283 lines
8 KiB
Coq
283 lines
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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Require Import context.
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Require Import morph.
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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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Include Context.
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Include Morph.
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Module Typing.
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(** Typing Derivation *)
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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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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 t τ x,
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(Γ |- s \is σ) ->
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((ctx_assign x σ Γ) |- t \is τ) ->
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(Γ |- (expr_let x s t) \is τ)
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| T_TypeAbs : forall Γ e τ 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 σ τ τ',
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Γ |- e \is (type_univ α τ) ->
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(type_subst1 α σ τ τ') ->
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Γ |- (expr_ty_app e σ) \is τ'
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| T_Abs : forall Γ x σ t τ,
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((ctx_assign x σ Γ) |- t \is τ) ->
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(Γ |- (expr_abs x σ t) \is (type_fun σ τ))
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| T_MorphAbs : forall Γ x σ e τ,
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((ctx_assign x σ Γ) |- e \is τ) ->
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Γ |- (expr_morph x σ e) \is (type_morph σ τ)
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| T_App : forall Γ f a σ' σ τ,
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(Γ |- f \is (type_fun σ τ)) ->
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(Γ |- a \is σ') ->
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(Γ |- σ' ~> σ) ->
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(Γ |- (expr_app f a) \is τ)
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| T_MorphFun : forall Γ f σ τ,
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Γ |- f \is (type_morph σ τ) ->
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Γ |- f \is (type_fun σ τ)
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| T_Ascend : forall Γ e τ τ',
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(Γ |- e \is τ) ->
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(Γ |- (expr_ascend τ' e) \is (type_ladder τ' τ))
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| T_DescendImplicit : 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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| T_Descend : forall Γ x τ τ',
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Γ |- x \is τ ->
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(τ :<= τ') ->
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Γ |- (expr_descend τ' x) \is τ'
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where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
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Definition is_well_typed (e:expr_term) : Prop :=
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forall Γ,
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exists τ,
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Γ |- e \is τ
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.
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Inductive translate_typing : context -> expr_term -> type_term -> expr_term -> Prop :=
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| Expand_Var : forall Γ x τ,
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(Γ |- (expr_var x) \is τ) ->
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(translate_typing Γ (expr_var x) τ (expr_var x))
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| Expand_Let : forall Γ x e e' t t' σ τ,
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(Γ |- e \is σ) ->
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((ctx_assign x σ Γ) |- t \is τ) ->
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(translate_typing Γ e σ e') ->
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(translate_typing (ctx_assign x σ Γ) t τ t') ->
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(translate_typing Γ (expr_let x e t) τ (expr_let x e' t'))
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| Expand_TypeAbs : forall Γ α e e' τ,
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(Γ |- e \is τ) ->
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(translate_typing Γ e τ e') ->
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(translate_typing Γ
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(expr_ty_abs α e)
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(type_univ α τ)
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(expr_ty_abs α e'))
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| Expand_TypeApp : forall Γ e e' σ α τ,
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(Γ |- e \is (type_univ α τ)) ->
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(translate_typing Γ e τ e') ->
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(translate_typing Γ (expr_ty_app e τ) (type_subst α σ τ) (expr_ty_app e' τ))
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| Expand_Abs : forall Γ x σ e e' τ,
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((ctx_assign x σ Γ) |- e \is τ) ->
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(Γ |- (expr_abs x σ e) \is (type_fun σ τ)) ->
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(translate_typing (ctx_assign x σ Γ) e τ e') ->
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(translate_typing Γ (expr_abs x σ e) (type_fun σ τ) (expr_abs x σ e'))
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| Expand_MorphAbs : forall Γ x σ e e' τ,
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((ctx_assign x σ Γ) |- e \is τ) ->
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(Γ |- (expr_abs x σ e) \is (type_fun σ τ)) ->
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(translate_typing (ctx_assign x σ Γ) e τ e') ->
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(translate_typing Γ (expr_morph x σ e) (type_morph σ τ) (expr_morph x σ e'))
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| Expand_App : forall Γ f f' a a' m σ τ σ',
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(Γ |- f \is (type_fun σ τ)) ->
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(Γ |- a \is σ') ->
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(Γ |- σ' ~> σ) ->
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(translate_typing Γ f (type_fun σ τ) f') ->
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(translate_typing Γ a σ' a') ->
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(translate_morphism_path Γ σ' σ m) ->
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(translate_typing Γ (expr_app f a) τ (expr_app f' (expr_app m a')))
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| Expand_MorphFun : forall Γ f f' σ τ,
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(Γ |- f \is (type_morph σ τ)) ->
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(Γ |- f \is (type_fun σ τ)) ->
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(translate_typing Γ f (type_morph σ τ) f') ->
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(translate_typing Γ f (type_fun σ τ) f')
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| Expand_Ascend : forall Γ e e' τ τ',
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(Γ |- e \is τ) ->
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(Γ |- (expr_ascend τ' e) \is (type_ladder τ' τ)) ->
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(translate_typing Γ e τ e') ->
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(translate_typing Γ (expr_ascend τ' e) (type_ladder τ' τ) (expr_ascend τ' e'))
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| Expand_Descend : forall Γ e e' τ τ',
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(Γ |- e \is τ) ->
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(τ :<= τ') ->
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(Γ |- e \is τ') ->
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(translate_typing Γ e τ e') ->
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(translate_typing Γ e τ' e')
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.
