[ papers ] Start implementing GCL

I am upset about the amount of computation done in that view...
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Thomas E. Hansen 2022-09-06 09:47:41 +02:00 committed by CodingCellist
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||| The content of this module is based on the paper
||| Applications of Applicative Proof Search
||| by Liam O'Connor
||| https://doi.org/10.1145/2976022.2976030
module Search.GCL
import Data.Nat
import Data.List.Lazy
import Data.List.Quantifiers
import Data.List.Lazy.Quantifiers
import public Search.Negation
import public Search.HDecidable
import public Search.Properties
import public Search.CTL
%default total
parameters (Sts : Type)
mutual
||| Guarded Command Language
public export
data GCL : Type where
IF : (gs : List GUARD) -> GCL
DOT : GCL -> GCL -> GCL
DO : (gs : List GUARD) -> GCL
UPDATE : (u : Sts -> Sts) -> GCL
||| Termination
SKIP : GCL
||| A predicate
public export
Pred : Type
Pred = (st : Sts) -> Bool
public export
record GUARD where
constructor MkGUARD
g : Pred
x : GCL
public export
Uninhabited (IF _ === SKIP) where
uninhabited Refl impossible
public export
Uninhabited (DOT _ _ === SKIP) where
uninhabited Refl impossible
public export
Uninhabited (DO _ === SKIP) where
uninhabited Refl impossible
public export
Uninhabited (UPDATE _ === SKIP) where
uninhabited Refl impossible
||| Prove that the given program terminated (i.e. reached a `SKIP`).
public export
isSkip : (l : GCL) -> Dec (l === SKIP)
isSkip (IF xs) = No absurd
isSkip (DOT y z) = No absurd
isSkip (DO xs) = No absurd
isSkip (UPDATE uf) = No absurd
isSkip SKIP = Yes Refl
||| Operational sematics of GCL
public export
covering
ops : (GCL, Sts) -> List (GCL, Sts)
ops (SKIP, st) = []
ops ((UPDATE u), st) = [(SKIP, u st)]
ops ((DOT SKIP y), st) = [(y, st)]
ops ((DOT x y), st) = map (\ (x, st') => ((DOT x y), st')) $
ops (x, st)
ops ((IF gs), st) = map (\ aGuard => (aGuard.x, st)) $
filter (\ aGuard => aGuard.g st) gs
ops ((DO gs), st) with (map (\ aG => ((DOT aG.x (DO gs)), st)) $
filter (\ aG => aG.g st) gs)
_ | [] = [(SKIP, st)]
_ | ys = ys
||| We can convert a GCL program to a transition digram by using the program
||| as the state and the operational semantics as the transition function.
public export
covering
gclToDiag : GCL -> Diagram GCL Sts
gclToDiag p = TD ops p