English
For a ∈ α × β with HasCompl on both components, the first projection of the complemented pair equals the complement of the first projection: (aᶜ)₁ = a₁ᶜ.
Русский
Для a ∈ α × β с комплементами на компоненте, первая проекция дополненного пары равна комплементу первой проекции: (aᶜ)₁ = a₁ᶜ.
LaTeX
$$$ a^\complement_1 = a_1^\complement $$$
Lean4
/-- Construct a Heyting algebra from the lattice structure and Heyting implication alone. -/
abbrev ofHImp [DistribLattice α] [BoundedOrder α] (himp : α → α → α) (le_himp_iff : ∀ a b c, a ≤ himp b c ↔ a ⊓ b ≤ c) :
HeytingAlgebra α :=
{ ‹DistribLattice α›, ‹BoundedOrder α› with himp, compl := fun a => himp a ⊥, le_himp_iff, himp_bot := fun _ => rfl }
-- See note [reducible non-instances]
