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From mathcomp Require Import ssreflect ssrbool eqtype.
Require Import Arith List String Lia.
From QuickChick Require Import QuickChick.
Import ListNotations.
(* Types *)
Inductive type : Type :=
| N : type
| Arrow : type -> type -> type.
Derive (Arbitrary, Show, EnumSized) for type.
Instance dec_type (t1 t2 : type) : Dec (t1 = t2).
Proof. dec_eq. Defined.
(* Terms *)
Definition var := nat.
Inductive term : Type :=
| Const : nat -> term
| Id : var -> term
| App : term -> term -> term
| Abs : type -> term -> term.
(* Environments *)
Definition env := list type.
Inductive bind : env -> nat -> type -> Prop :=
| BindNow : forall t G, bind (t :: G) 0 t
| BindLater : forall t t' x G,
bind G x t -> bind (t' :: G) (S x) t.
(* Generate variables of a specific type in an env. *)
Derive ArbitrarySizedSuchThat for (fun x => bind G x t).
(* Get the type of a given variable in an env. *)
Derive EnumSizedSuchThat for (fun t => bind G x t).
(* Check whether a variable has a given type in an env. *)
Derive DecOpt for (bind G e t).
(* Typing *)
Inductive typing (G : env) : term -> type -> Prop :=
| TId :
forall x t,
bind G x t ->
typing G (Id x) t
| TConst :
forall n,
typing G (Const n) N
| TAbs :
forall e t1 t2,
typing (t1 :: G) e t2 ->
typing G (Abs t1 e) (Arrow t1 t2)
| TApp :
forall e1 e2 t1 t2,
typing G e2 t1 ->
typing G e1 (Arrow t1 t2) ->
typing G (App e1 e2) t2.
Fixpoint typeOf G e : option type :=
match e with
| Id x => nth_error G x
| Const n => Some N
| Abs t e' =>
match typeOf (t::G) e' with
| Some t' => Some (Arrow t t')
| None => None
end
| App e1 e2 =>
match typeOf G e1, typeOf G e2 with
| Some (Arrow t1 t2), Some t1' =>
if t1 = t1'? then Some t2 else None
| _, _ => None
end
end.
(* Generate terms of a specific type in an env. *)
Derive ArbitrarySizedSuchThat for (fun e => typing G e t).
Derive EnumSizedSuchThat for (fun t => typing G e t).
(* Check whether a variable has a given type in an env. *)
Derive DecOpt for (typing G e t).
(* Small step CBV semantics *)
Inductive value : term -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall t e, value (Abs t e).
Derive DecOpt for (value e).
Definition is_value (e : term) : bool :=
match e with
| Const _ | Abs _ _ => true
| _ => false
end.
Fixpoint subst (y : var) (e1 : term) (e2 : term) : term :=
match e2 with
| Const n => Const n
| Id x =>
if eq_nat_dec x y then e1 else e2
| App e e' =>
App (subst y e1 e) (subst y e1 e')
| Abs t e =>
Abs t (subst (S y) e1 e)
end.
Fixpoint step (e : term) : option term :=
match e with
| Const _ | Id _ => None | Abs _ x => None
| App (Abs t e1) e2 =>
if is_value e2 then Some (subst 0 e2 e1)
else
match step e2 with
| Some e2' => Some (App (Abs t e1) e2')
| None => None
end
| App e1 e2 =>
match step e1 with
| Some e1' => Some (App e1' e2)
| None => None
end
end.
Eval compute in (step (App (Abs N (Id 0)) (Const 42))).
Eval compute in (step (App (Abs N (Abs N (Id 0))) (Const 42))).
Eval compute in (subst 0 (Const 42) (Abs N (Id 0))).
(* Printing *)
Open Scope string.
Fixpoint show_type (tau : type) :=
match tau with
| N => "N"
| Arrow tau1 tau2 =>
"(Arrow " ++ show_type tau1 ++ " -> " ++ show_type tau2 ++ ")"
end.
Instance showType : Show type := { show := show_type }.
Fixpoint show_term (e : term) :=
match e with
| Const n => "(Const " ++ show n ++ ")"
| Id x => "(Id " ++ show x ++ ")"
| App e1 e2 => "(App " ++ show_term e1 ++ " " ++ show_term e2 ++ ")"
| Abs t e => "(Abs " ++ show t ++ " " ++ show_term e ++ ")"
end.
Close Scope string.
Instance showTerm : Show term := { show := show_term }.
Instance dec_eq_opt_type : Dec_Eq (option type).
Proof. dec_eq. Defined.
Definition preservation (e : term) (t: type) : Checker :=
match step e with
| Some e' => checker ((typeOf nil e' = Some t)?)
| None => checker true
end.
Definition preservation' (e : term) (t: type) : Checker :=
match step e with
| Some e' => typing nil e' t ?? 10
| None => checker true
end.
Definition preservation_prop (c : term -> type -> Checker) :=
forAll (@arbitrary type _) (fun t =>
forAllMaybe ((genST (fun e => typing nil e t))) (fun e =>
c e t)).
Extract Constant defNumTests => "20000".
QuickChick (preservation_prop preservation).
QuickChick (preservation_prop preservation').
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