Goal: IsAtom aᶜ ↔ IsCoatom a
-- isCoatom_compl
@[simp] lemma isCoatom_compl : IsCoatom aᶜ ↔ IsAtom a 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a : α}
Goal: IsCoatom aᶜ ↔ IsAtom a
-- Set.isAtom_singleton
theorem isAtom_singleton (x : α) : IsAtom ({x} : Set α) 
Arguments: {α : Type u_2} (x : α)
Goal: IsAtom {x}
-- Set.isAtom_iff
theorem isAtom_iff {s : Set α} : IsAtom s ↔ ∃ x, s = {x} 
Arguments: {α : Type u_2} {s : Set α}
Goal: IsAtom s ↔ ∃ x, s = {x}
-- Set.isCoatom_iff
theorem isCoatom_iff (s : Set α) : IsCoatom s ↔ ∃ x, s = {x}ᶜ 
Arguments: {α : Type u_2} (s : Set α)
Goal: IsCoatom s ↔ ∃ x, s = {x}ᶜ
-- Set.isCoatom_singleton_compl
theorem isCoatom_singleton_compl (x : α) : IsCoatom ({x}ᶜ : Set α) 
Arguments: {α : Type u_2} (x : α)
Goal: IsCoatom {x}ᶜ
-- Monovary.monovaryOn
protected theorem Monovary.monovaryOn (h : Monovary f g) (s : Set ι) : MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} (h : Monovary f g) (s : Set ι)
Goal: MonovaryOn f g s
-- Antivary.antivaryOn
protected theorem Antivary.antivaryOn (h : Antivary f g) (s : Set ι) : AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} (h : Antivary f g) (s : Set ι)
Goal: AntivaryOn f g s
-- MonovaryOn.empty
@[simp]
theorem MonovaryOn.empty : MonovaryOn f g ∅ 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: MonovaryOn f g ∅
-- AntivaryOn.empty
@[simp]
theorem AntivaryOn.empty : AntivaryOn f g ∅ 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: AntivaryOn f g ∅
-- monovaryOn_univ
@[simp]
theorem monovaryOn_univ : MonovaryOn f g univ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: MonovaryOn f g Set.univ ↔ Monovary f g
-- antivaryOn_univ
@[simp]
theorem antivaryOn_univ : AntivaryOn f g univ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: AntivaryOn f g Set.univ ↔ Antivary f g
-- MonovaryOn.subset
protected theorem MonovaryOn.subset (hst : s ⊆ t) (h : MonovaryOn f g t) : MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s t : Set ι} (hst : s ⊆ t) (h : MonovaryOn f g t)
Goal: MonovaryOn f g s