(** Type-safety proofs for Fsub.  Authors: Brian Aydemir and Arthur Chargu\'eraud, with help from Aaron Bohannon, Jeffrey Vaughan, and Dimitrios Vytiniotis.  In parentheses are given the label of the corresponding lemma in the appendix (informal proofs) of the POPLmark Challenge.  Table of contents: - #<a href="##subtyping">Properties of subtyping</a># - #<a href="##typing">Properties of typing</a># - #<a href="##preservation">Preservation</a># - #<a href="##progress">Progress</a># *)
Require Export Fsub.Fsub_LetSum_Lemmas.
(* ********************************************************************** *)
(** * #<a name="subtyping"></a># Properties of subtyping *)
(* ********************************************************************** *)
(** ** Reflexivity (1) *)
Proof with eauto.
intros E T Ok Wf.
induction Wf...
pick fresh Y and apply sub_all...
Qed.
(* ********************************************************************** *)
(** ** Weakening (2) *)
Proof with simpl_env; auto using wf_typ_weakening.
intros E F G S T Sub Ok.
remember (G ++ E) as H.
generalize dependent G.
induction Sub; intros G Ok EQ; subst...
Case "sub_trans_tvar".
apply (sub_trans_tvar U)...
Case "sub_all".
pick fresh Y and apply sub_all...
rewrite <- app_assoc.
apply H0...
Qed.
(* ********************************************************************** *)
(** ** Narrowing and transitivity (3) *)
Definition transitivity_on Q := forall E S T,
sub E S Q -> sub E Q T -> sub E S T.
Proof with simpl_env; eauto using wf_typ_narrowing, wf_env_narrowing.
intros Q F E Z P S T TransQ SsubT PsubQ.
remember (F ++ Z ~ bind_sub Q ++ E) as G. generalize dependent F.
induction SsubT; intros F EQ; subst...
Case "sub_trans_tvar".
destruct (X == Z); subst.
SCase "X = Z".
apply (sub_trans_tvar P); [ eauto using fresh_mid_head | ].
apply TransQ.
SSCase "P <: Q".
rewrite_env (empty ++ (F ++ Z ~ bind_sub P) ++ E).
apply sub_weakening...
SSCase "Q <: T".
analyze_binds_uniq H.
injection BindsTacVal; intros; subst...
SCase "X <> Z".
apply (sub_trans_tvar U)...
Case "sub_all".
pick fresh Y and apply sub_all...
rewrite <- app_assoc.
apply H0...
Qed.
Proof with simpl_env; auto.
unfold transitivity_on.
intros Q E S T SsubQ QsubT.
assert (W : type Q) by auto.
generalize dependent T.
generalize dependent S.
generalize dependent E.
remember Q as Q' in |- *.
generalize dependent Q'.
induction W;
intros Q' EQ E S SsubQ;
induction SsubQ; try discriminate; inversion EQ; subst;
intros T' QsubT;
inversion QsubT; subst; eauto 4 using sub_trans_tvar.
Case "sub_all / sub_top".
assert (sub E (typ_all S1 S2) (typ_all T1 T2)).
SCase "proof of assertion".
pick fresh y and apply sub_all...
auto.
Case "sub_all / sub_all".
pick fresh Y and apply sub_all.
SCase "bounds".
eauto.
SCase "bodies".
lapply (H0 Y); [ intros K | auto ].
apply (K (open_tt T2 Y))...
rewrite_env (empty ++ Y ~ bind_sub T0 ++ E).
apply (sub_narrowing_aux T1)...
unfold transitivity_on.
auto using (IHW T1).
Qed.
(* ********************************************************************** *)
(** ** Type substitution preserves subtyping (10) *)
Proof with
simpl_env;
eauto 4 using wf_typ_subst_tb, wf_env_subst_tb, wf_typ_weaken_head.
intros Q E F Z S T P SsubT PsubQ.
remember (F ++ Z ~ bind_sub Q ++ E) as G.
generalize dependent F.
induction SsubT; intros G EQ; subst; simpl subst_tt...
Case "sub_top".
apply sub_top...
Case "sub_refl_tvar".
destruct (X == Z); subst.
SCase "X = Z".
apply sub_reflexivity...
SCase "X <> Z".
apply sub_reflexivity...
inversion H0; subst.
analyze_binds H3...
apply (wf_typ_var (subst_tt Z P U))...
Case "sub_trans_tvar".
destruct (X == Z); subst.
SCase "X = Z".
apply (sub_transitivity Q).
SSCase "left branch".
rewrite_env (empty ++ map (subst_tb Z P) G ++ E).
apply sub_weakening...
