Module Qcert.Translation.NRAtocNNRC


Section NRAtocNNRC.
  Require Import String.
  Require Import List.
  Require Import EquivDec.
  Require Import Compare_dec.
  Require Import Program.
  Require Import Utils.
  Require Import CommonRuntime.
  Require Import NRARuntime.
  Require Import cNNRCRuntime.

  Context {fruntime:foreign_runtime}.

Translation from NRA to Named Nested Relational Calculus

  Fixpoint nra_to_nnrc_core (op:nra) (var:var) : nnrc :=
    match op with
    | NRAGetConstant s =>
      NNRCGetConstant s
    | NRAID => NNRCVar var
    | NRAConst rd => NNRCConst rd
    | NRABinop bop op1 op2 =>
      NNRCBinop bop (nra_to_nnrc_core op1 var) (nra_to_nnrc_core op2 var)
    | NRAUnop uop op1 =>
      NNRCUnop uop (nra_to_nnrc_core op1 var)
    | NRAMap op1 op2 =>
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let t := fresh_var "tmap$" (var::nil) in
      NNRCFor t nnrc2 (nra_to_nnrc_core op1 t)
    | NRAMapProduct op1 op2 =>
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let (t1,t2) := fresh_var2 "tmc$" "tmc$" (var::nil) in
      NNRCUnop OpFlatten
               (NNRCFor t1 nnrc2
                        (NNRCFor t2 (nra_to_nnrc_core op1 t1)
                                 ((NNRCBinop OpRecConcat) (NNRCVar t1) (NNRCVar t2))))
    | NRAProduct op1 op2 =>
      let nnrc1 := (nra_to_nnrc_core op1 var) in
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let (t1,t2) := fresh_var2 "tprod$" "tprod$" (var::nil) in
      NNRCUnop OpFlatten
               (NNRCFor t1 nnrc1
                        (NNRCFor t2 nnrc2
                                 ((NNRCBinop OpRecConcat) (NNRCVar t1) (NNRCVar t2))))
    | NRASelect op1 op2 =>
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let t := fresh_var "tsel$" (var::nil) in
      let nnrc1 := (nra_to_nnrc_core op1 t) in
      NNRCUnop OpFlatten
               (NNRCFor t nnrc2
                        (NNRCIf nnrc1 (NNRCUnop OpBag (NNRCVar t)) (NNRCConst (dcoll nil))))
    | NRADefault op1 op2 =>
      let nnrc1 := (nra_to_nnrc_core op1 var) in
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let t := fresh_var "tdef$" (var::nil) in
      (NNRCLet t nnrc1
               (NNRCIf (NNRCBinop OpEqual
                                  (NNRCVar t)
                                  (NNRCUnop OpFlatten (NNRCConst (dcoll nil))))
                       nnrc2 (NNRCVar t)))
    | NRAEither opl opr =>
      let nnrcl := (nra_to_nnrc_core opl var) in
      let nnrcr := (nra_to_nnrc_core opr var) in
      NNRCEither (NNRCVar var) var nnrcl var nnrcr
    | NRAEitherConcat op1 op2 =>
      let nnrc1 := (nra_to_nnrc_core op1 var) in
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let t := fresh_var "ec$" (var::nil) in
      NNRCLet t nnrc2
              (NNRCEither nnrc1
                          var (NNRCUnop OpLeft (NNRCBinop OpRecConcat (NNRCVar var) (NNRCVar t)))
                          var (NNRCUnop OpRight (NNRCBinop OpRecConcat (NNRCVar var) (NNRCVar t))))
    | NRAApp op1 op2 =>
      let nnrc2 := (nra_to_nnrc_core op2 var) in
      let t := fresh_var "tapp$" (var::nil) in
      let nnrc1 := (nra_to_nnrc_core op1 t) in
      (NNRCLet t nnrc2 nnrc1)
    end.

Auxiliary lemmas used in the proof of correctness for the translation

  Lemma map_sem_correct (h:list (string*string)) (op:nra) cenv (l:list data) (env:bindings) (vid:var):
    (forall (did: data) (env : bindings) (vid : var),
        lookup equiv_dec env vid = Some did ->
        nnrc_core_eval h cenv env (nra_to_nnrc_core op vid) = (nra_eval h cenv op did)) ->
    lift_map
      (fun x : data =>
         nnrc_core_eval h cenv ((vid, x) :: env) (nra_to_nnrc_core op vid)) l
    =
    lift_map (nra_eval h cenv op) l.
Proof.
    intros.
    induction l.
    reflexivity.
    simpl in *.
    rewrite (H a ((vid, a) :: env) vid); simpl; trivial;
      try solve [match_destr; congruence].
  Qed.

