Finish the last proof obligation for trace walking

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-05-09 19:01:36 -07:00
parent 794c04eee9
commit f4392b32c0
3 changed files with 19 additions and 2 deletions

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@ -82,7 +82,7 @@ module WithProg (prog : Program) where
-- Finally, the map we care about is (state -> (variables -> value)). Bring that in.
module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
open StateVariablesFiniteMap
using (_[_]; []-∈; m₁≼m₂⇒m₁[ks]≼m₂[ks])
using (_[_]; []-∈; m₁≼m₂⇒m₁[ks]≼m₂[ks]; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂)
renaming
( FiniteMap to StateVariables
; isLattice to isLatticeᵐ
@ -124,7 +124,9 @@ module WithProg (prog : Program) where
variablesAt-∈ s sv = proj₂ (locateᵐ {s} {sv} (states-in-Map s sv))
variablesAt-≈ : s sv₁ sv₂ sv₁ ≈ᵐ sv₂ variablesAt s sv₁ ≈ᵛ variablesAt s sv₂
variablesAt-≈ = {!!}
variablesAt-≈ s sv₁ sv₂ sv₁≈sv₂ =
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ sv₁ sv₂ sv₁≈sv₂
(states-in-Map s sv₁) (states-in-Map s sv₂)
-- build up the 'join' function, which follows from Exercise 4.26's
--

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@ -32,6 +32,7 @@ open import Lattice.Map ≡-dec-A lB as Map
; _[_] to _[_]ᵐ
; []-∈ to []ᵐ-∈
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ to m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ᵐ
; locate to locateᵐ
; keys to keysᵐ
; _updating_via_ to _updatingᵐ_via_
@ -164,6 +165,11 @@ module WithKeys (ks : List A) where
fm₁ fm₂ (k , v₁) fm₁ (k , v₂) fm₂ v₁ ≼₂ v₂
m₁≼m₂⇒m₁[k]≼m₂[k] (m₁ , _) (m₂ , _) m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ : (fm₁ fm₂ : FiniteMap) {k : A}
fm₁ fm₂ (k∈kfm₁ : k ∈k fm₁) (k∈kfm₂ : k ∈k fm₂)
proj₁ (locate {fm = fm₁} k∈kfm₁) ≈₂ proj₁ (locate {fm = fm₂} k∈kfm₂)
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ (m₁ , _) (m₂ , _) = m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ᵐ m₁ m₂
module GeneralizedUpdate
{l} {L : Set l}
{_≈ˡ_ : L L Set l} {_⊔ˡ_ : L L L} {_⊓ˡ_ : L L L}

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@ -1142,3 +1142,12 @@ m₁≼m₂⇒k∈km₁⇒k∈km₂ m₁ m₂ m₁≼m₂ k∈km₁ =
(v' , (v≈v' , k,v'∈m₂)) = (proj₁ m₁≼m₂) _ _ k,v∈m₁m₂
in
forget k,v'∈m₂
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ : (m₁ m₂ : Map) {k : A}
m₁ m₂ (k∈km₁ : k ∈k m₁) (k∈km₂ : k ∈k m₂)
proj₁ (locate {m = m₁} k∈km₁) ≈₂ proj₁ (locate {m = m₂} k∈km₂)
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ m₁ m₂ {k} (m₁⊆m₂ , m₂⊆m₁) k∈km₁ k∈km₂
with (v₁ , k,v₁∈m₁) locate {m = m₁} k∈km₁
with (v₂ , k,v₂∈m₂) locate {m = m₂} k∈km₂
with (v₂' , (v₁≈v₂' , k,v₂'∈m₂)) m₁⊆m₂ k v₁ k,v₁∈m₁
rewrite Map-functional {m = m₂} k,v₂∈m₂ k,v₂'∈m₂ = v₁≈v₂'