SSCase "right branch".
rewrite (subst_tt_fresh Z P Q).
analyze_binds_uniq H.
inversion BindsTacVal; subst...
apply (notin_fv_wf E); eauto using fresh_mid_tail.
SCase "X <> Z".
apply (sub_trans_tvar (subst_tt Z P U))...
rewrite (map_subst_tb_id E Z P);
[ | auto | eapply fresh_mid_tail; eauto ].
analyze_binds H...
Case "sub_all".
pick fresh X and apply sub_all...
rewrite subst_tt_open_tt_var...
rewrite subst_tt_open_tt_var...
rewrite_env (map (subst_tb Z P) (X ~ bind_sub T1 ++ G) ++ E).
apply H0...
Qed.
(* ********************************************************************** *)
(** * #<a name="typing"></a># Properties of typing *)
(* ********************************************************************** *)
(** ** Weakening (5) *)
Proof with simpl_env;
eauto using wf_typ_weakening,
wf_typ_from_wf_env_typ,
wf_typ_from_wf_env_sub,
sub_weakening.
intros E F G e T Typ.
remember (G ++ E) as H.
generalize dependent G.
induction Typ; intros G EQ Ok; subst...
Case "typing_abs".
pick fresh x and apply typing_abs.
lapply (H x); [intros K | auto].
rewrite <- app_assoc.
apply (H0 x)...
Case "typing_tabs".
pick fresh X and apply typing_tabs.
lapply (H X); [intros K | auto].
rewrite <- app_assoc.
apply (H0 X)...
Case "typing_let".
pick fresh x and apply typing_let...
lapply (H0 x); [intros K | auto].
rewrite <- app_assoc.
apply H0...
Case "typing_case".
pick fresh x and apply typing_case...
SCase "inl branch".
lapply (H0 x); [intros K | auto].
rewrite <- app_assoc.
apply H0...
assert (J : wf_typ (G ++ F ++ E) (typ_sum T1 T2))...
inversion J...
SCase "inr branch".
lapply (H2 x); [intros K | auto].
rewrite <- app_assoc.
apply H2...
assert (J : wf_typ (G ++ F ++ E) (typ_sum T1 T2))...
inversion J...
Qed.
(* ********************************************************************** *)
(** ** Strengthening (6) *)
Proof with eauto using wf_typ_strengthening, wf_env_strengthening.
intros x U E F S T SsubT.
remember (F ++ x ~ bind_typ U ++ E) as E'.
generalize dependent F.
induction SsubT; intros F EQ; subst...
Case "sub_trans_tvar".
apply (sub_trans_tvar U0)...
analyze_binds H...
Case "sub_all".
pick fresh X and apply sub_all...
rewrite <- app_assoc.
apply H0...
Qed.
(************************************************************************ *)
(** ** Narrowing for typing (7) *)
Proof with eauto 6 using wf_env_narrowing, wf_typ_narrowing, sub_narrowing.
intros Q E F X P e T PsubQ Typ.
remember (F ++ X ~ bind_sub Q ++ E) as E'.
generalize dependent F.
induction Typ; intros F EQ; subst...
Case "typing_var".
analyze_binds H0...
Case "typing_abs".
pick fresh y and apply typing_abs.
rewrite <- app_assoc.
apply H0...
Case "typing_tabs".
pick fresh Y and apply typing_tabs.
rewrite <- app_assoc.
apply H0...
Case "typing_let".
pick fresh y and apply typing_let...
rewrite <- app_assoc.
apply H0...
Case "typing_case".
pick fresh y and apply typing_case...
SCase "inl branch".
rewrite <- app_assoc.
apply H0...
SCase "inr branch".
rewrite <- app_assoc.
apply H2...
Qed.
(************************************************************************ *)
(** ** Substitution preserves typing (8) *)
(* begin show *)
(** We provide detailed comments for the following proof, mainly to point out several useful tactics and proof techniques.  Starting a proof with "Proof with <some tactic>" allows us to specify a default tactic that should be used to solve goals.  To invoke this default tactic at the end of a proof step, we signal the end of the step with three periods instead of a single one, e.g., "apply typing_weakening...". *)
Proof with simpl_env;
eauto 4 using wf_typ_strengthening,
wf_env_strengthening,
sub_strengthening.
(** The proof proceeds by induction on the given typing derivation for e.  We use the [remember] tactic, along with [generalize dependent], to ensure that the goal is properly strengthened before we use induction. *)
intros U E F x T e u TypT TypU.
remember (F ++ x ~ bind_typ U ++ E) as E'.
generalize dependent F.
induction TypT; intros F EQ; subst; simpl subst_ee...
