123 lines
4.3 KiB
Coq
123 lines
4.3 KiB
Coq
(* Define the abstract syntax of the calculus
|
||
* by inductive definition of type-terms
|
||
* and expression-terms.
|
||
*)
|
||
|
||
From Coq Require Import Strings.String.
|
||
Module Terms.
|
||
|
||
(* types *)
|
||
Inductive type_term : Type :=
|
||
| type_id : string -> type_term
|
||
| type_var : string -> type_term
|
||
| type_fun : type_term -> type_term -> type_term
|
||
| type_univ : string -> type_term -> type_term
|
||
| type_spec : type_term -> type_term -> type_term
|
||
| type_morph : type_term -> type_term -> type_term
|
||
| type_ladder : type_term -> type_term -> type_term
|
||
.
|
||
|
||
(* expressions *)
|
||
Inductive expr_term : Type :=
|
||
| expr_var : string -> expr_term
|
||
| expr_ty_abs : string -> expr_term -> expr_term
|
||
| expr_ty_app : expr_term -> type_term -> expr_term
|
||
| expr_abs : string -> type_term -> expr_term -> expr_term
|
||
| expr_morph : string -> type_term -> expr_term -> expr_term
|
||
| expr_app : expr_term -> expr_term -> expr_term
|
||
| expr_let : string -> type_term -> expr_term -> expr_term -> expr_term
|
||
| expr_ascend : type_term -> expr_term -> expr_term
|
||
| expr_descend : type_term -> expr_term -> expr_term
|
||
.
|
||
|
||
|
||
(* TODO
|
||
|
||
Inductive type_DeBruijn : Type :=
|
||
| id : nat -> type_DeBruijn
|
||
| var : nat -> type_DeBruijn
|
||
| fun : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
|
||
| univ : type_DeBruijn -> type_DeBruijn
|
||
| spec : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
|
||
| morph : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
|
||
| ladder : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
|
||
|
||
Inductive expr_DeBruijn : Type :=
|
||
| var : nat -> expr_DeBruijn
|
||
| ty_abs : expr_DeBruijn -> expr_DeBruijn
|
||
| ty_app : expr_DeBruijn -> type_DeBruijn -> expr_Debruijn
|
||
| abs : type_DeBruijn -> expr_DeBruijn -> expr_DeBruijn
|
||
| morph : type_DeBruijn -> expr_DeBruijn -> expr_Debruijn
|
||
| app : expr_DeBruijn -> expr_DeBruijn -> expr_Debruijn
|
||
| let : type_DeBruijn -> expr_DeBruijn -> expr_Debruijn -> expr_Debruijn
|
||
| ascend : type_DeBruijn -> expr_DeBruijn -> expr_DeBruijn
|
||
| descend : type_DeBruijn -> expr_DeBruijn -> expr_DeBruijn
|
||
.
|
||
*)
|
||
|
||
|
||
|
||
(* values *)
|
||
Inductive is_value : expr_term -> Prop :=
|
||
| V_Abs : forall x τ e,
|
||
(is_value (expr_abs x τ e))
|
||
|
||
| V_TypAbs : forall τ e,
|
||
(is_value (expr_ty_abs τ e))
|
||
|
||
| V_Ascend : forall τ e,
|
||
(is_value e) ->
|
||
(is_value (expr_ascend τ e))
|
||
.
|
||
|
||
|
||
Declare Scope ladder_type_scope.
|
||
Declare Scope ladder_expr_scope.
|
||
Declare Custom Entry ladder_type.
|
||
Declare Custom Entry ladder_expr.
|
||
|
||
Notation "[< t >]" := t
|
||
(t custom ladder_type at level 80) : ladder_type_scope.
|
||
Notation "'∀' x ',' t" := (type_univ x t)
|
||
(t custom ladder_type at level 80, in custom ladder_type at level 80, x constr).
|
||
Notation "'<' σ τ '>'" := (type_spec σ τ)
|
||
(in custom ladder_type at level 80, left associativity) : ladder_type_scope.
|
||
Notation "'(' τ ')'" := τ
|
||
(in custom ladder_type at level 70) : ladder_type_scope.
|
||
Notation "σ '->' τ" := (type_fun σ τ)
|
||
(in custom ladder_type at level 75, right associativity) : ladder_type_scope.
|
||
Notation "σ '->morph' τ" := (type_morph σ τ)
|
||
(in custom ladder_type at level 75, right associativity, τ at level 80) : ladder_type_scope.
|
||
Notation "σ '~' τ" := (type_ladder σ τ)
|
||
(in custom ladder_type at level 70, right associativity) : ladder_type_scope.
|
||
Notation "'$' x '$'" := (type_id x%string)
|
||
(in custom ladder_type at level 0, x constr) : ladder_type_scope.
|
||
Notation "'%' x '%'" := (type_var x%string)
|
||
(in custom ladder_type at level 0, x constr) : ladder_type_scope.
|
||
|
||
Notation "[{ e }]" := e
|
||
(e custom ladder_expr at level 80) : ladder_expr_scope.
|
||
Notation "'%' x '%'" := (expr_var x%string)
|
||
(in custom ladder_expr at level 0, x constr) : ladder_expr_scope.
|
||
Notation "'λ' x τ '↦' e" := (expr_abs x τ e) (in custom ladder_expr at level 0, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99).
|
||
Notation "'Λ' t '↦' e" := (expr_ty_abs t e)
|
||
(in custom ladder_expr at level 0, t constr, e custom ladder_expr at level 80).
|
||
|
||
|
||
|
||
(* EXAMPLES *)
|
||
|
||
Open Scope ladder_type_scope.
|
||
Open Scope ladder_expr_scope.
|
||
|
||
Check [< ∀"α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) >].
|
||
|
||
Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x"%"T"% ↦ %"x"% }].
|
||
Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y"%"T"% ↦ %"y"% }].
|
||
|
||
Compute polymorphic_identity1.
|
||
|
||
Close Scope ladder_type_scope.
|
||
Close Scope ladder_expr_scope.
|
||
|
||
End Terms.
|