113 lines
4.0 KiB
Agda
113 lines
4.0 KiB
Agda
open import Lattice
|
||
|
||
module Lattice.IterProd {a} {A B : Set a}
|
||
(_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
|
||
(_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
|
||
(_⊓₁_ : A → A → A) (_⊓₂_ : B → B → B)
|
||
(lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
|
||
|
||
open import Agda.Primitive using (lsuc)
|
||
open import Data.Nat using (ℕ; suc; _+_)
|
||
open import Data.Product using (_×_)
|
||
open import Utils using (iterate)
|
||
|
||
open IsLattice lA renaming (FixedHeight to FixedHeight₁)
|
||
open IsLattice lB renaming (FixedHeight to FixedHeight₂)
|
||
|
||
IterProd : ℕ → Set a
|
||
IterProd k = iterate k (λ t → A × t) B
|
||
|
||
-- To make iteration more convenient, package the definitions in Lattice
|
||
-- records, perform the recursion, and unpackage.
|
||
|
||
module _ where
|
||
lattice : ∀ {k : ℕ} → Lattice (IterProd k)
|
||
lattice {0} = record
|
||
{ _≈_ = _≈₂_
|
||
; _⊔_ = _⊔₂_
|
||
; _⊓_ = _⊓₂_
|
||
; isLattice = lB
|
||
}
|
||
lattice {suc k'} = record
|
||
{ _≈_ = _≈_
|
||
; _⊔_ = _⊔_
|
||
; _⊓_ = _⊓_
|
||
; isLattice = isLattice
|
||
}
|
||
where
|
||
Right : Lattice (IterProd k')
|
||
Right = lattice {k'}
|
||
|
||
open import Lattice.Prod
|
||
_≈₁_ (Lattice._≈_ Right)
|
||
_⊔₁_ (Lattice._⊔_ Right)
|
||
_⊓₁_ (Lattice._⊓_ Right)
|
||
lA (Lattice.isLattice Right)
|
||
|
||
module _ (k : ℕ) where
|
||
open Lattice.Lattice (lattice {k}) public
|
||
|
||
module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
|
||
(h₁ h₂ : ℕ)
|
||
(fhA : FixedHeight₁ h₁) (fhB : FixedHeight₂ h₂) where
|
||
|
||
private module _ where
|
||
record FiniteHeightAndDecEq (A : Set a) : Set (lsuc a) where
|
||
field
|
||
height : ℕ
|
||
_≈_ : A → A → Set a
|
||
_⊔_ : A → A → A
|
||
_⊓_ : A → A → A
|
||
|
||
isFiniteHeightLattice : IsFiniteHeightLattice A height _≈_ _⊔_ _⊓_
|
||
≈-dec : IsDecidable _≈_
|
||
|
||
open IsFiniteHeightLattice isFiniteHeightLattice public
|
||
|
||
finiteHeightAndDec : ∀ {k : ℕ} → FiniteHeightAndDecEq (IterProd k)
|
||
finiteHeightAndDec {0} = record
|
||
{ height = h₂
|
||
; _≈_ = _≈₂_
|
||
; _⊔_ = _⊔₂_
|
||
; _⊓_ = _⊓₂_
|
||
; isFiniteHeightLattice = record
|
||
{ isLattice = lB
|
||
; fixedHeight = fhB
|
||
}
|
||
; ≈-dec = ≈₂-dec
|
||
}
|
||
finiteHeightAndDec {suc k'} = record
|
||
{ height = h₁ + FiniteHeightAndDecEq.height Right
|
||
; _≈_ = P._≈_
|
||
; _⊔_ = P._⊔_
|
||
; _⊓_ = P._⊓_
|
||
; isFiniteHeightLattice = isFiniteHeightLattice
|
||
≈₁-dec (FiniteHeightAndDecEq.≈-dec Right)
|
||
h₁ (FiniteHeightAndDecEq.height Right)
|
||
fhA (IsFiniteHeightLattice.fixedHeight (FiniteHeightAndDecEq.isFiniteHeightLattice Right))
|
||
; ≈-dec = ≈-dec ≈₁-dec (FiniteHeightAndDecEq.≈-dec Right)
|
||
}
|
||
where
|
||
Right = finiteHeightAndDec {k'}
|
||
|
||
open import Lattice.Prod
|
||
_≈₁_ (FiniteHeightAndDecEq._≈_ Right)
|
||
_⊔₁_ (FiniteHeightAndDecEq._⊔_ Right)
|
||
_⊓₁_ (FiniteHeightAndDecEq._⊓_ Right)
|
||
lA (FiniteHeightAndDecEq.isLattice Right) as P
|
||
|
||
module _ (k : ℕ) where
|
||
open FiniteHeightAndDecEq (finiteHeightAndDec {k}) using (isFiniteHeightLattice) public
|
||
|
||
private
|
||
FHD = finiteHeightAndDec {k}
|
||
|
||
finiteHeightLattice : FiniteHeightLattice (IterProd k)
|
||
finiteHeightLattice = record
|
||
{ height = FiniteHeightAndDecEq.height FHD
|
||
; _≈_ = FiniteHeightAndDecEq._≈_ FHD
|
||
; _⊔_ = FiniteHeightAndDecEq._⊔_ FHD
|
||
; _⊓_ = FiniteHeightAndDecEq._⊓_ FHD
|
||
; isFiniteHeightLattice = isFiniteHeightLattice
|
||
}
|