216 lines
5.8 KiB
Coq
216 lines
5.8 KiB
Coq
(* This module defines the typing relation
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* where each expression is assigned a type.
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*)
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From Coq Require Import Strings.String.
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Require Import terms.
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Require Import subst.
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Require Import equiv.
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Require Import subtype.
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Include Terms.
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Include Subst.
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Include Equiv.
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Include Subtype.
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Module Typing.
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(** Typing Derivation *)
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Inductive context : Type :=
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| ctx_assign : string -> type_term -> context -> context
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| ctx_empty : context
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.
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Inductive context_contains : context -> string -> type_term -> Prop :=
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| C_take : forall (x:string) (X:type_term) (Γ:context),
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(context_contains (ctx_assign x X Γ) x X)
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| C_shuffle : forall x X y Y Γ,
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(context_contains Γ x X) ->
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(context_contains (ctx_assign y Y Γ) x X).
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Reserved Notation "Gamma '|-' x '\is' X" (at level 101, x at next level, X at level 0).
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Reserved Notation "Gamma '|-' x '\compatible' X" (at level 101, x at next level, X at level 0).
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Inductive expr_type : context -> expr_term -> type_term -> Prop :=
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| T_Var : forall Γ x τ,
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(context_contains Γ x τ) ->
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(Γ |- (expr_var x) \is τ)
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| T_Let : forall Γ s (σ:type_term) t τ x,
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(Γ |- s \is σ) ->
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(Γ |- t \is τ) ->
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(Γ |- (expr_let x σ s t) \is τ)
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| T_TypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
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Γ |- e \is τ ->
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Γ |- (expr_ty_abs α e) \is (type_univ α τ)
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| T_TypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
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Γ |- e \is (type_univ α τ) ->
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Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)
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| T_Abs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
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(context_contains Γ x σ) ->
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Γ |- t \is τ ->
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Γ |- (expr_abs x σ t) \is (type_fun σ τ)
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| T_App : forall (Γ:context) (f:expr_term) (a:expr_term) (σ:type_term) (τ:type_term),
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Γ |- f \is (type_fun σ τ) ->
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Γ |- a \is σ ->
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Γ |- (expr_app f a) \is τ
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| T_Sub : forall Γ x τ τ',
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Γ |- x \is τ ->
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(τ :<= τ') ->
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Γ |- x \is τ'
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where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
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Inductive expr_type_compatible : context -> expr_term -> type_term -> Prop :=
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| T_CompatVar : forall Γ x τ,
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(context_contains Γ x τ) ->
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(Γ |- (expr_var x) \compatible τ)
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| T_CompatLet : forall Γ s (σ:type_term) t τ x,
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(Γ |- s \compatible σ) ->
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(Γ |- t \compatible τ) ->
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(Γ |- (expr_let x σ s t) \compatible τ)
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| T_CompatTypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
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Γ |- e \compatible τ ->
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Γ |- (expr_ty_abs α e) \compatible (type_univ α τ)
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| T_CompatTypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
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Γ |- e \compatible (type_univ α τ) ->
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Γ |- (expr_ty_app e σ) \compatible (type_subst α σ τ)
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| T_CompatMorphAbs : forall Γ x t τ τ',
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Γ |- t \compatible τ ->
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(τ ~<= τ') ->
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Γ |- (expr_morph x τ t) \compatible (type_morph τ τ')
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| T_CompatAbs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
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(context_contains Γ x σ) ->
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Γ |- t \compatible τ ->
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Γ |- (expr_abs x σ t) \compatible (type_fun σ τ)
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| T_CompatApp : forall Γ f a σ τ,
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(Γ |- f \compatible (type_fun σ τ)) ->
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(Γ |- a \compatible σ) ->
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(Γ |- (expr_app f a) \compatible τ)
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| T_CompatImplicitCast : forall Γ h x τ τ',
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(context_contains Γ h (type_morph τ τ')) ->
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(Γ |- x \compatible τ) ->
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(Γ |- x \compatible τ')
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| T_CompatSub : forall Γ x τ τ',
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(Γ |- x \compatible τ) ->
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(τ ~<= τ') ->
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(Γ |- x \compatible τ')
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where "Γ '|-' x '\compatible' τ" := (expr_type_compatible Γ x τ).
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Definition is_well_typed (e:expr_term) : Prop :=
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exists Γ τ,
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Γ |- e \compatible τ
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.
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(* Examples *)
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Example typing1 :
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forall Γ,
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(context_contains Γ "x" [< %"T"% >]) ->
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Γ |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
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(* Γ |- [ ΛT ↦ λx:T ↦ x ] : ∀T.(T->T) *)
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Proof.
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intros.
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apply T_TypeAbs.
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apply T_Abs.
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apply H.
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apply T_Var.
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apply H.
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Qed.
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Example typing2 :
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forall Γ,
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(context_contains Γ "x" [< %"T"% >]) ->
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Γ |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
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Proof.
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intros.
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apply T_Sub with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]).
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apply T_TypeAbs.
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apply T_Abs.
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apply H.
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apply T_Var.
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apply H.
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apply TSubRepr_Refl.
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apply TEq_Alpha.
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apply TAlpha_Rename.
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apply TSubst_Fun.
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apply TSubst_VarReplace.
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apply TSubst_VarReplace.
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Qed.
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Example typing3 :
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forall Γ,
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(context_contains Γ "x" [< %"T"% >]) ->
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(context_contains Γ "y" [< %"U"% >]) ->
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Γ |- [{
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Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"%
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}] \is [<
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∀"S",∀"T",(%"S"%->%"T"%->%"T"%)
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>].
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Proof.
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intros.
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apply T_Sub with (τ:=[< ∀"T",∀"U",(%"T"%->%"U"%->%"U"%) >]) (τ':=[< ∀"S",∀"T",(%"S"%->%"T"%->%"T"%) >]).
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apply T_TypeAbs, T_TypeAbs, T_Abs.
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apply H.
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apply T_Abs.
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apply H0.
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apply T_Var, H0.
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apply TSubRepr_Refl.
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apply TEq_Trans with (y:= [< ∀"S",∀"U",(%"S"%->%"U"%->%"U"%) >] ).
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apply TEq_Alpha.
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apply TAlpha_Rename.
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apply TSubst_UnivReplace. discriminate.
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easy.
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apply TSubst_Fun.
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apply TSubst_VarReplace.
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apply TSubst_Fun.
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apply TSubst_VarKeep. discriminate.
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apply TSubst_VarKeep. discriminate.
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apply TEq_Alpha.
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apply TAlpha_SubUniv.
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apply TAlpha_Rename.
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apply TSubst_Fun.
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apply TSubst_VarKeep. discriminate.
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apply TSubst_Fun.
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apply TSubst_VarReplace.
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apply TSubst_VarReplace.
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Qed.
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Example typing4 : (is_well_typed
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[{ Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"x"% }]
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).
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Proof.
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unfold is_well_typed.
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exists (ctx_assign "x" [< %"T"% >]
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(ctx_assign "y" [< %"U"% >] ctx_empty)).
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exists [< ∀"T",∀"U",%"T"%->%"U"%->%"T"% >].
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apply T_CompatTypeAbs.
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apply T_CompatTypeAbs.
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apply T_CompatAbs.
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apply C_take.
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apply T_CompatAbs.
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apply C_shuffle. apply C_take.
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apply T_CompatVar.
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apply C_take.
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Qed.
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End Typing.
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