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(** * Named Representation of Gallina *)
Require Import Coq.Classes.RelationClasses.
Require Import Crypto.Reflection.Syntax.
Require Import Crypto.Util.PointedProp.
Require Import Crypto.Util.Tuple.
Require Import Crypto.Util.Tactics.
Require Import Crypto.Util.Notations.
Class Context {base_type_code} (Name : Type) (var : base_type_code -> Type) :=
{ ContextT : Type;
lookupb : ContextT -> Name -> forall {t : base_type_code}, option (var t);
extendb : ContextT -> Name -> forall {t : base_type_code}, var t -> ContextT;
removeb : ContextT -> Name -> base_type_code -> ContextT;
empty : ContextT }.
Coercion ContextT : Context >-> Sortclass.
Arguments ContextT {_ _ _ _}, {_ _ _} _.
Arguments lookupb {_ _ _ _} _ _ {_}, {_ _ _ _} _ _ _.
Arguments extendb {_ _ _ _} _ _ [_] _.
Arguments removeb {_ _ _ _} _ _ _.
Arguments empty {_ _ _ _}.
Local Open Scope ctype_scope.
Local Open Scope expr_scope.
Delimit Scope nexpr_scope with nexpr.
Module Export Named.
Section language.
Context (base_type_code : Type)
(interp_base_type : base_type_code -> Type)
(op : flat_type base_type_code -> flat_type base_type_code -> Type)
(Name : Type).
Local Notation flat_type := (flat_type base_type_code).
Local Notation type := (type base_type_code).
Let Tbase := @Tbase base_type_code.
Local Coercion Tbase : base_type_code >-> Syntax.flat_type.
Local Notation interp_type := (interp_type interp_base_type).
Local Notation interp_flat_type_gen := interp_flat_type.
Local Notation interp_flat_type := (interp_flat_type interp_base_type).
Section expr_param.
Section expr.
Inductive exprf : flat_type -> Type :=
| Const {t : flat_type} : interp_type t -> exprf t
| Var {t : base_type_code} : Name -> exprf t
| Op {t1 tR} : op t1 tR -> exprf t1 -> exprf tR
| LetIn : forall {tx}, interp_flat_type_gen (fun _ => Name) tx -> exprf tx -> forall {tC}, exprf tC -> exprf tC
| Pair : forall {t1}, exprf t1 -> forall {t2}, exprf t2 -> exprf (Prod t1 t2).
Bind Scope nexpr_scope with exprf.
Inductive expr : type -> Type :=
| Return {t} : exprf t -> expr t
| Abs {src dst} : Name -> expr dst -> expr (Arrow src dst).
Bind Scope nexpr_scope with expr.
Global Coercion Return : exprf >-> expr.
(** [SmartVar] is like [Var], except that it inserts
pair-projections and [Pair] as necessary to handle
[flat_type], and not just [base_type_code] *)
Definition SmartVar {t} : interp_flat_type_gen (fun _ => Name) t -> exprf t
:= smart_interp_flat_map (f:=fun _ => Name) (g:=exprf) _ (fun t => Var) (fun A B x y => Pair x y).
Definition SmartConst {t} : interp_flat_type t -> @interp_flat_type_gen base_type_code exprf t
:= smart_interp_flat_map (g:=@interp_flat_type_gen base_type_code exprf) _ (fun t => Const (t:=t)) (fun A B x y => pair x y).
End expr.
Section with_context.
Context {var : base_type_code -> Type}
{Context : Context Name var}.
Fixpoint extend (ctx : Context) {t : flat_type}
(n : interp_flat_type_gen (fun _ => Name) t) (v : interp_flat_type_gen var t)
: Context
:= match t return interp_flat_type_gen (fun _ => Name) t -> interp_flat_type_gen var t -> Context with
| Syntax.Tbase t => fun n v => extendb ctx n v
| Prod A B => fun n v
=> let ctx := @extend ctx A (fst n) (fst v) in
let ctx := @extend ctx B (snd n) (snd v) in
ctx
end n v.
Fixpoint remove (ctx : Context) {t : flat_type}
(n : interp_flat_type_gen (fun _ => Name) t)
: Context
:= match t return interp_flat_type_gen (fun _ => Name) t -> Context with
| Syntax.Tbase t => fun n => removeb ctx n t
| Prod A B => fun n
=> let ctx := @remove ctx A (fst n) in
let ctx := @remove ctx B (snd n) in
ctx
end n.
Definition lookup (ctx : Context) {t}
: interp_flat_type_gen (fun _ => Name) t -> option (interp_flat_type_gen var t)
:= smart_interp_flat_map
base_type_code
(g := fun t => option (interp_flat_type_gen var t))
(fun t v => lookupb ctx v)
(fun A B x y => match x, y with
| Some x', Some y' => Some (x', y')%core
| _, _ => None
end).
Section wf.
