(* This module defines the typing relation
 * where each expression is assigned a type.
 *)
From Coq Require Import Strings.String.
Require Import terms.

Include Terms.


Module Typing.

Inductive context : Type :=
  | ctx_assign : string -> ladder_type -> context -> context
  | ctx_empty : context
.

Inductive context_contains : context -> string -> ladder_type -> Prop :=
  | C_take : forall (x:string) (X:ladder_type) (Γ:context),
    (context_contains (ctx_assign x X Γ) x X)
  | C_shuffle : forall x X y Y Γ,
    (context_contains Γ x X) ->
    (context_contains (ctx_assign y Y Γ) x X).

Reserved Notation "Gamma '|-' x '\in' X"  (at level 101, x at next level, X at level 0).

Inductive expr_type : context -> expr -> ladder_type -> Prop :=
  | T_Var : forall Γ x X,
    (context_contains Γ x X) ->
    Γ |- x \in X

  | T_Let : forall Γ s (σ:ladder_type) t τ x,
    Γ |- s \in σ ->
    Γ |- t \in τ ->
    Γ |- (expr_let x σ s t) \in τ

  | T_Abs : forall (Γ:context) (x:string) (X:ladder_type) (t:expr) (T:ladder_type),
    Γ |- t \in T ->
    Γ |- (expr_tm_abs x X t) \in (type_fun X T)

  | T_App : forall (Γ:context) (f:expr) (a:expr) (S:ladder_type) (T:ladder_type),
    Γ |- f \in (type_fun S T) ->
    Γ |- a \in S ->
    Γ |- (expr_tm_app f a) \in T

where "Γ '|-' x '\in' X" := (expr_type Γ x X).


Example typing1 :
  ctx_empty |-
  (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \in
    (type_abs "T" (type_fun (type_var "T") (type_var "T"))).
Proof.
Admitted.

End Typing.