Re-write the IterProd proofs to couple lattice and finite height lattice

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-03-01 21:02:56 -08:00
parent 29898e738b
commit ca90f6509c
3 changed files with 94 additions and 85 deletions

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@ -23,94 +23,97 @@ IterProd k = iterate k (λ t → A × t) B
-- To make iteration more convenient, package the definitions in Lattice -- To make iteration more convenient, package the definitions in Lattice
-- records, perform the recursion, and unpackage. -- records, perform the recursion, and unpackage.
--
module _ where -- If we prove isLattice and isFiniteHeightLattice by induction separately,
lattice : {k : } Lattice (IterProd k) -- we lose the connection between the operations (which should be the same)
lattice {0} = record -- that are built up by the two iterations. So, do everything in one iteration.
-- This requires some odd code.
private
record RequiredForFixedHeight : Set (lsuc a) where
field
≈₁-dec : IsDecidable _≈₁_
≈₂-dec : IsDecidable _≈₂_
h₁ h₂ :
fhA : FixedHeight₁ h₁
fhB : FixedHeight₂ h₂
record IsFiniteHeightAndDecEq {A : Set a} {_≈_ : A A Set a} {_⊔_ : A A A} {_⊓_ : A A A} (isLattice : IsLattice A _≈_ _⊔_ _⊓_) : Set (lsuc a) where
field
height :
fixedHeight : IsLattice.FixedHeight isLattice height
≈-dec : IsDecidable _≈_
record Everything (A : Set a) : Set (lsuc a) where
field
_≈_ : A A Set a
_⊔_ : A A A
_⊓_ : A A A
isLattice : IsLattice A _≈_ _⊔_ _⊓_
isFiniteHeightIfSupported : RequiredForFixedHeight IsFiniteHeightAndDecEq isLattice
everything : (k : ) Everything (IterProd k)
everything 0 = record
{ _≈_ = _≈₂_ { _≈_ = _≈₂_
; _⊔_ = _⊔₂_ ; _⊔_ = _⊔₂_
; _⊓_ = _⊓₂_ ; _⊓_ = _⊓₂_
; isLattice = lB ; isLattice = lB
; isFiniteHeightIfSupported = λ req record
{ height = RequiredForFixedHeight.h₂ req
; fixedHeight = RequiredForFixedHeight.fhB req
; ≈-dec = RequiredForFixedHeight.≈₂-dec req
}
} }
lattice {suc k'} = record everything (suc k') = record
{ _≈_ = _≈_ { _≈_ = P._≈_
; _⊔_ = _⊔_ ; _⊔_ = P._⊔_
; _⊓_ = _⊓_ ; _⊓_ = P._⊓_
; isLattice = isLattice ; isLattice = P.isLattice
; isFiniteHeightIfSupported = λ req
let
fhlRest = Everything.isFiniteHeightIfSupported everythingRest req
in
record
{ height = (RequiredForFixedHeight.h₁ req) + IsFiniteHeightAndDecEq.height fhlRest
; fixedHeight =
P.fixedHeight
(RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightAndDecEq.≈-dec fhlRest)
(RequiredForFixedHeight.h₁ req) (IsFiniteHeightAndDecEq.height fhlRest)
(RequiredForFixedHeight.fhA req) (IsFiniteHeightAndDecEq.fixedHeight fhlRest)
; ≈-dec = P.≈-dec (RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightAndDecEq.≈-dec fhlRest)
}
} }
where where
Right : Lattice (IterProd k') everythingRest = everything k'
Right = lattice {k'}
open import Lattice.Prod import Lattice.Prod
_≈₁_ (Lattice._≈_ Right) _≈₁_ (Everything._≈_ everythingRest)
_⊔₁_ (Lattice._⊔_ Right) _⊔₁_ (Everything._⊔_ everythingRest)
_⊓₁_ (Lattice._⊓_ Right) _⊓₁_ (Everything._⊓_ everythingRest)
lA (Lattice.isLattice Right) lA (Everything.isLattice everythingRest) as P
module _ (k : ) where module _ (k : ) where
open Lattice.Lattice (lattice {k}) public open Everything (everything k) using (_≈_; _⊔_; _⊓_; isLattice) public
open Lattice.IsLattice isLattice public
module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_) module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
(h₁ h₂ : ) (h₁ h₂ : )
(fhA : FixedHeight₁ h₁) (fhB : FixedHeight₂ h₂) where (fhA : FixedHeight₁ h₁) (fhB : FixedHeight₂ h₂) where
private module _ where
record FiniteHeightAndDecEq (A : Set a) : Set (lsuc a) where
field
height :
_≈_ : A A Set a
_⊔_ : A A A
_⊓_ : A A A
isFiniteHeightLattice : IsFiniteHeightLattice A height _≈_ _⊔_ _⊓_
≈-dec : IsDecidable _≈_
open IsFiniteHeightLattice isFiniteHeightLattice public
finiteHeightAndDec : {k : } 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 : ) where
open FiniteHeightAndDecEq (finiteHeightAndDec {k}) using (isFiniteHeightLattice) public
private private
FHD = finiteHeightAndDec {k} required : RequiredForFixedHeight
required = record
{ ≈₁-dec = ≈₁-dec
; ≈₂-dec = ≈₂-dec
; h₁ = h₁
; h₂ = h₂
; fhA = fhA
; fhB = fhB
}
finiteHeightLattice : FiniteHeightLattice (IterProd k) isFiniteHeightLattice = record
finiteHeightLattice = record { isLattice = isLattice
{ height = FiniteHeightAndDecEq.height FHD ; fixedHeight = IsFiniteHeightAndDecEq.fixedHeight (Everything.isFiniteHeightIfSupported (everything k) required)
; _≈_ = FiniteHeightAndDecEq._≈_ FHD
; _⊔_ = FiniteHeightAndDecEq._⊔_ FHD
; _⊓_ = FiniteHeightAndDecEq._⊓_ FHD
; isFiniteHeightLattice = isFiniteHeightLattice
} }

