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Add functions to reason about the 'monotonic state' operations

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-04-20 18:09:01 -07:00
parent 2f91ca113e
commit 855bf3f56c
1 changed files with 34 additions and 2 deletions
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36
MonotonicState.agda
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@ -87,6 +87,10 @@ Both : {T₁ T₂ : S → Set s} → DependentPredicate T₁ → DependentPredic
DependentPredicate (T₁ ⊗ T₂) DependentPredicate (T₁ ⊗ T₂)
Both P Q = (λ { s (t₁ , t₂) → (P s t₁ × Q s t₂) }) Both P Q = (λ { s (t₁ , t₂) → (P s t₁ × Q s t₂) })
And : {T : S → Set s} → DependentPredicate T → DependentPredicate T →
DependentPredicate T
And P Q = (λ { s t → (P s t × Q s t) })
-- Since monotnic functions keep adding on to the state, proofs of -- Since monotnic functions keep adding on to the state, proofs of
-- predicates over their outputs go stale fast (they describe old values of -- predicates over their outputs go stale fast (they describe old values of
-- the state). To keep them relevant, we need them to still hold on 'bigger -- the state). To keep them relevant, we need them to still hold on 'bigger
@ -103,14 +107,42 @@ record MonotonicPredicate {T : S → Set s} {{ r : Relaxable T }} (P : Dependent
always : ∀ {T : S → Set s} → DependentPredicate T → MonotonicState T → Set s always : ∀ {T : S → Set s} → DependentPredicate T → MonotonicState T → Set s
always P m = ∀ s₁ → let (s₂ , t , _) = m s₁ in P s₂ t always P m = ∀ s₁ → let (s₂ , t , _) = m s₁ in P s₂ t
⟨⊗⟩-reason : ∀ {T₁ T₂ : S → Set s} {{ _ : Relaxable T₁ }} infixr 4 _⟨⊗⟩-reason_
_⟨⊗⟩-reason_ : ∀ {T₁ T₂ : S → Set s} {{ _ : Relaxable T₁ }}
{P : DependentPredicate T₁} {Q : DependentPredicate T₂} {P : DependentPredicate T₁} {Q : DependentPredicate T₂}
{{P-Mono : MonotonicPredicate P}} {{P-Mono : MonotonicPredicate P}}
{m₁ : MonotonicState T₁} {m₂ : MonotonicState T₂} → {m₁ : MonotonicState T₁} {m₂ : MonotonicState T₂} →
always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂) always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂)
⟨⊗⟩-reason {{P-Mono = P-Mono}} {m₁ = m₁} {m₂ = m₂} aP aQ s _⟨⊗⟩-reason_ {{P-Mono = P-Mono}} {m₁ = m₁} {m₂ = m₂} aP aQ s
with p ← aP s with p ← aP s
with (s' , (t₁ , s≼s')) ← m₁ s with (s' , (t₁ , s≼s')) ← m₁ s
with q ← aQ s' with q ← aQ s'
with (s'' , (t₂ , s'≼s'')) ← m₂ s' = with (s'' , (t₂ , s'≼s'')) ← m₂ s' =
(MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p , q) (MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p , q)
infixl 4 _update-reason_
_update-reason_ : ∀ {T : S → Set s} {{ r : Relaxable T }} →
{P : DependentPredicate T} {Q : DependentPredicate T}
{{P-Mono : MonotonicPredicate P}}
{m : MonotonicState T} {mod : ∀ (s : S) → T s → Σ S (λ s' → s ≼ s')} →
always P m → (∀ (s : S) (t : T s) →
let (s' , s≼s') = mod s t
in P s t → Q s' (Relaxable.relax r s≼s' t)) →
always (And P Q) (m update mod)
_update-reason_ {{r = r}} {{P-Mono = P-Mono}} {m = m} {mod = mod} aP modQ s
with p ← aP s
with (s' , (t , s≼s')) ← m s
with q ← modQ s' t p
with (s'' , s'≼s'') ← mod s' t =
(MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p , q)
infixl 4 _map-reason_
_map-reason_ : ∀ {T₁ T₂ : S → Set s}
{P : DependentPredicate T₁} {Q : DependentPredicate T₂}
{m : MonotonicState T₁}
{f : ∀ (s : S) → T₁ s → T₂ s} →
always P m → (∀ (s : S) (t₁ : T₁ s) (t₂ : T₂ s) → P s t₁ → Q s t₂) →
always Q (m map f)
_map-reason_ {m = m} {f = f} aP P⇒Q s
with p ← aP s
with (s' , (t₁ , s≼s')) ← m s = P⇒Q s' t₁ (f s' t₁) p