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