ladder-calculus/coq/terms.v

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(* 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
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| 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 -> expr_term -> expr_term -> expr_term
| expr_ascend : type_term -> expr_term -> expr_term
| expr_descend : type_term -> expr_term -> expr_term
.
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(* values *)
Inductive is_abs_value : expr_term -> Prop :=
| VAbs_Var : forall x,
(is_abs_value (expr_var x))
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| VAbs_Abs : forall x τ e,
(is_abs_value (expr_abs x τ e))
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| VAbs_Morph : forall x τ e,
(is_abs_value (expr_morph x τ e))
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| VAbs_TypAbs : forall τ e,
(is_abs_value (expr_ty_abs τ e))
.
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Inductive is_value : expr_term -> Prop :=
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| V_Abs : forall e,
(is_abs_value e) ->
(is_value e)
| V_Ascend : forall τ e,
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(is_abs_value e) ->
(is_value (expr_ascend τ e))
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| V_Descend : forall τ e,
(is_abs_value e) ->
(is_value (expr_descend τ e))
.
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Declare Scope ladder_type_scope.
Declare Scope ladder_expr_scope.
Declare Custom Entry ladder_type.
Declare Custom Entry ladder_expr.
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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.
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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' t '↦' e" := (expr_ty_abs t e)
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(in custom ladder_expr at level 10, t constr, e custom ladder_expr at level 80) : ladder_expr_scope.
Notation "'λ' x τ '↦' e" := (expr_abs x τ e)
(in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope.
Notation "'λ' x τ '↦morph' e" := (expr_morph x τ e)
(in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope.
Notation "'let' x ':=' e 'in' t" := (expr_let x e t)
(in custom ladder_expr at level 20, x constr, e custom ladder_expr at level 99, t custom ladder_expr at level 99) : ladder_expr_scope.
Notation "e 'as' τ" := (expr_ascend τ e)
(in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
Notation "e 'des' τ" := (expr_descend τ e)
(in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
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Notation "e1 e2" := (expr_app e1 e2)
(in custom ladder_expr at level 50) : ladder_expr_scope.
Notation "'(' e ')'" := e
(in custom ladder_expr at level 0) : ladder_expr_scope.
(* EXAMPLES *)
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Open Scope ladder_type_scope.
Open Scope ladder_expr_scope.
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Check [< "α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) >].
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Definition polymorphic_identity1 : expr_term := [{ Λ"T" λ"x"%"T"% (%"x"%) }].
Definition polymorphic_identity2 : expr_term := [{ Λ"T" λ"y"%"T"% %"y"% }].
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Compute polymorphic_identity1.
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Close Scope ladder_type_scope.
Close Scope ladder_expr_scope.
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End Terms.