Re-write the IterProd proofs to couple lattice and finite height lattice
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -23,94 +23,97 @@ 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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--
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module _ where
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lattice : ∀ {k : ℕ} → Lattice (IterProd k)
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lattice {0} = record
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-- If we prove isLattice and isFiniteHeightLattice by induction separately,
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-- we lose the connection between the operations (which should be the same)
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-- that are built up by the two iterations. So, do everything in one iteration.
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-- This requires some odd code.
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private
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record RequiredForFixedHeight : Set (lsuc a) where
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field
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≈₁-dec : IsDecidable _≈₁_
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≈₂-dec : IsDecidable _≈₂_
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h₁ h₂ : ℕ
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fhA : FixedHeight₁ h₁
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fhB : FixedHeight₂ h₂
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record IsFiniteHeightAndDecEq {A : Set a} {_≈_ : A → A → Set a} {_⊔_ : A → A → A} {_⊓_ : A → A → A} (isLattice : IsLattice A _≈_ _⊔_ _⊓_) : Set (lsuc a) where
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field
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height : ℕ
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fixedHeight : IsLattice.FixedHeight isLattice height
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≈-dec : IsDecidable _≈_
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record Everything (A : Set a) : Set (lsuc a) where
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field
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_≈_ : A → A → Set a
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_⊔_ : A → A → A
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_⊓_ : A → A → A
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isLattice : IsLattice A _≈_ _⊔_ _⊓_
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isFiniteHeightIfSupported : RequiredForFixedHeight → IsFiniteHeightAndDecEq isLattice
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everything : ∀ (k : ℕ) → Everything (IterProd k)
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everything 0 = 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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; isFiniteHeightIfSupported = λ req → record
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{ height = RequiredForFixedHeight.h₂ req
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; fixedHeight = RequiredForFixedHeight.fhB req
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; ≈-dec = RequiredForFixedHeight.≈₂-dec req
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}
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}
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lattice {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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everything (suc k') = record
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{ _≈_ = P._≈_
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; _⊔_ = P._⊔_
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; _⊓_ = P._⊓_
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; isLattice = P.isLattice
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; isFiniteHeightIfSupported = λ req →
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let
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fhlRest = Everything.isFiniteHeightIfSupported everythingRest req
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in
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record
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{ height = (RequiredForFixedHeight.h₁ req) + IsFiniteHeightAndDecEq.height fhlRest
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; fixedHeight =
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P.fixedHeight
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(RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightAndDecEq.≈-dec fhlRest)
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(RequiredForFixedHeight.h₁ req) (IsFiniteHeightAndDecEq.height fhlRest)
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(RequiredForFixedHeight.fhA req) (IsFiniteHeightAndDecEq.fixedHeight fhlRest)
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; ≈-dec = P.≈-dec (RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightAndDecEq.≈-dec fhlRest)
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}
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}
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where
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Right : Lattice (IterProd k')
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Right = lattice {k'}
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everythingRest = everything k'
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open import Lattice.Prod
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_≈₁_ (Lattice._≈_ Right)
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_⊔₁_ (Lattice._⊔_ Right)
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_⊓₁_ (Lattice._⊓_ Right)
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lA (Lattice.isLattice Right)
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import Lattice.Prod
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_≈₁_ (Everything._≈_ everythingRest)
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_⊔₁_ (Everything._⊔_ everythingRest)
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_⊓₁_ (Everything._⊓_ everythingRest)
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lA (Everything.isLattice everythingRest) as P
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module _ (k : ℕ) where
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open Lattice.Lattice (lattice {k}) public
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open Everything (everything k) using (_≈_; _⊔_; _⊓_; isLattice) public
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open Lattice.IsLattice isLattice public
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module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
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(h₁ h₂ : ℕ)
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(fhA : FixedHeight₁ h₁) (fhB : FixedHeight₂ h₂) where
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private module _ where
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record FiniteHeightAndDecEq (A : Set a) : Set (lsuc a) where
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field
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height : ℕ
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_≈_ : A → A → Set a
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_⊔_ : A → A → A
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_⊓_ : A → A → A
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isFiniteHeightLattice : IsFiniteHeightLattice A height _≈_ _⊔_ _⊓_
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≈-dec : IsDecidable _≈_
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open IsFiniteHeightLattice isFiniteHeightLattice public
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finiteHeightAndDec : ∀ {k : ℕ} → FiniteHeightAndDecEq (IterProd k)
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finiteHeightAndDec {0} = record
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{ height = h₂
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; _≈_ = _≈₂_
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; _⊔_ = _⊔₂_
