open import Lattice

module Lattice.Bundles.IterProd {a} (A B : Set a) where
open import Data.Nat using (ℕ)

module _ (lA : Lattice A) (lB : Lattice B) where
    open Lattice.Lattice lA using () renaming
        ( _≈_ to _≈₁_; _⊔_ to _⊔₁_; _⊓_ to _⊓₁_
        ; isLattice to isLattice₁
        )

    open Lattice.Lattice lB using () renaming
        ( _≈_ to _≈₂_; _⊔_ to _⊔₂_; _⊓_ to _⊓₂_
        ; isLattice to isLattice₂
        )

    module _ (k : ℕ) where
        open import Lattice.IterProd _≈₁_ _≈₂_ _⊔₁_ _⊔₂_ _⊓₁_ _⊓₂_ isLattice₁ isLattice₂ using (lattice) public

module _ (fhA : FiniteHeightLattice A) (fhB : FiniteHeightLattice B) where
    open Lattice.FiniteHeightLattice fhA using () renaming
        ( _≈_ to _≈₁_; _⊔_ to _⊔₁_; _⊓_ to _⊓₁_
        ; height to height₁
        ; isLattice to isLattice₁
        ; fixedHeight to fixedHeight₁
        )

    open Lattice.FiniteHeightLattice fhB using () renaming
        ( _≈_ to _≈₂_; _⊔_ to _⊔₂_; _⊓_ to _⊓₂_
        ; height to height₂
        ; isLattice to isLattice₂
        ; fixedHeight to fixedHeight₂
        )

    module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_) (k : ℕ) where
        import Lattice.IterProd _≈₁_ _≈₂_ _⊔₁_ _⊔₂_ _⊓₁_ _⊓₂_ isLattice₁ isLattice₂ as IP

        finiteHeightLattice = IP.finiteHeightLattice k ≈₁-dec ≈₂-dec height₁ height₂ fixedHeight₁ fixedHeight₂