Boolean equality

ink

class PartialEquivBEq (α : Type u_1) [inst : BEq α] : Prop

• Symmetry for BEq. If a == b then b == a.

  symm : ∀ {a b : α}, (a == b) = true → (b == a) = true

• Transitivity for BEq. If a == b and b == c then a == c.

  trans : ∀ {a b c : α}, (a == b) = true → (b == c) = true → (a == c) = true

PartialEquivBEq α says that the BEq implementation is a partial equivalence relation, that is:

• it is symmetric: a == b → b == a→ b == a
• it is transitive: a == b → b == c → a == c→ b == c → a == c→ a == c.

ink

@[simp]

theorem beq_eq_false_iff_ne {α : Type u_1} [inst : BEq α] [inst : LawfulBEq α] (a : α) (b : α) : (a == b) = false ↔ a ≠ b