Library Fsub_Part1A

Require Import Metatheory.

Description of the Language


Representation of pre-types

Inductive typ : Set :=
  | typ_top : typ
  | typ_bvar : nat -> typ
  | typ_fvar : var -> typ
  | typ_arrow : typ -> typ -> typ
  | typ_all : typ -> typ -> typ.

Opening up a type binder occuring in a type

Fixpoint open_tt_rec (K : nat) (U : typ) (T : typ) {struct T} : typ :=
  match T with
  | typ_top => typ_top
  | typ_bvar J => if K === J then U else (typ_bvar J)
  | typ_fvar X => typ_fvar X
  | typ_arrow T1 T2 => typ_arrow (open_tt_rec K U T1) (open_tt_rec K U T2)
  | typ_all T1 T2 => typ_all (open_tt_rec K U T1) (open_tt_rec (S K) U T2)
  end.

Definition open_tt T U := open_tt_rec 0 U T.

Notation for opening up binders with type or term variables

Notation "T 'open_tt_var' X" := (open_tt T (typ_fvar X)) (at level 67).

Types as locally closed pre-types

Inductive type : typ -> Prop :=
  | type_top :
      type typ_top
  | type_var : forall X,
      type (typ_fvar X)
  | type_arrow : forall T1 T2,
      type T1 ->
      type T2 ->
      type (typ_arrow T1 T2)
  | type_all : forall L T1 T2,
      type T1 ->
      (forall X, X \notin L -> type (T2 open_tt_var X)) ->
      type (typ_all T1 T2).

Binding are either mapping type or term variables. X ~<: T is a subtyping asumption and x ~: T is a typing assumption

Inductive bind : Set :=
  | bind_sub : typ -> bind.

Notation "X ~<: T" := (X ~ bind_sub T)
  (at level 31, left associativity) : env_scope.

Environment is an associative list of bindings.

Definition env := Env.env bind.

Well-formedness of a pre-type T in an environment E: all the type variables of T must be bound via a subtyping relation in E. This predicates implies that T is a type

Inductive wft : env -> typ -> Prop :=
  | wft_top : forall E,
      wft E typ_top
  | wft_var : forall U E X,
      binds X (bind_sub U) E ->
      wft E (typ_fvar X)
  | wft_arrow : forall E T1 T2,
      wft E T1 ->
      wft E T2 ->
      wft E (typ_arrow T1 T2)
  | wft_all : forall L E T1 T2,
      wft E T1 ->
      (forall X, X \notin L ->
        wft (E & X ~<: T1) (T2 open_tt_var X)) ->
      wft E (typ_all T1 T2).

A environment E is well-formed if it contains no duplicate bindings and if each type in it is well-formed with respect to the environment it is pushed on to.

Inductive okt : env -> Prop :=
  | okt_empty :
      okt empty
  | okt_sub : forall E X T,
      okt E -> wft E T -> X # E -> okt (E & X ~<: T).

Subtyping relation

Inductive sub : env -> typ -> typ -> Prop :=
  | sub_top : forall E S,
      okt E ->
      wft E S ->
      sub E S typ_top
  | sub_refl_tvar : forall E X,
      okt E ->
      wft E (typ_fvar X) ->
      sub E (typ_fvar X) (typ_fvar X)
  | sub_trans_tvar : forall U E T X,
      binds X (bind_sub U) E ->
      sub E U T ->
      sub E (typ_fvar X) T
  | sub_arrow : forall E S1 S2 T1 T2,
      sub E T1 S1 ->
      sub E S2 T2 ->
      sub E (typ_arrow S1 S2) (typ_arrow T1 T2)
  | sub_all : forall L E S1 S2 T1 T2,
      sub E T1 S1 ->
      (forall X, X \notin L ->
          sub (E & X ~<: T1) (S2 open_tt_var X) (T2 open_tt_var X)) ->
      sub E (typ_all S1 S2) (typ_all T1 T2).

Additional Definitions Used in the Proofs


Computing free type variables in a type

Fixpoint fv_tt (T : typ) {struct T} : vars :=
  match T with
  | typ_top => {}
  | typ_bvar J => {}
  | typ_fvar X => {{X}}
  | typ_arrow T1 T2 => (fv_tt T1) \u (fv_tt T2)
  | typ_all T1 T2 => (fv_tt T1) \u (fv_tt T2)
  end.

Substitution for free type variables in types.

Fixpoint subst_tt (Z : var) (U : typ) (T : typ) {struct T} : typ :=
  match T with
  | typ_top => typ_top
  | typ_bvar J => typ_bvar J
  | typ_fvar X => if X == Z then U else (typ_fvar X)
  | typ_arrow T1 T2 => typ_arrow (subst_tt Z U T1) (subst_tt Z U T2)
  | typ_all T1 T2 => typ_all (subst_tt Z U T1) (subst_tt Z U T2)
  end.

Substitution for free type variables in environment.
Definition subst_tb (Z : var) (P : typ) (b : bind) : bind :=
  match b with
  | bind_sub T => bind_sub (subst_tt Z P T)
  end.