(** The [typing_var] case involves a case analysis on whether the variable is the same as the one being substituted for. *)
Case "typing_var".
destruct (x0 == x); try subst x0.
(** In the case where [x0=x], we first observe that hypothesis [H0] implies that [T=U], since [x] can only be bound once in the environment.  The conclusion then follows from hypothesis [TypU] and weakening.  We can use the [analyze_binds_uniq] tactic, described in the MetatheoryEnv library, with [H0] to obtain the fact that [T=U]. *)
SCase "x0 = x".
analyze_binds_uniq H0.
injection BindsTacVal; intros; subst.
(** In order to apply [typing_weakening], we need to rewrite the environment so that it has the right shape.  (We could also prove a corollary of typing_weakening.)  The [rewrite_env] tactic, described in the Environment library, is one way to perform this rewriting. *)
rewrite_env (empty ++ F ++ E).
apply typing_weakening...
(** In the case where [x0<>x], the result follows by an exhaustive case analysis on exactly where [x0] is bound in the environment.  We perform this case analysis by using the [analyze_binds] tactic, described in the MetatheoryEnv library. *)
SCase "x0 <> x".
analyze_binds H0.
eauto using wf_env_strengthening.
eauto using wf_env_strengthening.
(** Informally, the [typing_abs] case is a straightforward application of the induction hypothesis, which is called [H0] here. *)
Case "typing_abs".
(** We use the "pick fresh and apply" tactic to apply the rule [typing_abs] without having to calculate the appropriate finite set of atoms. *)
pick fresh y and apply typing_abs.
(** We cannot apply [H0] directly here.  The first problem is that the induction hypothesis has [(subst_ee open_ee)], whereas in the goal we have [(open_ee subst_ee)].  The lemma [subst_ee_open_ee_var] lets us swap the order of these two operations. *)
rewrite subst_ee_open_ee_var...
(** The second problem is how the concatenations are associated in the environments.  In the goal, we currently have  <<       (y ~ bind_typ V ++ F ++ E), >> where concatenation associates to the right.  In order to apply the induction hypothesis, we need  <<        ((y ~ bind_typ V ++ F) ++ E). >> We can use the [rewrite_env] tactic to perform this rewriting, or we can rewrite directly with an appropriate lemma from the MetatheoryEnv library. *)
rewrite <- app_assoc.
(** Now we can apply the induction hypothesis. *)
apply H0...
(** The remaining cases in this proof are straightforward, given everything that we have pointed out above. *)
Case "typing_tabs".
pick fresh Y and apply typing_tabs.
rewrite subst_ee_open_te_var...
rewrite <- app_assoc.
apply H0...
Case "typing_let".
pick fresh y and apply typing_let...
rewrite subst_ee_open_ee_var...
rewrite <- app_assoc.
apply H0...
Case "typing_case".
pick fresh y and apply typing_case...
rewrite subst_ee_open_ee_var...
rewrite <- app_assoc.
apply H0...
rewrite subst_ee_open_ee_var...
rewrite <- app_assoc.
apply H2...
Qed.
(* end show *)
(************************************************************************ *)
(** ** Type substitution preserves typing (11) *)
Proof with simpl_env;
eauto 6 using wf_env_subst_tb,
wf_typ_subst_tb,
sub_through_subst_tt.
intros Q E F Z e T P Typ PsubQ.
remember (F ++ Z ~ bind_sub Q ++ E) as G.
generalize dependent F.
induction Typ; intros F EQ; subst;
simpl subst_te in *; simpl subst_tt in *...
Case "typing_var".
apply typing_var...
rewrite (map_subst_tb_id E Z P);
[ | auto | eapply fresh_mid_tail; eauto ].
analyze_binds H0...
Case "typing_abs".
pick fresh y and apply typing_abs.
rewrite subst_te_open_ee_var...
rewrite_env (map (subst_tb Z P) (y ~ bind_typ V ++ F) ++ E).
apply H0...
Case "typing_tabs".
pick fresh Y and apply typing_tabs.
rewrite subst_te_open_te_var...
rewrite subst_tt_open_tt_var...
rewrite_env (map (subst_tb Z P) (Y ~ bind_sub V ++ F) ++ E).
apply H0...
Case "typing_tapp".
rewrite subst_tt_open_tt...
Case "typing_let".
pick fresh y and apply typing_let...
rewrite subst_te_open_ee_var...
rewrite_env (map (subst_tb Z P) (y ~ bind_typ T1 ++ F) ++ E).
apply H0...
Case "typing_case".
pick fresh y and apply typing_case...
SCase "inl branch".
rewrite subst_te_open_ee_var...
rewrite_env (map (subst_tb Z P) (y ~ bind_typ T1 ++ F) ++ E).
apply H0...
