ladder-calculus/coq/typing.v

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

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

Module Typing.

(** Typing Derivation *)

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

Inductive context_contains : context -> string -> type_term -> Prop :=
  | C_take : forall (x:string) (X:type_term) (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 '\is' X"  (at level 101, x at next level, X at level 0).
Reserved Notation "Gamma '|-' x '\compatible' X"  (at level 101, x at next level, X at level 0).

Inductive expr_type : context -> expr_term -> type_term -> Prop :=
  | T_Var : forall Γ x τ,
    (context_contains Γ x τ) ->
    (Γ |- (expr_var x) \is τ)

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

  | T_TypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
    Γ |- e \is τ ->
    Γ |- (expr_ty_abs α e) \is (type_univ α τ)

  | T_TypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
    Γ |- e \is (type_univ α τ) ->
    Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)

  | T_Abs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
    (context_contains Γ x σ) ->
    Γ |- t \is τ ->
    Γ |- (expr_abs x σ t) \is (type_fun σ τ)

  | T_App : forall (Γ:context) (f:expr_term) (a:expr_term) (σ:type_term) (τ:type_term),
    Γ |- f \is (type_fun σ τ) ->
    Γ |- a \is σ ->
    Γ |- (expr_app f a) \is τ

  | T_Sub : forall Γ x τ τ',
    Γ |- x \is τ ->
    (τ :<= τ') ->
    Γ |- x \is τ'

where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).


Inductive expr_type_compatible : context -> expr_term -> type_term -> Prop :=
  | T_CompatVar : forall Γ x τ,
    (context_contains Γ x τ) ->
    (Γ |- (expr_var x) \compatible τ)

  | T_CompatLet : forall Γ s (σ:type_term) t τ x,
    (Γ |- s \compatible σ) ->
    (Γ |- t \compatible τ) ->
    (Γ |- (expr_let x σ s t) \compatible τ)

  | T_CompatTypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
    Γ |- e \compatible τ ->
    Γ |- (expr_ty_abs α e) \compatible (type_univ α τ)

  | T_CompatTypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
    Γ |- e \compatible (type_univ α τ) ->
    Γ |- (expr_ty_app e σ) \compatible (type_subst α σ τ)

  | T_CompatMorphAbs : forall Γ x t τ τ',
    Γ |- t \compatible τ ->
    (τ ~<= τ') ->
    Γ |- (expr_morph x τ t) \compatible (type_morph τ τ')

  | T_CompatAbs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
    (context_contains Γ x σ) ->
    Γ |- t \compatible τ ->
    Γ |- (expr_abs x σ t) \compatible (type_fun σ τ)

  | T_CompatApp : forall Γ f a σ τ,
    (Γ |- f \compatible (type_fun σ τ)) ->
    (Γ |- a \compatible σ) ->
    (Γ |- (expr_app f a) \compatible τ)

  | T_CompatImplicitCast : forall Γ h x τ τ',
    (context_contains Γ h (type_morph τ τ')) ->
    (Γ |- x \compatible τ) ->
    (Γ |- x \compatible τ')

  | T_CompatSub : forall Γ x τ τ',
    (Γ |- x \compatible τ) ->
    (τ ~<= τ') ->
    (Γ |- x \compatible τ')

where "Γ '|-' x '\compatible' τ" := (expr_type_compatible Γ x τ).

Definition is_well_typed (e:expr_term) : Prop :=
  exists Γ τ,
  Γ |- e \compatible τ
.

(* Examples *)

Example typing1 :
  forall Γ,
  (context_contains Γ "x" [ %"T"% ]) ->
  Γ |- [[ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% ]] \is [ ∀"T", %"T"% -> %"T"% ].
   (* Γ |- [ ΛT ↦ λx:T ↦ x ] : ∀T.(T->T) *)
Proof.
  intros.
  apply T_TypeAbs.
  apply T_Abs.
  apply H.
  apply T_Var.
  apply H.
Qed.

Example typing2 :
  forall Γ,
  (context_contains Γ "x" [ %"T"% ]) ->
  Γ |- [[ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% ]] \is [ ∀"U", %"U"% -> %"U"% ].
Proof.
  intros.
  apply T_Sub with (τ:=[∀"T",(%"T"% -> %"T"%)]).
  apply T_TypeAbs.
  apply T_Abs.
  apply H.
  apply T_Var.
  apply H.

  apply TSubRepr_Refl.
  apply TEq_Alpha.
  apply TAlpha_Rename.
  apply TSubst_Fun.
  apply TSubst_VarReplace.
  apply TSubst_VarReplace.
Qed.

Example typing3 :
  forall Γ,
  (context_contains Γ "x" [ %"T"% ]) ->
  (context_contains Γ "y" [ %"U"% ]) ->
  Γ |- [[ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"% ]] \is [ ∀"S",∀"T",(%"S"%->%"T"%->%"T"%) ].
Proof.
  intros.
  apply T_Sub with (τ:=[∀"T",∀"U",(%"T"%->%"U"%->%"U"%)]) (τ':=[∀"S",∀"T",(%"S"%->%"T"%->%"T"%)]).
  apply T_TypeAbs, T_TypeAbs, T_Abs.
  apply H.
  apply T_Abs.
  apply H0.
  apply T_Var, H0.

  apply TSubRepr_Refl.
  apply TEq_Trans with (y:= [∀"S",∀"U",(%"S"%->%"U"%->%"U"%)] ).
  apply TEq_Alpha.
  apply TAlpha_Rename.
  apply TSubst_UnivReplace. discriminate.
  easy.
  apply TSubst_Fun.
  apply TSubst_VarReplace.
  apply TSubst_Fun.
  apply TSubst_VarKeep. discriminate.
  apply TSubst_VarKeep. discriminate.

  apply TEq_Alpha.
  apply TAlpha_SubUniv.
  apply TAlpha_Rename.
  apply TSubst_Fun.
  apply TSubst_VarKeep. discriminate.
  apply TSubst_Fun.
  apply TSubst_VarReplace.
  apply TSubst_VarReplace.
Qed.


Example typing4 : (is_well_typed
  [[ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% ]]
).
Proof.
  unfold is_well_typed.
  exists (ctx_assign "x" [%"T"%]
            (ctx_assign "y" [%"U"%] ctx_empty)).
  exists [ ∀"T",∀"U",%"T"%->%"U"%->%"T"% ].
  apply T_CompatTypeAbs.
  apply T_CompatTypeAbs.
  apply T_CompatAbs.
  apply C_take.
  apply T_CompatAbs.
  apply C_shuffle. apply C_take.
  apply T_CompatVar.
  apply C_take.
Qed.

End Typing.