Prove that 'updateAll' has preservation

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Analysis/Forward.agda

@@ -52,7 +52,6 @@ module WithProg (prog : Program) where
             ; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
             ; ∈k-dec to ∈k-decᵛ
             ; all-equal-keys to all-equal-keysᵛ
-            ; forget to forgetᵛ
             )
         public
     open IsLattice isLatticeᵛ
@@ -181,6 +180,7 @@ module WithProg (prog : Program) where
             renaming
                 ( f' to updateAll
                 ; f'-Monotonic to updateAll-Mono
+                ; f'-k∈ks-≡ to updateAll-k∈ks-≡
                 )
 
         -- Finally, the whole analysis consists of getting the 'join'
@@ -201,6 +201,12 @@ module WithProg (prog : Program) where
             renaming (aᶠ to result)
             public
 
+        variablesAt-updateAll : ∀ (s : State) (sv : StateVariables) →
+                                variablesAt s (updateAll sv) ≡ updateVariablesForState s sv
+        variablesAt-updateAll s sv
+            with (vs , s,vs∈usv) ← locateᵐ {s} {updateAll sv} (states-in-Map s (updateAll sv)) =
+            updateAll-k∈ks-≡ {l = sv} (states-complete s) s,vs∈usv
+
         module WithInterpretation (latticeInterpretationˡ : LatticeInterpretation isLatticeˡ) where
             open LatticeInterpretation latticeInterpretationˡ
                 using ()
@@ -237,3 +243,7 @@ module WithProg (prog : Program) where
 
                 updateVariablesForState-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ updateVariablesForState s sv ⟧ᵛ ρ₂
                 updateVariablesForState-matches = updateVariablesFromStmt-fold-matches
+
+                updateAll-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ variablesAt s (updateAll sv) ⟧ᵛ ρ₂
+                updateAll-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁ rewrite variablesAt-updateAll s sv =
+                    updateVariablesForState-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