Fixpoint wff (ctx : Context) {t} (e : exprf t) : option pointed_Prop
:= match e with
| Const _ x => Some trivial
| Var t n => match lookupb ctx n t return bool with
| Some _ => true
| None => false
end
| Op _ _ op args => @wff ctx _ args
| LetIn _ n ex _ eC => @wff ctx _ ex /\ inject (forall v, prop_of_option (@wff (extend ctx n v) _ eC))
| Pair _ ex _ ey => @wff ctx _ ex /\ @wff ctx _ ey
end%option_pointed_prop.
Fixpoint wf (ctx : Context) {t} (e : expr t) : Prop
:= match e with
| Return _ x => prop_of_option (wff ctx x)
| Abs src _ n f => forall v, @wf (extend ctx (t:=src) n v) _ f
end.
End wf.
Section interp_gen.
Context (output_interp_flat_type : flat_type -> Type)
(interp_const : forall t, interp_flat_type t -> output_interp_flat_type t)
(interp_var : forall t, var t -> output_interp_flat_type t)
(interp_op : forall src dst, op src dst -> output_interp_flat_type src -> output_interp_flat_type dst)
(interp_let : forall (tx : flat_type) (ex : output_interp_flat_type tx)
tC (eC : interp_flat_type_gen var tx -> output_interp_flat_type tC),
output_interp_flat_type tC)
(interp_pair : forall (tx : flat_type) (ex : output_interp_flat_type tx)
(ty : flat_type) (ey : output_interp_flat_type ty),
output_interp_flat_type (Prod tx ty)).
Fixpoint interp_genf (ctx : Context) {t} (e : exprf t)
: prop_of_option (wff ctx e) -> output_interp_flat_type t
:= match e in exprf t return prop_of_option (wff ctx e) -> output_interp_flat_type t with
| Const _ x => fun _ => interp_const _ x
| Var t' x => match lookupb ctx x t' as v
return prop_of_option (match v return bool with
| Some _ => true
| None => false
end)
-> output_interp_flat_type t'
with
| Some v => fun _ => interp_var _ v
| None => fun bad => match bad : False with end
end
| Op _ _ op args => fun good => @interp_op _ _ op (@interp_genf ctx _ args good)
| LetIn _ n ex _ eC => fun good => let goodxC := proj1 (@prop_of_option_and _ _) good in
let x := @interp_genf ctx _ ex (proj1 goodxC) in
interp_let _ x _ (fun x => @interp_genf (extend ctx n x) _ eC (proj2 goodxC x))
| Pair _ ex _ ey => fun good => let goodxy := proj1 (@prop_of_option_and _ _) good in
let x := @interp_genf ctx _ ex (proj1 goodxy) in
let y := @interp_genf ctx _ ey (proj2 goodxy) in
interp_pair _ x _ y
end.
End interp_gen.
End with_context.
Section with_val_context.
Context (Context : Context Name interp_base_type)
(interp_op : forall src dst, op src dst -> interp_flat_type src -> interp_flat_type dst).
Definition interpf
: forall (ctx : Context) {t} (e : exprf t),
prop_of_option (wff ctx e) -> interp_flat_type t
:= @interp_genf
interp_base_type Context interp_flat_type
(fun _ x => x) (fun _ x => x) interp_op (fun _ y _ f => let x := y in f x)
(fun _ x _ y => (x, y)%core).
Fixpoint interp (ctx : Context) {t} (e : expr t)
: wf ctx e -> interp_type t
:= match e in expr t return wf ctx e -> interp_type t with
| Return _ x => interpf ctx x
| Abs _ _ n f => fun good v => @interp _ _ f (good v)
end.
End with_val_context.
End expr_param.
End language.
End Named.
Global Arguments Const {_ _ _ _ _} _.
Global Arguments Var {_ _ _ _ _} _.
Global Arguments SmartVar {_ _ _ _ _} _.
Global Arguments SmartConst {_ _ _ _ _} _.
Global Arguments Op {_ _ _ _ _ _} _ _.
Global Arguments LetIn {_ _ _ _} _ _ _ {_} _.
Global Arguments Pair {_ _ _ _ _} _ {_} _.
Global Arguments Return {_ _ _ _ _} _.
Global Arguments Abs {_ _ _ _ _ _} _ _.
Global Arguments extend {_ _ _ _} ctx {_} _ _.
Global Arguments remove {_ _ _ _} ctx {_} _.
Global Arguments lookup {_ _ _ _} ctx {_} _, {_ _ _ _} ctx _ _.
Global Arguments wff {_ _ _ _ _ _} ctx {t} _.
Global Arguments wf {_ _ _ _ _ _} ctx {t} _.
Global Arguments interp_genf {_ _ _ _ var _} _ _ _ _ _ _ {ctx t} _ _.
Global Arguments interpf {_ _ _ _ _ interp_op ctx t} _ _.
Global Arguments interp {_ _ _ _ _ interp_op ctx t} _ _.
Notation "'slet' x := A 'in' b" := (LetIn _ x A%nexpr b%nexpr) : nexpr_scope.
Notation "'λn' x .. y , t" := (Abs x .. (Abs y t%nexpr) .. ) : nexpr_scope.
Notation "( x , y , .. , z )" := (Pair .. (Pair x%nexpr y%nexpr) .. z%nexpr) : nexpr_scope.
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