Library ML_Infrastructure

Require Import List Metatheory ML_Definitions.

Lemma ok_iter_push : forall (A:Set) n E xs (vs:list A),
  fresh (dom E) n xs ->
  ok E ->
  ok (E & iter_push xs vs).

Hint Resolve ok_iter_push.

Additional Definitions used in the Proofs


Free Type Variables


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).

Free Term Variables


Computing free variables of a term.

Fixpoint trm_fv (t : trm) {struct t} : vars :=
  match t with
  | trm_bvar i j => {}
  | trm_fvar x => {{x}}
  | trm_abs t1 => (trm_fv t1)
  | trm_fix t1 => (trm_fv t1)
  | trm_let t1 t2 => (trm_fv t1) \u (trm_fv t2)
  | trm_match t1 p1 b t2 => (trm_fv t1) \u (trm_fv b) \u (trm_fv t2)
  | trm_app t1 t2 => (trm_fv t1) \u (trm_fv t2)
  | trm_unit => {}
  | trm_nat n => {}
  | trm_add => {}
  | trm_pair t1 t2 => (trm_fv t1) \u (trm_fv t2)
  | trm_inj1 t1 => (trm_fv t1)
  | trm_inj2 t1 => (trm_fv t1)
  | trm_loc l => {}
  | trm_ref t1 => (trm_fv t1)
  | trm_get t1 => (trm_fv t1)
  | trm_set t1 t2 => (trm_fv t1) \u (trm_fv t2)
  | trm_raise t1 => (trm_fv t1)
  | trm_catch t1 t2 => (trm_fv t1) \u (trm_fv t2)
  end.

Free Variables in Environments


Computing free type variables of the values of an environment.

Definition env_fv :=
  fv_in sch_fv.

Computing free type variables of the values for store typing.

Definition phi_fv :=
  fv_in typ_fv.

Type substitution for free names


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)
  | typ_unit => typ_unit
  | typ_nat => typ_nat
  | typ_prod T1 T2 => typ_prod (typ_subst Z U T1) (typ_subst Z U T2)
  | typ_sum T1 T2 => typ_sum (typ_subst Z U T1) (typ_subst Z U T2)
  | typ_ref T1 => typ_ref (typ_subst Z U T1)
  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)).

Term substitution for free names


Computing free variables of a list of terms.

Definition trm_fv_list :=
  List.fold_right (fun t acc => trm_fv t \u acc) {}.

Substitution for names

Fixpoint trm_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 (trm_subst z u t1)
  | trm_fix t1 => trm_fix (trm_subst z u t1)
  | trm_let t1 t2 => trm_let (trm_subst z u t1) (trm_subst z u t2)
  | trm_match t1 p1 e t2 => trm_match (trm_subst z u t1) p1
                               (trm_subst z u e)
                               (trm_subst z u t2)
  | trm_app t1 t2 => trm_app (trm_subst z u t1) (trm_subst z u t2)
  | trm_unit => trm_unit
  | trm_nat n => trm_nat n
  | trm_add => trm_add
  | trm_pair t1 t2 => trm_pair (trm_subst z u t1) (trm_subst z u t2)
  | trm_inj1 t1 => trm_inj1 (trm_subst z u t1)
  | trm_inj2 t1 => trm_inj2 (trm_subst z u t1)
  | trm_loc l => trm_loc l
  | trm_ref t1 => trm_ref (trm_subst z u t1)
  | trm_get t1 => trm_get (trm_subst z u t1)
  | trm_set t1 t2 => trm_set (trm_subst z u t1) (trm_subst z u t2)
  | trm_raise t1 => trm_raise (trm_subst z u t1)
  | trm_catch t1 t2 => trm_catch (trm_subst z u t1) (trm_subst z u t2)
  end.

Notation "[ z ~> u ] t" := (trm_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


Iterated typing

Inductive typings (E : env) (P : phi) : list trm -> list typ -> Prop :=
  | typings_nil : typings E P nil nil
  | typings_cons : forall ts Us t U,
      typings E P ts Us ->
      typing E P t U ->
      typings E P (t::ts) (U::Us).

Tactics


Instanciation of Tactics


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 trm => trm_fv_list x) in
  let G := gather_vars_with (fun x : list typ => typ_fv_list x) in
  let H := gather_vars_with (fun x : env => env_fv x) in
  let I := gather_vars_with (fun x : sch => sch_fv x) in
  let J := gather_vars_with (fun x : phi => phi_fv x) in
  constr:(A \u B \u C \u D \u E \u F \u G \u H \u I \u J).

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) :=
  apply_fresh_base_simple T gather_vars.

Tactic Notation "apply_fresh" "*" constr(T) :=
  apply_fresh T; auto*.

Automation


Hint Constructors type term phi_ok sto_ok pat_typing typing value red typings.

Lemma trm_def_fresh : trm_fv trm_def = {}.
Lemma typ_def_fresh : typ_fv typ_def = {}.

Hint Extern 1 (_ \notin trm_fv trm_def) =>
  rewrite trm_def_fresh.
Hint Extern 1 (_ \notin typ_fv typ_def) =>
  rewrite typ_def_fresh.
Hint Extern 1 (_ \notin typ_fv typ_exn) =>
  rewrite typ_exn_fresh.

Hint Extern 1 (terms _ _) => split; auto.
Hint Extern 1 (types _ _) => split; auto.