Theorem 5.2: proof of correctness for the translation

  Opaque append.
  Theorem nra_sem_correct (h:list (string*string)) (cenv:bindings) (op:nra) (env:bindings) (vid:var) (did:data) :
    lookup equiv_dec env vid = Some did ->
    nnrc_core_eval h cenv env (nra_to_nnrc_core op vid) = hop @ₐ didcenv.
Proof.
    Opaque fresh_var.
    Hint Resolve fresh_var_fresh1 fresh_var_fresh2 fresh_var_fresh3 fresh_var2_distinct.
    revert did env vid.
    nra_cases (induction op) Case; intros; simpl.
    - Case "NRAGetConstant"%string.
      reflexivity.
    - Case "NRAID"%string.
      assumption.
    - Case "NRAConst"%string.
      reflexivity.
    - Case "NRABinop"%string.
      rewrite (IHop1 did env vid H); trivial.
      rewrite (IHop2 did env vid H); trivial.
    - Case "NRAUnop"%string.
       simpl; rewrite (IHop did env vid H); trivial.
    - Case "NRAMap"%string.
      simpl; rewrite (IHop2 did env vid H); clear IHop2.
      destruct (hop2 @ₐ didcenv); try reflexivity.
      destruct d; try reflexivity.
      rewrite (map_sem_correct h op1 cenv l env); try reflexivity; intros.
      auto.
    - Case "NRAMapProduct"%string.
      generalize (fresh_var2_distinct "tmc$" "tmc$" [vid]).
      simpl; intros dis.
      repeat (dest_eqdec; try congruence).
      rewrite (IHop2 did env vid H).
      simpl.
      destruct (hop2 @ₐ didcenv); try reflexivity; clear op2 IHop2.
      destruct d; try reflexivity.
      autorewrite with alg in *.
      apply lift_dcoll_inversion.
      induction l; try reflexivity; simpl.
      unfold omap_product in *; simpl.
      unfold oncoll_map_concat in *.
      rewrite <- IHl; clear IHl.
      rewrite (IHop1 a (((fresh_var "tmc$" [vid], a)) :: env) (fresh_var "tmc$" [vid])) at 1; clear IHop1; trivial.
      destruct (hop1 @ₐ acenv); try reflexivity.
      destruct d; try reflexivity.
      unfold omap_concat, orecconcat, rec_concat_sort.
      generalize (lift_map
       (fun d1 : data =>
        match a with
        | drec r1 =>
            match d1 with
            | drec r2 => Some (drec (rec_sort (r1 ++ r2)))
            | _ => None
            end
        | _ => None
        end) l0); intros.
      destruct o; try reflexivity.
      + rewrite oflatten_through_match.
        reflexivity.
      + simpl.
        dest_eqdec; try congruence.
    - Case "NRAProduct"%string.
      generalize (fresh_var2_distinct "tprod$" "tprod$" [vid]).
      simpl; rewrite (IHop1 did env vid H).
      intros dis.
      repeat (dest_eqdec; try congruence).
      simpl.
      destruct (hop1 @ₐ didcenv); try reflexivity; clear op1 IHop1.
      destruct d; try (destruct (hop2 @ₐ didcenv); reflexivity).
      autorewrite with alg in *.
      apply lift_dcoll_inversion.
      induction l; try reflexivity; simpl.
      unfold omap_product in *; simpl.
      unfold oncoll_map_concat in *.
      rewrite <- IHl; clear IHl.
      rewrite (IHop2 did ((fresh_var "tprod$" [vid], a) :: env) vid) at 1; clear IHop2; trivial.
      destruct (hop2 @ₐ didcenv); try reflexivity.
      destruct d; try reflexivity.
      unfold omap_concat, orecconcat, rec_concat_sort.
      generalize (lift_map
       (fun d1 : data =>
        match a with
        | drec r1 =>
            match d1 with
            | drec r2 => Some (drec (rec_sort (r1 ++ r2)))
            | _ => None
            end
        | _ => None
        end) l0); intros.
      destruct o; try reflexivity.
      + rewrite oflatten_through_match.
        reflexivity.
      + simpl.
        match_destr.
        elim (fresh_var_fresh1 _ _ _ e1).
    - Case "NRASelect"%string.
      simpl.
      rewrite (IHop2 did env vid H).
      destruct (hop2 @ₐ didcenv); try reflexivity. clear IHop2 op2.
      destruct d; try reflexivity.
      autorewrite with alg.
      apply lift_dcoll_inversion.
      induction l; try reflexivity; simpl.
      repeat (dest_eqdec; try congruence).
      simpl.
      rewrite <- IHl; clear IHl.
      rewrite (IHop1 a) at 1.
      destruct (hop1 @ₐ acenv); try (simpl;reflexivity).
      destruct d; simpl in *; try congruence.
      destruct b.
      rewrite oflatten_lift_cons_dcoll.
      reflexivity.
      rewrite match_lift_id.
      rewrite oflatten_lift_empty_dcoll.
      reflexivity.
      simpl.
      trivial.
      trivial.
      simpl. match_destr; congruence.
    - Case "NRADefault"%string.
      simpl. rewrite (IHop1 did env vid H).
      case_eq (hop1 @ₐ didcenv); intros; try reflexivity.
      repeat (dest_eqdec; try congruence).
      simpl.
      destruct (data_eq_dec d (dcoll [])); subst; simpl.
      + rewrite (IHop2 did); trivial.
        simpl; match_destr.
        elim (fresh_var_fresh1 _ _ _ e0).
      + destruct d; trivial. destruct l; congruence.
    - Case "NRAEither"%string.
      simpl. rewrite H. match_destr.
      + apply IHop1. simpl; trivial; match_destr; try congruence.
      + apply IHop2. simpl; trivial; match_destr; try congruence.
    - Case "NRAEitherConcat"%string.
      rewrite H.
      rewrite (IHop2 _ _ _ H).
      match_destr; [| repeat (match_destr; trivial)].
      rewrite <- (IHop1 did ((fresh_var "ec$" (vid :: nil), d) :: env) vid); simpl; trivial.
      + unfold var in *.
        destruct (nnrc_core_eval h cenv (_ :: env)); trivial.
        dest_eqdec; [| congruence].
        dest_eqdec; [elim (fresh_var_fresh1 _ _ _ (symmetry e0)) | ].
        dest_eqdec; [| congruence].
        match_destr; simpl; match_destr; match_destr.
      + match_destr.
        elim (fresh_var_fresh1 _ _ _ e).
    - Case "NRAApp"%string.
      simpl. rewrite (IHop2 did env vid H).
      case (hop2 @ₐ didcenv); intros; try reflexivity; simpl.
      rewrite (IHop1 d); simpl; try reflexivity.
      trivial. match_destr; trivial.
      congruence.
  Qed.