SCase "inr branch".
rewrite subst_te_open_ee_var...
rewrite_env (map (subst_tb Z P) (y ~ bind_typ T2 ++ F) ++ E).
apply H2...
Qed.
(* ********************************************************************** *)
(** * #<a name="preservation"></a># Preservation *)
(* ********************************************************************** *)
(** ** Inversion of typing (13) *)
Proof with auto.
intros E S1 e1 T Typ.
remember (exp_abs S1 e1) as e.
generalize dependent e1.
generalize dependent S1.
induction Typ; intros S1 b1 EQ U1 U2 Sub; inversion EQ; subst.
Case "typing_abs".
inversion Sub; subst.
split...
exists T1. exists L...
Case "typing_sub".
auto using (sub_transitivity T).
Qed.
Proof with simpl_env; auto.
intros E S1 e1 T Typ.
remember (exp_tabs S1 e1) as e.
generalize dependent e1.
generalize dependent S1.
induction Typ; intros S1 e0 EQ U1 U2 Sub; inversion EQ; subst.
Case "typing_tabs".
inversion Sub; subst.
split...
exists T1.
exists (L0 `union` L).
intros Y Fr.
split...
rewrite_env (empty ++ Y ~ bind_sub U1 ++ E).
apply (typing_narrowing S1)...
Case "typing_sub".
auto using (sub_transitivity T).
Qed.
Proof with eauto.
intros E e1 T Typ.
remember (exp_inl e1) as e. generalize dependent e1.
induction Typ; intros e' EQ U1 U2 Sub; inversion EQ; subst.
Case "typing_sub".
eauto using (sub_transitivity T).
Case "typing_inl".
inversion Sub; subst...
Qed.
Proof with eauto.
intros E e1 T Typ.
remember (exp_inr e1) as e. generalize dependent e1.
induction Typ; intros e' EQ U1 U2 Sub; inversion EQ; subst.
Case "typing_sub".
eauto using (sub_transitivity T).
Case "typing_inr".
inversion Sub; subst...
Qed.
(* ********************************************************************** *)
(** ** Preservation (20) *)
Proof with simpl_env; eauto.
intros E e e' T Typ. generalize dependent e'.
induction Typ; intros e' Red; try solve [ inversion Red; subst; eauto ].
Case "typing_app".
inversion Red; subst...
SCase "red_abs".
destruct (typing_inv_abs _ _ _ _ Typ1 T1 T2) as [P1 [S2 [L P2]]].
apply sub_reflexivity...
pick fresh x.
destruct (P2 x) as [? ?]...
rewrite (subst_ee_intro x)...
rewrite_env (empty ++ E).
apply (typing_through_subst_ee T).
apply (typing_sub S2)...
rewrite_env (empty ++ x ~ bind_typ T ++ E).
apply sub_weakening...
eauto.
Case "typing_tapp".
inversion Red; subst...
SCase "red_tabs".
destruct (typing_inv_tabs _ _ _ _ Typ T1 T2) as [P1 [S2 [L P2]]].
apply sub_reflexivity...
pick fresh X.
destruct (P2 X) as [? ?]...
rewrite (subst_te_intro X)...
rewrite (subst_tt_intro X)...
rewrite_env (map (subst_tb X T) empty ++ E).
apply (typing_through_subst_te T1)...
Case "typing_let".
inversion Red; subst.
SCase "red_let_1".
eapply typing_let; eauto.
SCase "red_let".
pick fresh x.
rewrite (subst_ee_intro x)...
rewrite_env (empty ++ E).
apply (typing_through_subst_ee T1)...
Case "typing_case".
inversion Red; subst.
SCase "red_case_1".
eapply typing_case; eauto.
SCase "red_case_inl".
destruct (typing_inv_inl _ _ _ Typ T1 T2) as [S1 [J2 J3]].
apply sub_reflexivity...
pick fresh x.
rewrite (subst_ee_intro x)...
rewrite_env (empty ++ E).
apply (typing_through_subst_ee T1)...
SCase "red_case_inr".
destruct (typing_inv_inr _ _ _ Typ T1 T2) as [S1 [J2 J3]].
apply sub_reflexivity...
pick fresh x.
rewrite (subst_ee_intro x)...
rewrite_env (empty ++ E).
apply (typing_through_subst_ee T2)...
Qed.
(* ********************************************************************** *)
(** * #<a name="progress"></a># Progress *)
(* ********************************************************************** *)
(** ** Canonical forms (14) *)

Lemma canonical_form_abs : forall e U1 U2,
value e ->
typing empty e (typ_arrow U1 U2) ->
exists V, exists e1, e = exp_abs V e1.