Properties of iterated operator


Lemma typ_fv_list_map : forall ts1 ts2,
  typ_fv_list (ts1 ++ ts2) = typ_fv_list ts1 \u typ_fv_list ts2.

Lemma trm_fv_list_map : forall ts1 ts2,
  trm_fv_list (ts1 ++ ts2) = trm_fv_list ts1 \u trm_fv_list ts2.

Lemma typings_concat : forall E P ts1 Us1 ts2 Us2,
  typings E P ts1 Us1 ->
  typings E P ts2 Us2 ->
  typings E P (ts1++ts2) (Us1++Us2).

Properties of terms


Properties of substitution



Substitution for a fresh name is identity.

Lemma subst_fresh : forall x t u,
  x \notin trm_fv t ->
  [x ~> u] t = t.

Lemma subst_fresh_list : forall z u ts,
  z \notin trm_fv_list ts ->
  ts = List.map (trm_subst z u) ts.

Lemma subst_fresh_trm_fvars : forall z u xs,
  fresh ({{z}}) (length xs) xs ->
  trm_fvars xs = List.map (trm_subst z u) (trm_fvars xs).

Lemma substs_fresh : forall xs us t,
  fresh (trm_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 (trm_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 (trm_fv t \u trm_fv_list vs \u trm_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 (trm_fv t \u trm_fv_list us) (length xs) xs ->
  terms (length xs) us ->
  t ^^ us = substs xs us (t ^ xs).

Terms are stable through substitutions


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).

Lemma subst_bodys : forall z u n t,
  term u -> bodys n t -> bodys n ([z ~> u]t).

Hint Resolve subst_bodys.

Terms are stable through open


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_let_to_body : forall t1 t2,
  term (trm_let t1 t2) -> bodys 1 t2.

Lemma body_to_term_let : forall t1 t2,
  term t1 -> bodys 1 t2 -> term (trm_let t1 t2).

Lemma term_match_to_body : forall t1 t2 b p,
  term (trm_match t1 p b t2) -> bodys (pat_arity p) b.

Lemma body_to_term_match : forall t1 t2 b p,
  term t1 -> term t2 -> bodys (pat_arity p) b ->
  term (trm_match t1 p b t2).

Hint Resolve body_to_term_abs term_abs_to_body
             body_to_term_fix term_fix_to_body
             body_to_term_match
             body_to_term_let.

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.

Hint Extern 1 (bodys 1 ?t2) =>
  match goal with H: context [trm_let ?t1 t2] |- _ =>
    apply (@term_let_to_body t1) end.

Opening a body with a term gives a term


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.

Properties of types


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).

Opening a body with a list of types gives a type


Lemma typ_open_types : forall T Us,
  typ_body (length Us) T ->
  types (length Us) Us ->
  type (typ_open T Us).

Properties of schemes


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.

Opening a scheme with a list of types gives a type


Lemma sch_open_types : forall M Us,
  scheme M ->
  types (sch_arity M) Us ->
  type (sch_open M Us).

Hint Resolve sch_open_types.

Properties of store


Lemma phi_ok_binds : forall P l T,
   phi_ok P -> binds l T P -> type T.

Lemma sto_ok_binds : forall mu l t,
   sto_ok mu -> binds l t mu -> term t.

Hint Resolve phi_ok_binds sto_ok_binds.

Properties of judgments


Regularity of relations


The value predicate only holds on locally-closed terms.

Lemma value_regular : forall e,
  value e -> term e.

A typing relation is restricted to well-formed objects.

Tactic Notation "split4" "in" ident(H) :=
  let H1 := fresh in let H2 := fresh in
  let H3 := fresh in let H4 := fresh in
  destruct H as [H1 [H2 [H3 H4]]].

Lemma typing_regular : forall E P e T,
  typing E P e T -> ok E /\ phi_ok P /\ term e /\ type T.

A fails relation only holds on pairs of locally-closed terms.

Lemma fails_regular : forall t e,
  fails t e -> term t /\ term e.

A reduction relation only holds only on well-formed objects.

Lemma red_regular : forall c c',
  red c c' ->
     (term (fst c) /\ term (fst c'))
  /\ (sto_ok (snd c) /\ sto_ok (snd c')).

Automation for well-formedness


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 (proj43 (typing_regular H))
  | H: red (t,_) _ |- _ => apply (proj41 (red_regular H))
  | H: red _ (t,_) |- _ => apply (proj42 (red_regular H))
  | H: value t |- _ => apply (value_regular H)
  | H: fails t _ |- _ => apply (proj1 (fails_regular H))
  | H: fails _ t |- _ => apply (proj2 (fails_regular H))
  end.

Hint Extern 1 (sto_ok ?mu) =>
  match goal with
  | H: red (_,mu) _ |- _ => apply (proj1 (proj2 (red_regular H)))
  | H: red _ (_,mu) |- _ => apply (proj2 (proj2 (red_regular H)))
  | H: sto_typing _ mu |- _ => apply (proj42 H)
  end.

Hint Extern 1 (phi_ok ?P) =>
  match goal with
  | H: typing _ P _ _ |- _ => apply (proj42 (typing_regular H))
  | H: sto_typing P _ |- _ => apply (proj41 H)
  end.

Hint Extern 1 (type ?T) => match goal with
  | H: typing _ _ _ T |- _ => apply (proj44 (typing_regular H))
  end.