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Author SHA1 Message Date
53a08b8f79 Prove that 'first' presrves equality
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-02-25 18:08:03 -08:00
d6064ff752 Expose 'locate' and 'forget' from Map
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-02-25 18:07:50 -08:00
2 changed files with 63 additions and 12 deletions

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@ -16,20 +16,23 @@ module Lattice.FiniteValueMap (A : Set) (B : Set)
(lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
open import Data.List using (List; length; []; _∷_)
open import Utils using (Unique; push; empty)
open import Data.Product using (Σ; proj₁; proj₂; _×_)
open import Data.Empty using (⊥-elim)
open import Utils using (Unique; push; empty; All¬-¬Any)
open import Data.Product using (_,_)
open import Data.List.Properties using (∷-injectiveʳ)
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)
open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB using (subset-impl; locate; forget; _∈_; Map-functional)
open import Lattice.FiniteMap A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB public
module IterProdIsomorphism where
open import Data.Unit using (; tt)
open import Lattice.Unit using () renaming (_≈_ to _≈ᵘ_; _⊔_ to _⊔ᵘ_; _⊓_ to _⊓ᵘ_; ≈-dec to ≈ᵘ-dec; isLattice to isLatticeᵘ)
open import Lattice.Unit using () renaming (_≈_ to _≈ᵘ_; _⊔_ to _⊔ᵘ_; _⊓_ to _⊓ᵘ_; ≈-dec to ≈ᵘ-dec; isLattice to isLatticeᵘ; ≈-equiv to ≈ᵘ-equiv)
open import Lattice.IterProd _≈₂_ _≈ᵘ_ _⊔₂_ _⊔ᵘ_ _⊓₂_ _⊓ᵘ_ lB isLatticeᵘ as IP using (IterProd)
open IsLattice lB using () renaming (≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym)
from : {ks : List A} FiniteMap ks IterProd (length ks)
from {[]} (([] , _) , _) = tt
@ -54,6 +57,9 @@ module IterProdIsomorphism where
_≈ᵐ_ : {ks : List A} FiniteMap ks FiniteMap ks Set
_≈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)
@ -91,3 +97,42 @@ module IterProdIsomorphism where
m₂⊆m₁ k' v' (there k',v'∈kvs'₂) =
let (v'' , (v'≈v'' , k',v''∈kvs'₁)) = kvs'₂⊆kvs'₁ k' v' k',v'∈kvs'₂
in (v'' , (v'≈v'' , there k',v''∈kvs'₁))
private
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
pop : {k : A} {ks : List A} FiniteMap (k ks) FiniteMap ks
pop (((_ kvs') , push _ ukvs') , refl) = ((kvs' , ukvs') , refl)
pop-≈ : {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ks)) fm₁ ≈ᵐ fm₂ pop fm₁ ≈ᵐ pop fm₂
pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) = (narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
where
narrow₁ : {fm₁ fm₂ : FiniteMap (k ks)} fm₁ ⊆ᵐ fm₂ pop fm₁ ⊆ᵐ fm₂
narrow₁ {(_ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈kvs'₁ = kvs₁⊆kvs₂ k' v' (there k',v'∈kvs'₁)
narrow₂ : {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ks)} fm₁ ⊆ᵐ fm₂ fm₁ ⊆ᵐ pop fm₂
narrow₂ {fm₁} {fm₂ = (_ kvs'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈kvs'₁
with kvs₁⊆kvs₂ k' v' k',v'∈kvs'₁
... | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) = ⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈kvs'₁))
... | (v'' , (v'≈v'' , there k',v'∈kvs'₂)) = (v'' , (v'≈v'' , k',v'∈kvs'₂))
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)
from-rest : {k : A} {ks : List A} (fm : FiniteMap (k ks)) proj₂ (from fm) from (pop fm)
from-rest (((_ kvs') , push _ ukvs') , refl) = refl
from-preserves-≈ : {ks : List A} (fm₁ fm₂ : FiniteMap ks) fm₁ ≈ᵐ fm₂ (_≈ⁱᵖ_ {ks}) (from fm₁) (from fm₂)
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
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))

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@ -73,9 +73,9 @@ private module _ where
