57 lines
1.7 KiB
Agda
57 lines
1.7 KiB
Agda
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open import Lattice
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module Lattice.IterProd {a} {A B : Set a}
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(_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
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(_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
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(_⊓₁_ : A → A → A) (_⊓₂_ : B → B → B)
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(lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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open import Data.Nat using (ℕ; suc)
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open import Data.Product using (_×_)
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open import Utils using (iterate)
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IterProd : ℕ → Set a
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IterProd k = iterate k (λ t → A × t) B
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-- To make iteration more convenient, package the definitions in Lattice
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-- records, perform the recursion, and unpackage.
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private module _ where
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ALattice : Lattice A
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ALattice = record
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{ _≈_ = _≈₁_
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; _⊔_ = _⊔₁_
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; _⊓_ = _⊓₁_
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; isLattice = lA
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}
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BLattice : Lattice B
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BLattice = record
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{ _≈_ = _≈₂_
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; _⊔_ = _⊔₂_
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; _⊓_ = _⊓₂_
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; isLattice = lB
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}
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IterProdLattice : ∀ {k : ℕ} → Lattice (IterProd k)
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IterProdLattice {0} = BLattice
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IterProdLattice {suc k'} = record
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{ _≈_ = _≈_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isLattice = isLattice
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}
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where
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RightLattice : Lattice (IterProd k')
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RightLattice = IterProdLattice {k'}
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open import Lattice.Prod
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_≈₁_ (Lattice._≈_ RightLattice)
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_⊔₁_ (Lattice._⊔_ RightLattice)
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_⊓₁_ (Lattice._⊓_ RightLattice)
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lA (Lattice.isLattice RightLattice)
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-- Expose the computed definition in public.
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module _ (k : ℕ) where
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open Lattice.Lattice (IterProdLattice {k}) public
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