Generic.v

(* Copyright (c) 2008, Adam Chlipala
 * 
 * This work is licensed under a
 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
 * Unported License.
 * The license text is available at:
 *   http://creativecommons.org/licenses/by-nc-nd/3.0/
 *)

(* begin hide *)
Require Import String List.

Require Import Tactics DepList.

Set Implicit Arguments.
(* end hide *)


(** %\part{Chapters to be Moved Earlier}

\chapter{Generic Programming}% *)

(** TODO: Prose for this chapter *)


(** * Reflecting Datatype Definitions *)

(* EX: Define a reflected representation of simple algebraic datatypes. *)

(* begin thide *)
Record constructor : Type := Con {
  nonrecursive : Type;
  recursive : nat
}.

Definition datatype := list constructor.

Definition Empty_set_dt : datatype := nil.
Definition unit_dt : datatype := Con unit 0 :: nil.
Definition bool_dt : datatype := Con unit 0 :: Con unit 0 :: nil.
Definition nat_dt : datatype := Con unit 0 :: Con unit 1 :: nil.
Definition list_dt (A : Type) : datatype := Con unit 0 :: Con A 1 :: nil.
(* end thide *)

Section tree.
  Variable A : Type.

  Inductive tree : Type :=
  | Leaf : A -> tree
  | Node : tree -> tree -> tree.
End tree.

(* begin thide *)
Definition tree_dt (A : Type) : datatype := Con A 0 :: Con unit 2 :: nil.

Section denote.
  Variable T : Type.

  Definition constructorDenote (c : constructor) :=
    nonrecursive c -> ilist T (recursive c) -> T.

  Definition datatypeDenote := hlist constructorDenote.
End denote.
(* end thide *)

Notation "[ ! , ! ~> x ]" := ((fun _ _ => x) : constructorDenote _ (Con _ _)).
Notation "[ v , ! ~> x ]" := ((fun v _ => x) : constructorDenote _ (Con _ _)).
Notation "[ ! , r # n ~> x ]" := ((fun _ r => x) : constructorDenote _ (Con _ n)).
Notation "[ v , r # n ~> x ]" := ((fun v r => x) : constructorDenote _ (Con _ n)).

(* begin thide *)
Definition Empty_set_den : datatypeDenote Empty_set Empty_set_dt :=
  hnil.
Definition unit_den : datatypeDenote unit unit_dt :=
  [!, ! ~> tt] ::: hnil.
Definition bool_den : datatypeDenote bool bool_dt :=
  [!, ! ~> true] ::: [!, ! ~> false] ::: hnil.
Definition nat_den : datatypeDenote nat nat_dt :=
  [!, ! ~> O] ::: [!, r # 1 ~> S (hd r)] ::: hnil.
Definition list_den (A : Type) : datatypeDenote (list A) (list_dt A) :=
  [!, ! ~> nil] ::: [x, r # 1 ~> x :: hd r] ::: hnil.
Definition tree_den (A : Type) : datatypeDenote (tree A) (tree_dt A) :=
  [v, ! ~> Leaf v] ::: [!, r # 2 ~> Node (hd r) (hd (tl r))] ::: hnil.
(* end thide *)


(** * Recursive Definitions *)

(* EX: Define a generic [size] function. *)

(* begin thide *)
Definition fixDenote (T : Type) (dt : datatype) :=
  forall (R : Type), datatypeDenote R dt -> (T -> R).

Definition size T dt (fx : fixDenote T dt) : T -> nat :=
  fx nat (hmake (B := constructorDenote nat) (fun _ _ r => foldr plus 1 r) dt).

Definition Empty_set_fix : fixDenote Empty_set Empty_set_dt :=
  fun R _ emp => match emp with end.
Eval compute in size Empty_set_fix.

Definition unit_fix : fixDenote unit unit_dt :=
  fun R cases _ => (fst cases) tt inil.
Eval compute in size unit_fix.

Definition bool_fix : fixDenote bool bool_dt :=
  fun R cases b => if b
    then (fst cases) tt inil
    else (fst (snd cases)) tt inil.
Eval compute in size bool_fix.

Definition nat_fix : fixDenote nat nat_dt :=
  fun R cases => fix F (n : nat) : R :=
    match n with
      | O => (fst cases) tt inil
      | S n' => (fst (snd cases)) tt (icons (F n') inil)
    end.
Eval cbv beta iota delta -[plus] in size nat_fix.

Definition list_fix (A : Type) : fixDenote (list A) (list_dt A) :=
  fun R cases => fix F (ls : list A) : R :=
    match ls with
      | nil => (fst cases) tt inil
      | x :: ls' => (fst (snd cases)) x (icons (F ls') inil)
    end.
Eval cbv beta iota delta -[plus] in fun A => size (@list_fix A).