∈-cong f (here c≡c') = here (cong f c≡c')
∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
locate : {k : A} {l : List (A × B)} k keys l Σ B (λ v (k , v) l)
locate {k} {(k' , v) xs} (here k≡k') rewrite k≡k' = (v , here refl)
locate {k} {(k' , v) xs} (there k∈kxs) = let (v , k,v∈xs) = locate k∈kxs in (v , there k,v∈xs)
locate-impl : {k : A} {l : List (A × B)} k keys l Σ B (λ v (k , v) l)
locate-impl {k} {(k' , v) xs} (here k≡k') rewrite k≡k' = (v , here refl)
locate-impl {k} {(k' , v) xs} (there k∈kxs) = let (v , k,v∈xs) = locate-impl k∈kxs in (v , there k,v∈xs)
private module ImplRelation where
open MemProp using (_∈_)
@ -476,6 +476,12 @@ _∈_ p (kvs , _) = MemProp._∈_ p kvs
_∈k_ : A Map Set a
_∈k_ k m = MemProp._∈_ k (keys m)
locate : {k : A} {m : Map} k ∈k m Σ B (λ v (k , v) m)
locate k∈km = locate-impl k∈km
forget : {k : A} {v : B} {m : Map} (k , v) m k ∈k m
forget = ∈-cong proj₁
Map-functional : {k : A} {v v' : B} {m : Map} (k , v) m (k , v') m v v'
Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,v'∈m
@ -549,7 +555,7 @@ data Provenance (k : A) : B → Expr → Set (a ⊔ℓ b) where
bothⁱ : {v₁ v₂ : B} {e₁ e₂ : Expr} Provenance k v₁ e₁ Provenance k v₂ e₂ Provenance k (v₁ ⊓₂ v₂) (e₁ e₂)
Expr-Provenance : (k : A) (e : Expr) k ∈k e Σ B (λ v (Provenance k v e × (k , v) e ))
Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate-impl k∈km in (v , (single k,v∈m , k,v∈m))
Expr-Provenance k (e₁ e₂) k∈ke₁e₂
with ∈k-dec k (proj₁ e₁ ) | ∈k-dec k (proj₁ e₂ )
... | yes k∈ke₁ | yes k∈ke₂ =
@ -582,7 +588,7 @@ module _ (≈₂-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈₂ b₂)) where
SubsetInfo-to-dec : {m₁ m₂ : Map} SubsetInfo m₁ m₂ Dec (m₁ m₂)
SubsetInfo-to-dec (extra k k∈km₁ k∉km₂) =
let (v , k,v∈m₁) = locate k∈km₁
let (v , k,v∈m₁) = locate-impl k∈km₁
in no (λ m₁⊆m₂
let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
in k∉km₂ (∈-cong proj₁ k,v'∈m₂))
@ -601,7 +607,7 @@ module _ (≈₂-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈₂ b₂)) where
mismatch k' v₁ v₂ (there k',v₁∈xs₁) k',v₂∈m₂ v₁̷≈v₂
... | fine xs₁⊆m₂ with ∈k-dec k l₂
... | no k∉km₂ = extra k (here refl) k∉km₂
... | yes k∈km₂ with locate k∈km₂
... | yes k∈km₂ with locate-impl k∈km₂
... | (v' , k,v'∈m₂) with ≈₂-dec v v'
... | no v̷≈v' = mismatch k v v' (here refl) (k,v'∈m₂) v̷≈v'
... | yes v≈v' = fine m₁⊆m₂
@ -635,7 +641,7 @@ private module I⊓ = ImplInsert _⊓₂_
where
≈-∉-cong : {m₁ m₂ : Map} {k : A} m₁ m₂ ¬ k ∈k m₁ ¬ k ∈k m₂
≈-∉-cong (m₁⊆m₂ , m₂⊆m₁) k∉km₁ k∈km₂ =
let (v₂ , k,v₂∈m₂) = locate k∈km₂
let (v₂ , k,v₂∈m₂) = locate-impl k∈km₂
(_ , (_ , k,v₁∈m₁)) = m₂⊆m₁ _ v₂ k,v₂∈m₂
in k∉km₁ (∈-cong proj₁ k,v₁∈m₁)
@ -825,7 +831,7 @@ absorb-⊓-⊔ m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (absorb-⊓-⊔¹ , absor
absorb-⊓-⊔² k v k,v∈m₁
with ∈k-dec k l₂
... | yes k∈km₂ =
let (v₂ , k,v₂∈m₂) = locate k∈km₂
let (v₂ , k,v₂∈m₂) = locate-impl k∈km₂
in (v ⊓₂ (v ⊔₂ v₂) , (≈₂-sym (absorb-⊓₂-⊔₂ v v₂) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
... | no k∉km₂ = (v ⊓₂ v , (≈₂-sym (⊓₂-idemp v) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-preserves-∈₁ u₁ k,v∈m₁ k∉km₂)))
@ -852,7 +858,7 @@ absorb-⊔-⊓ m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (absorb-⊔-⊓¹ , absor
absorb-⊔-⊓² k v k,v∈m₁
with ∈k-dec k l₂
... | yes k∈km₂ =
let (v₂ , k,v₂∈m₂) = locate k∈km₂
let (v₂ , k,v₂∈m₂) = locate-impl k∈km₂
in (v ⊔₂ (v ⊓₂ v₂) , (≈₂-sym (absorb-⊔₂-⊓₂ v v₂) , I⊔.union-combines u₁ (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) k,v∈m₁ (I⊓.intersect-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
... | no k∉km₂ = (v , (≈₂-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂)))