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