Definition tree_fix (A : Type) : fixDenote (tree A) (tree_dt A) :=
  fun R cases => fix F (t : tree A) : R :=
    match t with
      | Leaf x => (fst cases) x inil
      | Node t1 t2 => (fst (snd cases)) tt (icons (F t1) (icons (F t2) inil))
    end.
Eval cbv beta iota delta -[plus] in fun A => size (@tree_fix A).
(* end thide *)


(** ** Pretty-Printing *)

(* EX: Define a generic pretty-printing function. *)

(* begin thide *)
Record print_constructor (c : constructor) : Type := PI {
  printName : string;
  printNonrec : nonrecursive c -> string
}.

Notation "^" := (PI (Con _ _)).

Definition print_datatype := hlist print_constructor.

Open Local Scope string_scope.

Definition print T dt (pr : print_datatype dt) (fx : fixDenote T dt) : T -> string :=
  fx string (hmap (B1 := print_constructor) (B2 := constructorDenote string)
    (fun _ pc x r => printName pc ++ "(" ++ printNonrec pc x
      ++ foldr (fun s acc => ", " ++ s ++ acc) ")" r) pr).
(* end thide *)

Eval compute in print hnil Empty_set_fix.
Eval compute in print (^ "tt" (fun _ => "") ::: hnil) unit_fix.
Eval compute in print (^ "true" (fun _ => "")
  ::: ^ "false" (fun _ => "")
  ::: hnil) bool_fix.

Definition print_nat := print (^ "O" (fun _ => "")
  ::: ^ "S" (fun _ => "")
  ::: hnil) nat_fix.
Eval cbv beta iota delta -[append] in print_nat.
Eval simpl in print_nat 0.
Eval simpl in print_nat 1.
Eval simpl in print_nat 2.

Eval cbv beta iota delta -[append] in fun A (pr : A -> string) =>
  print (^ "nil" (fun _ => "")
  ::: ^ "cons" pr
  ::: hnil) (@list_fix A).
Eval cbv beta iota delta -[append] in fun A (pr : A -> string) =>
  print (^ "Leaf" pr
  ::: ^ "Node" (fun _ => "")
  ::: hnil) (@tree_fix A).


(** ** Mapping *)

(* EX: Define a generic [map] function. *)

(* begin thide *)
Definition map T dt (dd : datatypeDenote T dt) (fx : fixDenote T dt) (f : T -> T) : T -> T :=
  fx T (hmap (B1 := constructorDenote T) (B2 := constructorDenote T)
    (fun _ c x r => f (c x r)) dd).
(* end thide *)

Eval compute in map Empty_set_den Empty_set_fix.
Eval compute in map unit_den unit_fix.
Eval compute in map bool_den bool_fix.
Eval compute in map nat_den nat_fix.
Eval compute in fun A => map (list_den A) (@list_fix A).
Eval compute in fun A => map (tree_den A) (@tree_fix A).

Definition map_nat := map nat_den nat_fix.
Eval simpl in map_nat S 0.
Eval simpl in map_nat S 1.
Eval simpl in map_nat S 2.


(** * Proving Theorems about Recursive Definitions *)

(* begin thide *)
Section ok.
  Variable T : Type.
  Variable dt : datatype.

  Variable dd : datatypeDenote T dt.
  Variable fx : fixDenote T dt.

  Definition datatypeDenoteOk :=
    forall P : T -> Prop,
      (forall c (m : member c dt) (x : nonrecursive c) (r : ilist T (recursive c)),
        (forall i : index (recursive c), P (get r i))
        -> P ((hget dd m) x r))
      -> forall v, P v.

  Definition fixDenoteOk :=
    forall (R : Type) (cases : datatypeDenote R dt)
      c (m : member c dt)
      (x : nonrecursive c) (r : ilist T (recursive c)),
      fx R cases ((hget dd m) x r)
      = (hget cases m) x (imap (fx R cases) r).
End ok.

Implicit Arguments datatypeDenoteOk [T dt].

Lemma foldr_plus : forall n (ils : ilist nat n),
  foldr plus 1 ils > 0.
  induction n; crush.
  generalize (IHn b); crush.
Qed.
(* end thide *)

Theorem size_positive : forall T dt
  (dd : datatypeDenote T dt) (fx : fixDenote T dt)
  (dok : datatypeDenoteOk dd) (fok : fixDenoteOk dd fx)
  (v : T),
  size fx v > 0.
(* begin thide *)
  Hint Rewrite hget_hmake : cpdt.
  Hint Resolve foldr_plus.
 
  unfold size; intros; pattern v; apply dok; crush.
Qed.
(* end thide *)

Theorem map_id : forall T dt
  (dd : datatypeDenote T dt) (fx : fixDenote T dt)
  (dok : datatypeDenoteOk dd) (fok : fixDenoteOk dd fx)
  (v : T),
  map dd fx (fun x => x) v = v.
(* begin thide *)
  Hint Rewrite hget_hmap : cpdt.

  unfold map; intros; pattern v; apply dok; crush.
  match goal with
    | [ |- hget _ _ _ ?R1 = hget _ _ _ ?R2 ] => replace R1 with R2
  end; crush.

  induction (recursive c); crush.
  destruct r; reflexivity.
  destruct r; crush.
  rewrite (H None).
  unfold icons.
  f_equal.
  apply IHn; crush.
  apply (H (Some i0)).
Qed.
(* end thide *)