Library ML_Core_Infrastructure
Require Import List Metatheory ML_Core_Definitions.
Computing free variables of a type.
Fixpoint typ_fv (T : typ) {struct T} : vars :=
match T with
| typ_bvar i => {}
| typ_fvar x => {{x}}
| typ_arrow T1 T2 => (typ_fv T1) \u (typ_fv T2)
end.
Computing free variables of a list of terms.
Definition typ_fv_list :=
List.fold_right (fun t acc => typ_fv t \u acc) {}.
Computing free variables of a type scheme.
Definition sch_fv M :=
typ_fv (sch_type M).
Computing free type variables of the values of an environment.
Definition env_fv :=
fv_in sch_fv.
Computing free variables of a term.
Fixpoint trm_fv (t : trm) {struct t} : vars :=
match t with
| trm_bvar i => {}
| trm_fvar x => {{x}}
| trm_abs t1 => (trm_fv t1)
| trm_let t1 t2 => (trm_fv t1) \u (trm_fv t2)
| trm_app t1 t2 => (trm_fv t1) \u (trm_fv t2)
end.
Substitution for names for types
Fixpoint typ_subst (Z : var) (U : typ) (T : typ) {struct T} : typ :=
match T with
| typ_bvar i => typ_bvar i
| typ_fvar X => if X == Z then U else (typ_fvar X)
| typ_arrow T1 T2 => typ_arrow (typ_subst Z U T1) (typ_subst Z U T2)
end.
Iterated substitution for types
Fixpoint typ_substs (Zs : list var) (Us : list typ) (T : typ)
{struct Zs} : typ :=
match Zs, Us with
| Z::Zs', U::Us' => typ_substs Zs' Us' (typ_subst Z U T)
| _, _ => T
end.
Substitution for names for schemes.
Definition sch_subst Z U M :=
Sch (sch_arity M) (typ_subst Z U (sch_type M)).
Iterated substitution for schemes.
Definition sch_substs Zs Us M :=
Sch (sch_arity M) (typ_substs Zs Us (sch_type M)).
Substitution for name in a term.
Fixpoint trm_subst (z : var) (u : trm) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i => trm_bvar i
| trm_fvar x => if x == z then u else (trm_fvar x)
| trm_abs t1 => trm_abs (trm_subst z u t1)
| trm_let t1 t2 => trm_let (trm_subst z u t1) (trm_subst z u t2)
| trm_app t1 t2 => trm_app (trm_subst z u t1) (trm_subst z u t2)
end.
Notation "[ z ~> u ] t" := (trm_subst z u t) (at level 68).
Ltac gather_vars :=
let A := gather_vars_with (fun x : vars => x) in
let B := gather_vars_with (fun x : var => {{ x }}) in
let C := gather_vars_with (fun x : env => dom x) in
let D := gather_vars_with (fun x : trm => trm_fv x) in
let E := gather_vars_with (fun x : typ => typ_fv x) in
let F := gather_vars_with (fun x : list typ => typ_fv_list x) in
let G := gather_vars_with (fun x : env => env_fv x) in
let H := gather_vars_with (fun x : sch => sch_fv x) in
constr:(A \u B \u C \u D \u E \u F \u G \u H).
Tactic Notation "pick_fresh" ident(x) :=
let L := gather_vars in (pick_fresh_gen L x).
Tactic Notation "pick_freshes" constr(n) ident(x) :=
let L := gather_vars in (pick_freshes_gen L n x).
Tactic Notation "apply_fresh" constr(T) "as" ident(x) :=
apply_fresh_base T gather_vars x.
Tactic Notation "apply_fresh" "*" constr(T) "as" ident(x) :=
apply_fresh T as x; auto*.
Hint Constructors type term typing value red.
Lemma typ_def_fresh : typ_fv typ_def = {}.
Hint Extern 1 (_ \notin typ_fv typ_def) =>
rewrite typ_def_fresh.
Hint Extern 1 (types _ _) => split; auto.
Lemma fv_list_map : forall ts1 ts2,
typ_fv_list (ts1 ++ ts2) = typ_fv_list ts1 \u typ_fv_list ts2.
Substitution for a fresh name is identity.
Lemma trm_subst_fresh : forall x t u,
x \notin trm_fv t -> [x ~> u] t = t.
Substitution distributes on the open operation.
Lemma trm_subst_open : forall x u t1 t2, term u ->
[x ~> u] (t1 ^^ t2) = ([x ~> u]t1) ^^ ([x ~> u]t2).
Substitution and open_var for distinct names commute.
Lemma trm_subst_open_var : forall x y u e, y <> x -> term u ->
([x ~> u]e) ^ y = [x ~> u] (e ^ y).
Opening up an abstraction of body t with a term u is the same as opening
up the abstraction with a fresh name x and then substituting u for x.
Lemma trm_subst_intro : forall x t u,
x \notin (trm_fv t) -> term u ->
t ^^ u = [x ~> u](t ^ x).
Terms are stable by substitution
Lemma trm_subst_term : forall t z u,
term u -> term t -> term ([z ~> u]t).
Hint Resolve trm_subst_term.
Conversion from locally closed abstractions and bodies
Lemma term_abs_to_body : forall t1,
term (trm_abs t1) -> term_body t1.
Lemma body_to_term_abs : forall t1,
term_body t1 -> term (trm_abs t1).
Lemma term_let_to_body : forall t1 t2,
term (trm_let t1 t2) -> term_body t2.
Lemma body_to_term_let : forall t1 t2,
term_body t2 -> term t1 -> term (trm_let t1 t2).