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; _⊓_ = _⊓₂_
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; isFiniteHeightLattice = record
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{ isLattice = lB
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; fixedHeight = fhB
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}
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; ≈-dec = ≈₂-dec
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}
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finiteHeightAndDec {suc k'} = record
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{ height = h₁ + FiniteHeightAndDecEq.height Right
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; _≈_ = P._≈_
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; _⊔_ = P._⊔_
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; _⊓_ = P._⊓_
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; isFiniteHeightLattice = isFiniteHeightLattice
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≈₁-dec (FiniteHeightAndDecEq.≈-dec Right)
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h₁ (FiniteHeightAndDecEq.height Right)
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fhA (IsFiniteHeightLattice.fixedHeight (FiniteHeightAndDecEq.isFiniteHeightLattice Right))
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; ≈-dec = ≈-dec ≈₁-dec (FiniteHeightAndDecEq.≈-dec Right)
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}
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where
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Right = finiteHeightAndDec {k'}
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open import Lattice.Prod
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_≈₁_ (FiniteHeightAndDecEq._≈_ Right)
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_⊔₁_ (FiniteHeightAndDecEq._⊔_ Right)
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_⊓₁_ (FiniteHeightAndDecEq._⊓_ Right)
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lA (FiniteHeightAndDecEq.isLattice Right) as P
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module _ (k : ℕ) where
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open FiniteHeightAndDecEq (finiteHeightAndDec {k}) using (isFiniteHeightLattice) public
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module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
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(h₁ h₂ : ℕ)
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(fhA : FixedHeight₁ h₁) (fhB : FixedHeight₂ h₂) where
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private
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FHD = finiteHeightAndDec {k}
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required : RequiredForFixedHeight
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required = record
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{ ≈₁-dec = ≈₁-dec
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; ≈₂-dec = ≈₂-dec
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; h₁ = h₁
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; h₂ = h₂
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; fhA = fhA
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; fhB = fhB
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}
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finiteHeightLattice : FiniteHeightLattice (IterProd k)
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finiteHeightLattice = record
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{ height = FiniteHeightAndDecEq.height FHD
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; _≈_ = FiniteHeightAndDecEq._≈_ FHD
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; _⊔_ = FiniteHeightAndDecEq._⊔_ FHD
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; _⊓_ = FiniteHeightAndDecEq._⊓_ FHD
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; isFiniteHeightLattice = isFiniteHeightLattice
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isFiniteHeightLattice = record
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{ isLattice = isLattice
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; fixedHeight = IsFiniteHeightAndDecEq.fixedHeight (Everything.isFiniteHeightIfSupported (everything k) required)
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}
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@ -171,18 +171,21 @@ module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
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, ≤-stepsˡ 1 (subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂))
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))
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fixedHeight : IsLattice.FixedHeight isLattice (h₁ + h₂)
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fixedHeight =
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( ( ((amin , bmin) , (amax , bmax))
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, concat
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(ChainMapping₁.Chain-map (λ a → (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ fhA)))
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(ChainMapping₂.Chain-map (λ b → (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ fhB)))
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)
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, λ a₁b₁a₂b₂ → let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
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in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ fhA a₁a₂) (proj₂ fhB b₁b₂))
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)
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isFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_
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isFiniteHeightLattice = record
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{ isLattice = isLattice
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; fixedHeight =
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( ( ((amin , bmin) , (amax , bmax))
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, concat
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(ChainMapping₁.Chain-map (λ a → (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ fhA)))
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(ChainMapping₂.Chain-map (λ b → (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ fhB)))
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)
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, λ a₁b₁a₂b₂ → let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
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in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ fhA a₁a₂) (proj₂ fhB b₁b₂))
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)
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; fixedHeight = fixedHeight
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}
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finiteHeightLattice : FiniteHeightLattice (A × B)
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@ -107,10 +107,13 @@ private
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isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl)
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isLongest (done _) = z≤n
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fixedHeight : IsLattice.FixedHeight isLattice 0
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fixedHeight = (((tt , tt) , longestChain) , isLongest)
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isFiniteHeightLattice : IsFiniteHeightLattice ⊤ 0 _≈_ _⊔_ _⊓_
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isFiniteHeightLattice = record
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{ isLattice = isLattice
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; fixedHeight = (((tt , tt) , longestChain) , isLongest)
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; fixedHeight = fixedHeight
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}
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finiteHeightLattice : FiniteHeightLattice ⊤
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