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Add most of the proof of from distributivity.

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-02-25 20:28:07 -08:00
parent 3ad7db738a
commit b083561629
1 changed files with 35 additions and 5 deletions
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40
Lattice/FiniteValueMap.agda
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@ -25,7 +25,7 @@ open import Data.List.Relation.Unary.All using (All)
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Relation.Nullary using (¬_)
open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB using (subset-impl; locate; forget; _∈_; Map-functional)
open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB using (subset-impl; locate; forget; _∈_; Map-functional; Expr-Provenance; _∩_; _∪_; `_; in₁; in₂; bothᵘ; single)
open import Lattice.FiniteMap A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB public
module IterProdIsomorphism where
@ -57,12 +57,18 @@ module IterProdIsomorphism where
_≈ᵐ_ : ∀ {ks : List A} → FiniteMap ks → FiniteMap ks → Set
_≈ᵐ_ {ks} = _≈_ ks
_⊔ᵐ_ : ∀ {ks : List A} → FiniteMap ks → FiniteMap ks → FiniteMap ks
_⊔ᵐ_ {ks} = _⊔_ ks
_⊆ᵐ_ : ∀ {ks₁ ks₂ : List A} → FiniteMap ks₁ → FiniteMap ks₂ → Set
_⊆ᵐ_ fm₁ fm₂ = subset-impl (proj₁ (proj₁ fm₁)) (proj₁ (proj₁ fm₂))
_≈ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → Set
_≈ⁱᵖ_ {ks} = IP._≈_ (length ks)
_⊔ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
_⊔ⁱᵖ_ {ks} = IP._⊔_ (length ks)
from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
Inverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- from (to x) = x
from-to-inverseˡ {[]} _ _ = IsEquivalence.≈-refl (IP.≈-equiv 0)
@ -102,9 +108,13 @@ module IterProdIsomorphism where
first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → Σ B (λ v → (k , v) ∈ proj₁ fm)
first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)
from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₁ (from fm) ≈₂ proj₁ (first-key-in-map fm)
from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = IsLattice.≈-refl lB
from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = refl
-- We need pop because reasoning about two distinct 'refl' pattern
-- matches is giving us unification errors. So, stash the 'refl' pattern
-- matching into a helper functions, and write solutions in terms
-- of that.
pop : ∀ {k : A} {ks : List A} → FiniteMap (k ∷ ks) → FiniteMap ks
pop (((_ ∷ kvs') , push _ ukvs') , refl) = ((kvs' , ukvs') , refl)
@ -123,6 +133,9 @@ module IterProdIsomorphism where
narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ pop fm₂
narrow {fm₁} {fm₂} x = narrow₂ {pop fm₁} (narrow₁ {fm₂ = fm₂} x)
pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
pop-⊔-distr = {!!} -- pop (fm₁ ⊔ fm₂) ⊆ pop fm₁ ⊔ pop fm₂ etc.
from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₂ (from fm) ≡ from (pop fm)
from-rest (((_ ∷ kvs') , push _ ukvs') , refl) = refl
@ -130,12 +143,12 @@ module IterProdIsomorphism where
from-preserves-≈ {[]} (([] , _) , _) (([] , _) , _) _ = IsEquivalence.≈-refl ≈ᵘ-equiv
from-preserves-≈ {k ∷ ks'} fm₁@(m₁ , _) fm₂@(m₂ , _) fm₁≈fm₂@(kvs₁⊆kvs₂ , kvs₂⊆kvs₁)
with first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value fm₁ | from-first-value fm₂
... | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | fv₁≈v₁ | fv₂≈v₂
... | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl
with kvs₁⊆kvs₂ _ _ k,v₁∈fm₁
... | (v₁' , (v₁≈v₁' , k,v₁'∈fm₂))
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₁'∈fm₂
rewrite from-rest fm₁ rewrite from-rest fm₂
= (≈₂-trans fv₁≈v₁ (≈₂-trans v₁≈v₁' (≈₂-sym fv₂≈v₂)) , from-preserves-≈ (pop fm₁) (pop fm₂) (pop-≈ fm₁ fm₂ fm₁≈fm₂))
= (v₁≈v₁' , from-preserves-≈ (pop fm₁) (pop fm₂) (pop-≈ fm₁ fm₂ fm₁≈fm₂))
to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) (ip₁ ip₂ : IterProd (length ks)) → _≈ⁱᵖ_ {ks} ip₁ ip₂ → to uks ip₁ ≈ᵐ to uks ip₂
to-preserves-≈ {[]} empty tt tt _ = ((λ k v ()), (λ k v ()))
@ -156,3 +169,20 @@ module IterProdIsomorphism where
with k,v∈kvs₂
... | here refl = (v₁ , (IsLattice.≈-sym lB v₁≈v₂ , here refl))
... | there k,v∈kvs'₂ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₂)) = proj₂ (to-preserves-≈ uks' rest₁ rest₂ rest₁≈rest₂) k v k,v∈kvs'₂ in (v' , (v≈v' , there k,v'∈kvs₂))
from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → _≈ⁱᵖ_ {ks} (from (fm₁ ⊔ᵐ fm₂)) (_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
from-⊔-distr {[]} fm₁ fm₂ = IsEquivalence.≈-refl ≈ᵘ-equiv
from-⊔-distr {k ∷ ks} fm₁@(m₁ , _) fm₂@(m₂ , _)
with first-key-in-map (fm₁ ⊔ᵐ fm₂) | first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value (fm₁ ⊔ᵐ fm₂) | from-first-value fm₁ | from-first-value fm₂
... | (v , k,v∈fm₁fm₂) | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl | refl
with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
... | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget {m = m₂} k,v₂∈fm₂))
... | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget {m = m₁} k,v₁∈fm₁))
... | (v₁⊔v₂ , (bothᵘ {v₁'} {v₂'} (single k,v₁'∈m₁) (single k,v₂'∈m₂) , k,v₁⊔v₂∈m₁m₂))
rewrite Map-functional {m = m₁} k,v₁∈fm₁ k,v₁'∈m₁
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₂'∈m₂
rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂
rewrite from-rest (fm₁ ⊔ᵐ fm₂) rewrite from-rest fm₁ rewrite from-rest fm₂
= ( IsLattice.≈-refl lB
, IsEquivalence.≈-trans (IP.≈-equiv (length ks)) (from-preserves-≈ (pop (fm₁ ⊔ᵐ fm₂)) (pop fm₁ ⊔ᵐ pop fm₂) (pop-⊔-distr fm₁ fm₂)) ((from-⊔-distr (pop fm₁) (pop fm₂)))
)