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Add a lemma about the effect of joinAll

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-05-08 22:34:02 -07:00
parent 5837fdf19b
commit 3f5551d70c
1 changed files with 18 additions and 6 deletions
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24
Analysis/Forward.agda
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@ -95,6 +95,17 @@ module WithProg (prog : Program) where
≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec ≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
-- We now have our (state -> (variables -> value)) map.
-- Define a couple of helpers to retrieve values from it. Specifically,
-- since the State type is as specific as possible, it's always possible to
-- retrieve the variable values at each state.
states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
variablesAt : State → StateVariables → VariableValues
variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv))
-- build up the 'join' function, which follows from Exercise 4.26's -- build up the 'join' function, which follows from Exercise 4.26's
-- --
-- L₁ → (A → L₂) -- L₁ → (A → L₂)
@ -118,8 +129,15 @@ module WithProg (prog : Program) where
renaming renaming
( f' to joinAll ( f' to joinAll
; f'-Monotonic to joinAll-Mono ; f'-Monotonic to joinAll-Mono
; f'-k∈ks-≡ to joinAll-k∈ks-≡
) )
variablesAt-joinAll : ∀ (s : State) (sv : StateVariables) →
variablesAt s (joinAll sv) ≡ joinForKey s sv
variablesAt-joinAll s sv
with (vs , s,vs∈usv) ← locateᵐ {s} {joinAll sv} (states-in-Map s (joinAll sv)) =
joinAll-k∈ks-≡ {l = sv} (states-complete s) s,vs∈usv
-- With 'join' in hand, we need to perform abstract evaluation. -- With 'join' in hand, we need to perform abstract evaluation.
module WithEvaluator (eval : Expr → VariableValues → L) module WithEvaluator (eval : Expr → VariableValues → L)
(eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ˡ_ (eval e)) where (eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ˡ_ (eval e)) where
@ -138,9 +156,6 @@ module WithProg (prog : Program) where
) )
public public
states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
-- The per-state update function makes use of the single-key setter, -- The per-state update function makes use of the single-key setter,
-- updateVariablesFromExpression, for the case where the statement -- updateVariablesFromExpression, for the case where the statement
-- is an assignment. -- is an assignment.
@ -157,9 +172,6 @@ module WithProg (prog : Program) where
updateVariablesFromStmt-Monoʳ (k ← e) {vs₁} {vs₂} vs₁≼vs₂ = updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂ updateVariablesFromStmt-Monoʳ (k ← e) {vs₁} {vs₂} vs₁≼vs₂ = updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
updateVariablesFromStmt-Monoʳ noop vs₁≼vs₂ = vs₁≼vs₂ updateVariablesFromStmt-Monoʳ noop vs₁≼vs₂ = vs₁≼vs₂
variablesAt : State → StateVariables → VariableValues
variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv))
updateVariablesForState : State → StateVariables → VariableValues updateVariablesForState : State → StateVariables → VariableValues
updateVariablesForState s sv = updateVariablesForState s sv =
foldl (flip updateVariablesFromStmt) (variablesAt s sv) (code s) foldl (flip updateVariablesFromStmt) (variablesAt s sv) (code s)