Module Qcert.NNRC.Typing.TNNRC


Section TNNRC.
  Require Import String.
  Require Import List.
  Require Import Arith.
  Require Import Program.
  Require Import EquivDec.
  Require Import Morphisms.
  Require Import Utils.
  Require Import CommonSystem.
  Require Import cNNRC.
  Require Import NNRC.
  Require Import TcNNRC.

Typing rules for NNRC
  Section typ.
    Context {m:basic_model}.
    Contextconstants:tbindings).

    Definition nnrc_type (env:tbindings) (n:nnrc) (t:rtype) : Prop :=
      nnrc_core_type τconstants env (nnrc_to_nnrc_base n) t.
  End typ.
  
Main lemma for the type correctness of NNNRC

  Theorem typed_nnrc_yields_typed_data {m:basic_model} {τcenv} {τ} (cenv env:bindings) (tenv:tbindings) (e:nnrc) :
    bindings_type cenv τcenv ->
    bindings_type env tenv ->
    nnrc_type τcenv tenv e τ ->
    (exists x, (@nnrc_eval _ brand_relation_brands cenv env e) = Some x /\ (data_type x τ)).
Proof.
    intros.
    unfold nnrc_eval.
    unfold nnrc_type in H1.
    apply (@typed_nnrc_core_yields_typed_data _ τcenv _ cenv env tenv).
    assumption.
    assumption.
    assumption.
  Qed.

  Global Instance nnrc_type_lookup_equiv_prop {m:basic_model} :
    Proper (eq ==> lookup_equiv ==> eq ==> eq ==> iff) nnrc_type.
Proof.
    generalize nnrc_core_type_lookup_equiv_prop; intro Hnnrc_prop.
    unfold Proper, respectful, lookup_equiv, iff, impl in *; intros; subst.
    apply Hnnrc_prop; try reflexivity;
      try assumption.
  Qed.

End TNNRC.

Ltac nnrc_inverter :=
  unfold nnrc_type, nnrc_to_nnrc_base in *; simpl in *; try nnrc_core_inverter.

Ltac nnrc_input_well_typed :=
  repeat progress
         match goal with
         | [HO:nnrc_typec ?Γ ?opout,
               HE:bindings_type ?env
            |- context [(nnrc_eval brand_relation_brands ?cenv ?env ?op)]] =>
           let xout := fresh "dout" in
           let xtype := freshout" in
           let xeval := fresh "eout" in
           destruct (typed_nnrc_yields_typed_data env Γ op HE HO)
             as [xout [xeval xtype]]; rewrite xeval in *; simpl
         end.