Hint Resolve body_to_term_abs body_to_term_let.
Hint Extern 1 (term_body ?t) =>
match goal with
| H: context [trm_abs t] |- _ =>
apply term_abs_to_body
| H: context [trm_let ?t1 t] |- _ =>
apply (@term_let_to_body t1)
end.
Lemma trm_open_term : forall t u,
term_body t -> term u -> term (t ^^ u).
Hint Resolve trm_open_term.
Open on a type is the identity.
Lemma typ_open_type : forall T Us,
type T -> T = typ_open T Us.
Substitution for a fresh name is identity.
Lemma typ_subst_fresh : forall X U T,
X \notin typ_fv T ->
typ_subst X U T = T.
Lemma typ_subst_fresh_list : forall z u ts,
z \notin typ_fv_list ts ->
ts = List.map (typ_subst z u) ts.
Lemma typ_subst_fresh_trm_fvars : forall z u xs,
fresh ({{z}}) (length xs) xs ->
typ_fvars xs = List.map (typ_subst z u) (typ_fvars xs).
Lemma typ_substs_fresh : forall xs us t,
fresh (typ_fv t) (length xs) xs ->
typ_substs xs us t = t.
Substitution distributes on the open operation.
Lemma typ_subst_open : forall X U T1 T2, type U ->
typ_subst X U (typ_open T1 T2) =
typ_open (typ_subst X U T1) (List.map (typ_subst X U) T2).
Substitution and open_var for distinct names commute.
Lemma typ_subst_open_vars : forall X Ys U T,
fresh {{X}} (length Ys) Ys ->
type U ->
typ_open_vars (typ_subst X U T) Ys
= typ_subst X U (typ_open_vars T Ys).
Opening up an abstraction of body t with a term u is the same as opening
up the abstraction with a fresh name x and then substituting u for x.
Lemma typ_substs_intro_ind : forall T Xs Us Vs,
fresh (typ_fv T \u typ_fv_list Vs \u typ_fv_list Us) (length Xs) Xs ->
types (length Xs) Us ->
types (length Vs) Vs ->
typ_open T (Vs ++ Us) = typ_substs Xs Us (typ_open T (Vs ++ (typ_fvars Xs))).
Lemma typ_substs_intro : forall Xs Us T,
fresh (typ_fv T \u typ_fv_list Us) (length Xs) Xs ->
types (length Xs) Us ->
(typ_open T Us) = typ_substs Xs Us (typ_open_vars T Xs).
Types are stable by type substitution
Lemma typ_subst_type : forall T Z U,
type U -> type T -> type (typ_subst Z U T).
Hint Resolve typ_subst_type.
Types are stable by iterated type substitution
Lemma typ_substs_types : forall Xs Us T,
types (length Xs) Us ->
type T ->
type (typ_substs Xs Us T).
List of types are stable by type substitution
Lemma typ_subst_type_list : forall Z U Ts n,
type U -> types n Ts ->
types n (List.map (typ_subst Z U) Ts).
Lemma typ_open_types : forall T Us,
typ_body (length Us) T ->
types (length Us) Us ->
type (typ_open T Us).
Substitution for a fresh name is identity.
Lemma sch_subst_fresh : forall X U M,
X \notin sch_fv M ->
sch_subst X U M = M.
Trivial lemma to unfolding definition of
sch_subst
by rewriting.
Lemma sch_subst_fold : forall Z U T n,
Sch n (typ_subst Z U T) = sch_subst Z U (Sch n T).
Distributivity of type substitution on opening of scheme.
Lemma sch_subst_open : forall Z U M Us,
type U ->
typ_subst Z U (sch_open M Us)
= sch_open (sch_subst Z U M) (List.map (typ_subst Z U) Us).
Distributivity of type substitution on opening of scheme with variables.
Lemma sch_subst_open_vars : forall Z U M Xs,
fresh ({{Z}}) (length Xs) Xs ->
type U ->
typ_subst Z U (sch_open_vars M Xs)
= sch_open_vars (sch_subst Z U M) Xs.
Schemes are stable by type substitution.
Lemma sch_subst_type : forall Z U M,
type U -> scheme M -> scheme (sch_subst Z U M).
Hint Resolve sch_subst_type.
Scheme arity is preserved by type substitution.
Lemma sch_subst_arity : forall X U M,
sch_arity (sch_subst X U M) = sch_arity M.
Lemma sch_open_types : forall M Us,
scheme M ->
types (sch_arity M) Us ->
type (sch_open M Us).
Hint Resolve sch_open_types.
A typing relation is restricted to well-formed objects.
Lemma typing_regular : forall E e T,
typing E e T -> ok E /\ term e /\ type T.
The value predicate only holds on locally-closed terms.
Lemma value_regular : forall e,
value e -> term e.
A reduction relation only holds on pairs of locally-closed terms.
Lemma red_regular : forall e e',
red e e' -> term e /\ term e'.
Automation for reasoning on well-formedness.
Hint Extern 1 (ok ?E) =>
match goal with
| H: typing E _ _ |- _ => apply (proj31 (typing_regular H))
end.
Hint Extern 1 (term ?t) =>
match goal with
| H: typing _ t _ |- _ => apply (proj32 (typing_regular H))
| H: red t _ |- _ => apply (proj1 (red_regular H))
| H: red _ t |- _ => apply (proj2 (red_regular H))
| H: value t |- _ => apply (value_regular H)
end.
Hint Extern 1 (type ?T) => match goal with
| H: typing _ _ T |- _ => apply (proj33 (typing_regular H))
end.