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@ -171,18 +171,21 @@ module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
, ≤-stepsˡ 1 (subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂)) , ≤-stepsˡ 1 (subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂))
)) ))
fixedHeight : IsLattice.FixedHeight isLattice (h₁ + h₂)
fixedHeight =
( ( ((amin , bmin) , (amax , bmax))
, concat
(ChainMapping₁.Chain-map (λ a (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ fhA)))
(ChainMapping₂.Chain-map (λ b (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ fhB)))
)
, λ a₁b₁a₂b₂ let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ fhA a₁a₂) (proj₂ fhB b₁b₂))
)
isFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_ isFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_
isFiniteHeightLattice = record isFiniteHeightLattice = record
{ isLattice = isLattice { isLattice = isLattice
; fixedHeight = ; fixedHeight = fixedHeight
( ( ((amin , bmin) , (amax , bmax))
, concat
(ChainMapping₁.Chain-map (λ a (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ fhA)))
(ChainMapping₂.Chain-map (λ b (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ fhB)))
)
, λ a₁b₁a₂b₂ let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ fhA a₁a₂) (proj₂ fhB b₁b₂))
)
} }
finiteHeightLattice : FiniteHeightLattice (A × B) finiteHeightLattice : FiniteHeightLattice (A × B)

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@ -107,10 +107,13 @@ private
isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl) isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl)
isLongest (done _) = z≤n isLongest (done _) = z≤n
fixedHeight : IsLattice.FixedHeight isLattice 0
fixedHeight = (((tt , tt) , longestChain) , isLongest)
isFiniteHeightLattice : IsFiniteHeightLattice 0 _≈_ _⊔_ _⊓_ isFiniteHeightLattice : IsFiniteHeightLattice 0 _≈_ _⊔_ _⊓_
isFiniteHeightLattice = record isFiniteHeightLattice = record
{ isLattice = isLattice { isLattice = isLattice
; fixedHeight = (((tt , tt) , longestChain) , isLongest) ; fixedHeight = fixedHeight
} }
finiteHeightLattice : FiniteHeightLattice finiteHeightLattice : FiniteHeightLattice