Library STLC_Data_Infrastructure
Require Import List Metatheory STLC_Data_Definitions.
Computing free variables of a term.
Fixpoint fv (t : trm) {struct t} : vars :=
match t with
| trm_bvar i j => {}
| trm_fvar x => {{x}}
| trm_abs t1 => (fv t1)
| trm_fix t1 => (fv t1)
| trm_app t1 t2 => (fv t1) \u (fv t2)
| trm_match t1 p1 e t2 => (fv t1) \u (fv e) \u (fv t2)
| trm_unit => {}
| trm_pair t1 t2 => (fv t1) \u (fv t2)
| trm_inj1 t1 => (fv t1)
| trm_inj2 t1 => (fv t1)
end.
Computing free variables of a list of terms.
Definition fv_list :=
List.fold_right (fun t acc => fv t \u acc) {}.
Free variables of the default term.
Lemma trm_def_fresh : fv trm_def = {}.
Substitution for names
Fixpoint subst (z : var) (u : trm) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i j => trm_bvar i j
| trm_fvar x => if x == z then u else (trm_fvar x)
| trm_abs t1 => trm_abs (subst z u t1)
| trm_fix t1 => trm_fix (subst z u t1)
| trm_match t1 p1 e t2 => trm_match (subst z u t1) p1
(subst z u e)
(subst z u t2)
| trm_app t1 t2 => trm_app (subst z u t1) (subst z u t2)
| trm_unit => trm_unit
| trm_pair t1 t2 => trm_pair (subst z u t1) (subst z u t2)
| trm_inj1 t1 => trm_inj1 (subst z u t1)
| trm_inj2 t1 => trm_inj2 (subst z u t1)
end.
Notation "[ z ~> u ] t" := (subst z u t) (at level 68).
Iterated substitution
Fixpoint substs (zs : list var) (us : list trm) (t : trm)
{struct zs} : trm :=
match zs, us with
| z::zs', u::us' => substs zs' us' ([z ~> u]t)
| _, _ => t
end.
Predicate caraterizing lists of a given number of terms
Definition terms := list_for_n term.
Iterated typing
Inductive typings (E : env) : list trm -> list typ -> Prop :=
| typings_nil : typings E nil nil
| typings_cons : forall ts Us t U,
typings E ts Us ->
typing E t U ->
typings E (t::ts) (U::Us).
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 => fv x) in
let E := gather_vars_with (fun x : list trm => fv_list x) in
constr:(A \u B \u C \u D \u E).
Ltac pick_fresh Y :=
let L := gather_vars in (pick_fresh_gen L Y).
Ltac pick_freshes n Y :=
let L := gather_vars in (pick_freshes_gen L n Y).
Tactic Notation "apply_fresh" constr(T) :=
apply_fresh_base_simple T gather_vars.
Tactic Notation "apply_fresh" "*" constr(T) :=
apply_fresh T; auto*.
Hint Constructors term value red typing typings.
Hint Extern 1 (_ \notin fv trm_def) =>
rewrite trm_def_fresh.
Hint Extern 1 (terms _ _) => split; auto.
Hint Extern 1 (fresh (dom (_ & _)) _ _) => simpl_env.
Lemma fv_list_map : forall ts1 ts2,
fv_list (ts1 ++ ts2) = fv_list ts1 \u fv_list ts2.
Lemma typings_concat : forall E ts1 Us1 ts2 Us2,
typings E ts1 Us1 ->
typings E ts2 Us2 ->
typings E (ts1++ts2) (Us1++Us2).
Substitution for a fresh name is identity.
Lemma subst_fresh : forall x t u,
x \notin fv t ->
[x ~> u] t = t.
Lemma subst_fresh_list : forall z u ts,
z \notin fv_list ts ->
ts = List.map (subst z u) ts.
Lemma subst_fresh_trm_fvars : forall z u xs,
fresh ({{z}}) (length xs) xs ->
trm_fvars xs = List.map (subst z u) (trm_fvars xs).
Lemma substs_fresh : forall xs us t,
fresh (fv t) (length xs) xs ->
substs xs us t = t.
Substitution distributes on the open operation.
Lemma subst_open : forall x u t1 t2, term u ->
[x ~> u] (t1 ^^ t2) = ([x ~> u]t1) ^^ (List.map (subst x u) t2).
Substitution and open_var for distinct names commute.
Lemma subst_open_vars : forall x ys u t,
fresh {{x}} (length ys) ys ->
term u ->
([x ~> u]t) ^ ys = [x ~> u] (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 substs_intro_ind : forall t xs us vs,
fresh (fv t \u fv_list vs \u fv_list us) (length xs) xs ->
terms (length xs) us ->
terms (length vs) vs ->
t ^^ (vs ++ us) = substs xs us (t ^^ (vs ++ (trm_fvars xs))).
Lemma substs_intro : forall xs t us,
fresh (fv t \u fv_list us) (length xs) xs ->
terms (length xs) us ->
t ^^ us = substs xs us (t ^ xs).
Terms are stable by substitution
Lemma subst_term : forall t z u,
term u -> term t -> term ([z ~> u]t).
Hint Resolve subst_term.
Terms are stable by iterated substitution
Lemma substs_terms : forall xs us t,
terms (length xs) us ->
term t ->
term (substs xs us t).
Conversion from locally closed abstractions and bodies
Lemma term_abs_to_body : forall t1,
term (trm_abs t1) -> bodys 1 t1.
Lemma body_to_term_abs : forall t1,
bodys 1 t1 -> term (trm_abs t1).
Lemma term_fix_to_body : forall t1,
term (trm_fix t1) -> bodys 2 t1.
Lemma body_to_term_fix : forall t1,
bodys 2 t1 -> term (trm_fix t1).
Lemma term_match_to_body : forall t1 t2 e p,
term (trm_match t1 p e t2) -> bodys (pat_arity p) e.
Lemma body_to_term_match : forall t1 t2 e p,
term t1 -> term t2 -> bodys (pat_arity p) e ->
term (trm_match t1 p e t2).
Hint Resolve body_to_term_abs term_abs_to_body
body_to_term_fix term_fix_to_body
body_to_term_match.
Hint Extern 1 (bodys (pat_arity ?p) ?e) =>
match goal with H: context [trm_match ?t1 p e ?t2] |- _ =>
apply (@term_match_to_body t1 t2) end.
Lemma open_terms : forall t us,
bodys (length us) t ->
terms (length us) us ->
term (t ^^ us).
Hint Resolve open_terms.
The matching function returns a list of terms of the expected length.
Lemma pat_match_terms : forall p t ts,
(pat_match p t) = Some ts -> term t ->
terms (pat_arity p) ts.
A typing relation holds only if the environment has no
duplicated keys and the pre-term is locally-closed.
Lemma typing_regular : forall E e T,
typing E e T -> ok E /\ term e.
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 (proj1 (typing_regular H))
end.
Hint Extern 1 (term ?t) =>
match goal with
| H: typing _ t _ |- _ => apply (proj2 (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.