agda-spa/MonotonicState.agda

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open import Agda.Primitive using (lsuc)
module MonotonicState {s} {S : Set s}
(_≼_ : S S Set s)
(≼-trans : {s₁ s₂ s₃ : S} s₁ s₂ s₂ s₃ s₁ s₃) where
open import Data.Product using (Σ; _×_; _,_)
open import Utils using (_⊗_; _,_)
-- Sometimes, we need a state monad whose values depend on the state. However,
-- one trouble with such monads is that as the state evolves, old values
-- in scope are over the 'old' state, and don't get updated accordingly.
-- Apparently, a related version of this problem is called 'demonic bind'.
--
-- One solution to the problem is to also witness some kind of relationtion
-- between the input and output states. Using this relationship makes it possible
-- to 'bring old values up to speed'.
--
-- Motivated primarily by constructing a Control Flow Graph, the 'relationship'
-- I've chosen is a 'less-than' relation. Thus, 'MonotonicState' is just
-- a (dependent) state "monad" that also witnesses that the state keeps growing.
MonotonicState : (S Set s) Set s
MonotonicState T = (s₁ : S) Σ S (λ s₂ T s₂ × s₁ s₂)
-- It's not a given that the (arbitrary) _≼_ relationship can be used for
-- updating old values. The Relaxable typeclass represents type constructor
-- that support the operation.
record Relaxable (T : S Set s) : Set (lsuc s) where
field relax : {s₁ s₂ : S} s₁ s₂ T s₁ T s₂
instance
ProdRelaxable : {P : S Set s} {Q : S Set s}
{{ PRelaxable : Relaxable P }} {{ QRelaxable : Relaxable Q }}
Relaxable (P Q)
ProdRelaxable {{pr}} {{qr}} = record
{ relax = (λ { g₁≼g₂ (p , q)
( Relaxable.relax pr g₁≼g₂ p
, Relaxable.relax qr g₁≼g₂ q) }
)
}
-- In general, the "MonotonicState monad" is not even a monad; it's not
-- even applicative. The trouble is that functions in general cannot be
-- 'relaxed', and to apply an 'old' function to a 'new' value, you'd thus
-- need to un-relax the value (which also isn't possible in general).
--
-- However, we _can_ combine pairs from two functions into a tuple, which
-- would equivalent to the applicative operation if functions were relaxable.
--
-- TODO: Now that I think about it, the swapped version of the applicative
-- operation is possible, since it doesn't require lifting functions.
infixr 4 _⟨⊗⟩_
_⟨⊗⟩_ : {T₁ T₂ : S Set s} {{ _ : Relaxable T₁ }}
MonotonicState T₁ MonotonicState T₂ MonotonicState (T₁ T₂)
_⟨⊗⟩_ {{r}} f₁ f₂ s
with (s' , (t₁ , s≼s')) f₁ s
with (s'' , (t₂ , s'≼s'')) f₂ s' =
(s'' , ((Relaxable.relax r s'≼s'' t₁ , t₂) , ≼-trans s≼s' s'≼s''))
infixl 4 _update_
_update_ : {T : S Set s} {{ _ : Relaxable T }}
MonotonicState T ( (s : S) T s Σ S (λ s' s s'))
MonotonicState T
_update_ {{r}} f mod s
with (s' , (t , s≼s')) f s
with (s'' , s'≼s'') mod s' t =
(s'' , ((Relaxable.relax r s'≼s'' t , ≼-trans s≼s' s'≼s'')))
infixl 4 _map_
_map_ : {T₁ T₂ : S Set s}
MonotonicState T₁ ( (s : S) T₁ s T₂ s) MonotonicState T₂
_map_ f fn s = let (s' , (t₁ , s≼s')) = f s in (s' , (fn s' t₁ , s≼s'))
-- To reason about MonotonicState instances, we need predicates over their
-- values. But such values are dependent, so our predicates need to accept
-- the state as argument, too.
DependentPredicate : (S Set s) Set (lsuc s)
DependentPredicate T = (s₁ : S) T s₁ Set s
Both : {T₁ T₂ : S Set s} DependentPredicate T₁ DependentPredicate T₂
DependentPredicate (T₁ T₂)
Both P Q = (λ { s (t₁ , t₂) (P s t₁ × Q s t₂) })
-- Since monotnic functions keep adding on to the state, proofs 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
-- states'. We call such predicates monotonic as well, since they respect the
-- ordering relation.
record MonotonicPredicate {T : S Set s} {{ r : Relaxable T }} (P : DependentPredicate T) : Set s where
field relaxPredicate : (s₁ s₂ : S) (t₁ : T s₁) (s₁≼s₂ : s₁ s₂)
P s₁ t₁ P s₂ (Relaxable.relax r s₁≼s₂ t₁)
-- A MonotonicState "monad" m has a certain property if its ouputs satisfy that
-- property for all inputs.
always : {T : S Set s} DependentPredicate T MonotonicState T Set s
always P m = s₁ let (s₂ , t , _) = m s₁ in P s₂ t
⟨⊗⟩-reason : {T₁ T₂ : S Set s} {{ _ : Relaxable T₁ }}
{P : DependentPredicate T₁} {Q : DependentPredicate T₂}
{{P-Mono : MonotonicPredicate P}}
{m₁ : MonotonicState T₁} {m₂ : MonotonicState T₂}
always P m₁ always Q m₂ always (Both P Q) (m₁ ⟨⊗⟩ m₂)
⟨⊗⟩-reason {{P-Mono = P-Mono}} {m₁ = m₁} {m₂ = m₂} aP aQ s
with p aP s
with (s' , (t₁ , s≼s')) m₁ s
with q aQ s'
with (s'' , (t₂ , s'≼s'')) m₂ s' =
(MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p , q)