Prove a property of multi-key lookup

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-05-09 17:56:26 -07:00
parent 20dc99ba1f
commit a22c0c9252
2 changed files with 18 additions and 0 deletions

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@ -30,6 +30,7 @@ open import Lattice.Map ≡-dec-A lB as Map
; absorb-⊓-⊔ to absorb-⊓ᵐ-⊔ᵐ
; ≈-dec to ≈ᵐ-dec
; _[_] to _[_]ᵐ
; []-∈ to []ᵐ-∈
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
; locate to locateᵐ
; keys to keysᵐ
@ -104,6 +105,10 @@ module WithKeys (ks : List A) where
_[_] : FiniteMap List A List B
_[_] (m₁ , _) ks = m₁ [ ks ]ᵐ
[]-∈ : {k : A} {v : B} {ks' : List A} (fm : FiniteMap)
k ∈ˡ ks' (k , v) fm v ∈ˡ (fm [ ks' ])
[]-∈ {k} {v} {ks'} (m , _) k∈ks' k,v∈fm = []ᵐ-∈ m k,v∈fm k∈ks'
≈-equiv : IsEquivalence FiniteMap _≈_
≈-equiv = record
{ ≈-refl =

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@ -1112,6 +1112,19 @@ _[_] m (k ∷ ks)
... | yes k∈km = proj₁ (locate {m = m} k∈km) (m [ ks ])
... | no _ = m [ ks ]
[]-∈ : {k : A} {v : B} {ks : List A} (m : Map)
(k , v) m k ∈ˡ ks v ∈ˡ (m [ ks ])
[]-∈ {k} {v} {ks} m k,v∈m (here refl)
with ∈k-dec k (proj₁ m)
... | no k∉km = ⊥-elim (k∉km (forget k,v∈m))
... | yes k∈km
with (v' , k,v'∈m) locate {m = m} k∈km
rewrite Map-functional {m = m} k,v'∈m k,v∈m = here refl
[]-∈ {k} {v} {k' ks'} m k,v∈m (there k∈ks')
with ∈k-dec k' (proj₁ m)
... | no _ = []-∈ m k,v∈m k∈ks'
... | yes _ = there ([]-∈ m k,v∈m k∈ks')
m₁≼m₂⇒m₁[k]≼m₂[k] : (m₁ m₂ : Map) {k : A} {v₁ v₂ : B}
m₁ m₂ (k , v₁) m₁ (k , v₂) m₂ v₁ ≼₂ v₂
m₁≼m₂⇒m₁[k]≼m₂[k] m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