  Lemma nra_to_nnrc_core_is_core :
    forall q:nra, forall init_vid,
        nnrcIsCore (nra_to_nnrc_core q init_vid).
Proof.
    intro q; simpl.
    induction q; simpl; auto.
  Qed.

Lemma and proof of linear size translation

  Section Size.

    Require Import Omega.
    Require Import NNRC NNRCSize.

    Theorem nraToNNRC_size op v :
      nnrc_size (nra_to_nnrc_core op v) <= 10 * nra_size op.
Proof.
      revert v.
      induction op; simpl in *; intros; trivial.
      - omega.
      - omega.
      - omega.
      - specialize (IHop1 v); specialize (IHop2 v); omega.
      - specialize (IHop v); omega.
      - generalize (IHop1 (fresh_var "tmap$" [v]));
          generalize (IHop2 (fresh_var "tmap$" [v])).
        specialize (IHop1 v); specialize (IHop2 v); omega.
      - generalize (IHop1 (fresh_var "tmc$" [v])); generalize (IHop2 v).
        specialize (IHop1 v); specialize (IHop2 v); omega.
      - specialize (IHop1 v); specialize (IHop2 v); omega.
      - generalize (IHop1 (fresh_var "tsel$" [v])); generalize (IHop2 v).
        specialize (IHop1 v); specialize (IHop2 v); omega.
      - specialize (IHop1 v); specialize (IHop2 v); omega.
      - specialize (IHop1 v); specialize (IHop2 v); omega.
      - specialize (IHop1 v); specialize (IHop2 v); omega.
      - specialize (IHop2 v); specialize (IHop1 (fresh_var "tapp$" [v])); omega.
    Qed.

  End Size.

  Section Top.
    Definition init_vid := "id"%string.
    
    Definition nra_to_nnrc_base_top (q:nra) : nnrc :=
      (NNRCLet init_vid (NNRCConst dunit)
               (nra_to_nnrc_core q init_vid)).

    Lemma nra_to_nnrc_base_top_is_core :
      forall q:nra, nnrcIsCore (nra_to_nnrc_base_top q).
Proof.
      intros; simpl.
      split; [trivial| ].
      apply nra_to_nnrc_core_is_core.
    Qed.

    Program Definition nra_to_nnrc_core_top (q:nra) : nnrc_core :=
      exist _ (nra_to_nnrc_base_top q) _.
Next Obligation.
      apply nra_to_nnrc_base_top_is_core.
    Qed.

    Theorem nra_to_nnrc_core_top_correct
            (h:list (string*string)) (q:nra) (env:bindings) :
      nnrc_core_eval_top h (nra_to_nnrc_core_top q) env = nra_eval_top h q env.
Proof.
      intros.
      unfold nnrc_core_eval_top.
      unfold nra_eval_top.
      unfold nra_to_nnrc_core_top.
      unfold lift_nnrc_core.
      simpl.
      rewrite (nra_sem_correct h (rec_sort env) q
                               ((init_vid,dunit)::nil)
                               init_vid
                               dunit); try reflexivity.
    Qed.

  End Top.
  
End NRAtocNNRC.