import Spa.Lattice.Prod
import Spa.Lattice.Unit

namespace Spa

universe u

def IterProd (A B : Type u) : ℕ → Type u
  | 0 => B
  | k + 1 => A × IterProd A B k

namespace IterProd

variable {A B : Type u}

instance instLattice [Lattice A] [Lattice B] :
    ∀ k, Lattice (IterProd A B k)
  | 0 => inferInstanceAs (Lattice B)
  | k + 1 => @Prod.instLattice A (IterProd A B k) _ (instLattice k)

instance instDecidableEq [DecidableEq A] [DecidableEq B] :
    ∀ k, DecidableEq (IterProd A B k)
  | 0 => inferInstanceAs (DecidableEq B)
  | k + 1 => @instDecidableEqProd A (IterProd A B k) _ (instDecidableEq k)

def build (a : A) (b : B) : (k : ℕ) → IterProd A B k
  | 0 => b
  | k + 1 => (a, build a b k)

variable [Lattice A] [Lattice B]

def fixedHeight [FiniteHeightLattice A] [FiniteHeightLattice B] :
    ∀ k, FiniteHeightLattice (IterProd A B k)
  | 0 => inferInstanceAs (FiniteHeightLattice B)
  | k + 1 => @Spa.prod A (IterProd A B k) _ (instLattice k) _ (fixedHeight k)

instance instFiniteHeight [FiniteHeightLattice A] [FiniteHeightLattice B] (k : ℕ) :
    FiniteHeightLattice (IterProd A B k) := fixedHeight k

theorem bot_fixedHeight [FiniteHeightLattice A] [FiniteHeightLattice B] :
    ∀ k, (fixedHeight (A := A) (B := B) k).bot = build (⊥ : A) (⊥ : B) k
  | 0 => rfl
  | k + 1 => by
    show ((⊥ : A), (fixedHeight (A := A) (B := B) k).bot)
        = ((⊥ : A), build (⊥ : A) (⊥ : B) k)
    rw [bot_fixedHeight k]

end IterProd

end Spa