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(* Examples *)
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Example typing1 :
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ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"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 T_Var.
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apply C_take.
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Qed.
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Example typing2 :
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ctx_empty |- [{ 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_DescendImplicit with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]).
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apply T_TypeAbs.
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apply T_Abs.
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apply T_Var.
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apply C_take.
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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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ctx_empty |- [{
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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_DescendImplicit 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 T_Abs.
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apply T_Var.
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apply C_take.
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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 [< ∀"T",∀"U",%"T"%->%"U"%->%"T"% >].
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apply T_TypeAbs.
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apply T_TypeAbs.
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apply T_Abs.
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apply T_Abs.
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apply T_Var.
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apply C_shuffle, C_take.
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Qed.
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Open Scope ladder_expr_scope.
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Example typing5 :
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(ctx_assign "60" [< $"ℝ"$ >]
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(ctx_assign "360" [< $"ℝ"$ >]
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(ctx_assign "/" [< $"ℝ"$ -> $"ℝ"$ -> $"ℝ"$ >]
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ctx_empty)))
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|-
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[{
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let "deg2turns" :=
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(λ"x" $"Angle"$~$"Degrees"$~$"ℝ"$
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↦morph (((%"/"% %"x"%) %"360"%) as $"Angle"$~$"Turns"$))
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in ( %"deg2turns"% (%"60"% as $"Angle"$~$"Degrees"$) )
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}]
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\is
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[<
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$"Angle"$~$"Turns"$~$"ℝ"$
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>]
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.
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Proof.
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apply T_Let with (σ:=[< $"Angle"$~$"Degrees"$~$"ℝ"$ ->morph $"Angle"$~$"Turns"$~$"ℝ"$ >]).
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apply T_MorphAbs.
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apply T_DescendImplicit with (τ:=(type_ladder [<$"Angle"$~$"Turns"$>] [<$"ℝ"$>])).
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2: apply TSubRepr_Refl, TEq_LadderAssocLR.
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apply T_Ascend with (τ:=[<$"ℝ"$>]) (τ':=[<$"Angle"$~$"Turns"$>]).
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apply T_App with (σ := [< $"ℝ"$ >]) (σ' := [< $"ℝ"$ >]).
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apply T_App with (σ := [< $"ℝ"$ >]) (σ' := [< $"ℝ"$ >]).
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apply T_Var.
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apply C_shuffle. apply C_shuffle. apply C_shuffle. apply C_take.
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apply T_DescendImplicit with (τ := [< $"Angle"$~$"Degrees"$~$"ℝ"$ >]).
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apply T_Var.
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apply C_take.
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apply TSubRepr_Ladder. apply TSubRepr_Ladder.
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apply TSubRepr_Refl. apply TEq_Refl.
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apply M_Sub.
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apply TSubRepr_Refl. apply TEq_Refl.
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apply T_Var.
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apply C_shuffle, C_shuffle, C_take.
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apply M_Sub.
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apply TSubRepr_Refl.
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apply TEq_Refl.
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apply T_App with (σ:=[<$"Angle"$~$"Degrees"$~$"ℝ"$>]) (σ':=[<$"Angle"$~$"Degrees"$~$"ℝ"$>]).
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apply T_MorphFun.
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apply T_Var.
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apply C_take.
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apply T_DescendImplicit with (τ:=(type_ladder [<$"Angle"$~$"Degrees"$>] [<$"ℝ"$>])).
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2: apply TSubRepr_Refl, TEq_LadderAssocLR.
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apply T_Ascend.
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apply T_Var.
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apply C_shuffle. apply C_take.
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apply M_Sub. apply TSubRepr_Refl. apply TEq_Refl.
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Qed.
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End Typing.
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