Tactics


Constructors as hints.

Hint Constructors type wft ok okt.

Hint Resolve
  sub_top sub_refl_tvar sub_arrow.

Gathering free names already used in the proofs

Ltac gather_vars :=
  let A := gather_vars_with (fun x : vars => x) in
  let B := gather_vars_with (fun x : var => {{ x }}) in
  let E := gather_vars_with (fun x : typ => fv_tt x) in
  let F := gather_vars_with (fun x : env => dom x) in
  constr:(A \u B \u E \u F).

"pick_fresh x" tactic create a fresh variable with name x

Ltac pick_fresh x :=
  let L := gather_vars in (pick_fresh_gen L x).

"apply_fresh T as x" is used to apply inductive rule which use an universal quantification over a cofinite set

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*.

These tactics help applying a lemma which conclusion mentions an environment (E & F) in the particular case when F is empty

Ltac get_env :=
  match goal with
  | |- wft ?E _ => E
  | |- sub ?E _ _ => E
  end.

Tactic Notation "apply_empty_bis" tactic(get_env) constr(lemma) :=
  let E := get_env in rewrite <- (concat_empty E);
  eapply lemma; try rewrite concat_empty.

Tactic Notation "apply_empty" constr(F) :=
  apply_empty_bis (get_env) F.

Tactic Notation "apply_empty" "*" constr(F) :=
  apply_empty F; auto*.

Tactic to undo when Coq does too much simplification

Ltac unsimpl_map_bind :=
  match goal with |- context [ ?B (subst_tt ?Z ?P ?U) ] =>
    unsimpl ((subst_tb Z P) (B U)) end.

Tactic Notation "unsimpl_map_bind" "*" :=
  unsimpl_map_bind; auto*.

Properties of well-formedness of a type in an environment


If a type is well-formed in an environment then it is locally closed.

Lemma type_from_wft : forall E T,
  wft E T -> type T.

Through weakening

Lemma wft_weaken : forall G T E F,
  wft (E & G) T ->
  ok (E & F & G) ->
  wft (E & F & G) T.

Through narrowing

Lemma wft_narrow : forall V F U T E X,
  wft (E & X ~<: V & F) T ->
  ok (E & X ~<: U & F) ->
  wft (E & X ~<: U & F) T.

Relations between well-formed environment and types well-formed

in environments

If an environment is well-formed, then it does not contain duplicated keys.

Lemma ok_from_okt : forall E,
  okt E -> ok E.

Hint Extern 1 (ok _) => apply ok_from_okt.

Extraction from a subtyping assumption in a well-formed environments

Lemma wft_from_env_has_sub : forall x U E,
  okt E -> binds x (bind_sub U) E -> wft E U.

Hint Immediate wft_from_env_has_sub.

Extraction from a well-formed environment

Lemma wft_from_okt_sub : forall x T E,
  okt (E & x ~<: T) -> wft E T.
Hint Resolve wft_from_okt_sub.

Properties of well-formedness of an environment


Through narrowing

Lemma okt_narrow : forall V E F U X,
  okt (E & X ~<: V & F) ->
  wft E U ->
  okt (E & X ~<: U & F).

Automation

Hint Resolve okt_narrow wft_weaken.

Regularity of relations


The subtyping relation is restricted to well-formed objects.

Lemma sub_regular : forall E S T,
  sub E S T -> okt E /\ wft E S /\ wft E T.


Automation

Hint Extern 1 (okt ?E) =>
  match goal with
  | H: sub _ _ _ |- _ => apply (proj31 (sub_regular H))
  end.

Hint Extern 1 (wft ?E ?T) =>
  match goal with
  | H: sub E T _ |- _ => apply (proj32 (sub_regular H))
  | H: sub E _ T |- _ => apply (proj33 (sub_regular H))
  end.

Hint Extern 1 (type ?T) =>
  let go E := apply (@type_from_wft E); auto in
  match goal with
  | H: sub ?E T _ |- _ => go E
  | H: sub ?E _ T |- _ => go E
  end.

Properties of Subtyping


Reflexivity (1)

Lemma sub_reflexivity : forall E T,
  okt E ->
  wft E T ->
  sub E T T .

Weakening (2)

Lemma sub_weakening : forall E F G S T,
   sub (E & G) S T ->
   okt (E & F & G) ->
   sub (E & F & G) S T.

Narrowing and transitivity (3)

Section NarrowTrans.

Definition transitivity_on Q := forall E S T,
  sub E S Q -> sub E Q T -> sub E S T.

Hint Unfold transitivity_on.

Hint Resolve wft_narrow.

Lemma sub_narrowing_aux : forall Q F E Z P S T,
  transitivity_on Q ->
  sub (E & Z ~<: Q & F) S T ->
  sub E P Q ->
  sub (E & Z ~<: P & F) S T.

Lemma sub_transitivity : forall Q,
  transitivity_on Q.

Lemma sub_narrowing : forall Q E F Z P S T,
  sub E P Q ->
  sub (E & Z ~<: Q & F) S T ->
  sub (E & Z ~<: P & F) S T.

End NarrowTrans.