From Coq Require Import Strings.String.
Require Import terms.
Require Import subst.
Require Import equiv.
Require Import subtype.
Require Import typing.

Include Terms.
Include Subst.
Include Equiv.
Include Subtype.
Include Typing.

Module Morph.


Reserved Notation "Γ '|-' '[[' e ']]' '=' t"  (at level 101, e at next level, t at level 0).

Inductive expand_morphisms : context -> expr_term -> expr_term -> Prop :=

  | Expand_Transform : forall Γ e h τ τ',
    (context_contains Γ h (type_morph τ τ')) ->
    (Γ |- e \is τ) ->
    (Γ |- [[ e ]] = (expr_app (expr_var h) e))

  | Expand_Otherwise : forall Γ e,
    (Γ |- [[ e ]] = e)

  | Expand_App : forall Γ f f' a a' σ τ,
    (Γ |- f \compatible (type_fun σ τ)) ->
    (Γ |- a \compatible σ) ->
    (Γ |- [[ f ]] = f') ->
    (Γ |- [[ a ]] = a') ->
    (Γ |- [[ (expr_app f a) ]] = (expr_app f a'))

where "Γ '|-' '[[' e ']]' '=' t" := (expand_morphisms Γ e t).


(* For every well-typed term, there is an expanded exactly typed term *)
Lemma well_typed_is_expandable :
  forall (e:expr_term),
  (is_well_typed e) ->
  exists Γ e', (expand_morphisms Γ e e') -> (is_exactly_typed e')
.
Proof.
Admitted.


Example expand1 :
  (ctx_assign "morph-radix"
      [< <%"PosInt"% %"16"%> ->morph <%"PosInt"% %"10"%> >]
  (ctx_assign "fn-decimal"
      [< <%"PosInt"% %"10"%> -> <%"PosInt"% %"10"%> >]
  (ctx_assign "x"
      [< <%"PosInt"% %"16"%> >]
   ctx_empty)))

  |- [[ (expr_app (expr_var "fn-decimal") (expr_var "x")) ]]

  =  (expr_app (expr_var "fn-decimal")
      (expr_app (expr_var "morph-radix") (expr_var "x")))
.
Proof.
  apply Expand_App with (f':=(expr_var "fn-decimal")) (σ:=[< < %"PosInt"% %"10"% > >]) (τ:=[< < %"PosInt"% %"10"% > >]).
  apply TCompat_Native.
Admitted.

End Morph.