594 lines
18 KiB
Text
594 lines
18 KiB
Text
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(** Type-safety proofs for Fsub.
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Authors: Brian Aydemir and Arthur Charguéraud, with help from
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Aaron Bohannon, Jeffrey Vaughan, and Dimitrios Vytiniotis.
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In parentheses are given the label of the corresponding lemma in
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the appendix (informal proofs) of the POPLmark Challenge.
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Table of contents:
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- #<a href="##subtyping">Properties of subtyping</a>#
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- #<a href="##typing">Properties of typing</a>#
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- #<a href="##preservation">Preservation</a>#
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- #<a href="##progress">Progress</a># *)
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Require Export Fsub_Lemmas.
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(* ********************************************************************** *)
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(** * #<a name="subtyping"></a># Properties of subtyping *)
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(* ********************************************************************** *)
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(** ** Reflexivity (1) *)
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Lemma sub_reflexivity : forall E T,
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wf_env E ->
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wf_typ E T ->
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sub E T T.
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Proof with eauto.
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intros E T Ok Wf.
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induction Wf...
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pick fresh Y and apply sub_all...
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Qed.
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(* ********************************************************************** *)
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(** ** Weakening (2) *)
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Lemma sub_weakening : forall E F G S T,
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sub (G ++ E) S T ->
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wf_env (G ++ F ++ E) ->
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sub (G ++ F ++ E) S T.
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Proof with simpl_env; auto using wf_typ_weakening.
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intros E F G S T Sub Ok.
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remember (G ++ E) as H.
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generalize dependent G.
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induction Sub; intros G Ok EQ; subst...
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Case "sub_trans_tvar".
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apply (sub_trans_tvar U)...
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Case "sub_all".
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pick fresh Y and apply sub_all...
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rewrite <- concat_assoc.
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apply H0...
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Qed.
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(* ********************************************************************** *)
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(** ** Narrowing and transitivity (3) *)
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Definition transitivity_on Q := forall E S T,
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sub E S Q -> sub E Q T -> sub E S T.
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Lemma sub_narrowing_aux : forall Q F E Z P S T,
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transitivity_on Q ->
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sub (F ++ [(Z, bind_sub Q)] ++ E) S T ->
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sub E P Q ->
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sub (F ++ [(Z, bind_sub P)] ++ E) S T.
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Proof with simpl_env; eauto using wf_typ_narrowing, wf_env_narrowing.
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intros Q F E Z P S T TransQ SsubT PsubQ.
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remember (F ++ [(Z, bind_sub Q)] ++ E) as G. generalize dependent F.
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induction SsubT; intros F EQ; subst...
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Case "sub_top".
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apply sub_top...
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Case "sub_refl_tvar".
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apply sub_refl_tvar...
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Case "sub_trans_tvar".
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destruct (X == Z); subst.
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SCase "X = Z".
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apply (sub_trans_tvar P); [ eauto using fresh_mid_head | ].
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apply TransQ.
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SSCase "P <: Q".
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rewrite_env (empty ++ (F ++ [(Z, bind_sub P)]) ++ E).
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apply sub_weakening...
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SSCase "Q <: T".
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binds_get H.
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inversion H1; subst...
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SCase "X <> Z".
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apply (sub_trans_tvar U)...
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binds_cases H...
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Case "sub_all".
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pick fresh Y and apply sub_all...
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rewrite <- concat_assoc.
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apply H0...
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Qed.
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Lemma sub_transitivity : forall Q,
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transitivity_on Q.
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Proof with simpl_env; auto.
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unfold transitivity_on.
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intros Q E S T SsubQ QsubT.
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assert (W : type Q) by auto.
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generalize dependent T.
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generalize dependent S.
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generalize dependent E.
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remember Q as Q' in |-.
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generalize dependent Q'.
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induction W;
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intros Q' EQ E S SsubQ;
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induction SsubQ; try discriminate; inversion EQ; subst;
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intros T' QsubT;
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inversion QsubT; subst; eauto 4 using sub_trans_tvar.
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Case "sub_all / sub_top".
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assert (sub E (typ_all S1 S2) (typ_all T1 T2)).
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SCase "proof of assertion".
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pick fresh y and apply sub_all...
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auto.
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Case "sub_all / sub_all".
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pick fresh Y and apply sub_all.
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SCase "bounds".
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eauto.
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SCase "bodies".
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lapply (H0 Y); [ intros K | auto ].
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apply (K (open_tt T2 Y))...
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rewrite_env (empty ++ [(Y, bind_sub T0)] ++ E).
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apply (sub_narrowing_aux T1)...
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unfold transitivity_on.
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auto using (IHW T1).
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Qed.
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Lemma sub_narrowing : forall Q E F Z P S T,
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sub E P Q ->
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sub (F ++ [(Z, bind_sub Q)] ++ E) S T ->
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sub (F ++ [(Z, bind_sub P)] ++ E) S T.
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Proof.
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intros.
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eapply sub_narrowing_aux; eauto.
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apply sub_transitivity.
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Qed.
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(* ********************************************************************** *)
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(** ** Type substitution preserves subtyping (10) *)
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Lemma sub_through_subst_tt : forall Q E F Z S T P,
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sub (F ++ [(Z, bind_sub Q)] ++ E) S T ->
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sub E P Q ->
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sub (map (subst_tb Z P) F ++ E) (subst_tt Z P S) (subst_tt Z P T).
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Proof with
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simpl_env;
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eauto 4 using wf_typ_subst_tb, wf_env_subst_tb, wf_typ_weaken_head.
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intros Q E F Z S T P SsubT PsubQ.
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remember (F ++ [(Z, bind_sub Q)] ++ E) as G.
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generalize dependent F.
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induction SsubT; intros G EQ; subst; simpl subst_tt...
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Case "sub_top".
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apply sub_top...
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Case "sub_refl_tvar".
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destruct (X == Z); subst.
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SCase "X = Z".
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apply sub_reflexivity...
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SCase "X <> Z".
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apply sub_reflexivity...
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inversion H0; subst.
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binds_cases H3...
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apply (wf_typ_var (subst_tt Z P U))...
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Case "sub_trans_tvar".
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destruct (X == Z); subst.
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SCase "X = Z".
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apply (sub_transitivity Q).
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SSCase "left branch".
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rewrite_env (empty ++ map (subst_tb Z P) G ++ E).
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apply sub_weakening...
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SSCase "right branch".
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rewrite (subst_tt_fresh Z P Q).
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binds_get H.
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inversion H1; subst...
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apply (notin_fv_wf E); eauto using fresh_mid_tail.
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SCase "X <> Z".
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apply (sub_trans_tvar (subst_tt Z P U))...
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rewrite (map_subst_tb_id E Z P);
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[ | auto | eapply fresh_mid_tail; eauto ].
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binds_cases H...
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Case "sub_all".
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pick fresh X and apply sub_all...
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rewrite subst_tt_open_tt_var...
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rewrite subst_tt_open_tt_var...
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rewrite_env (map (subst_tb Z P) ([(X, bind_sub T1)] ++ G) ++ E).
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apply H0...
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Qed.
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(* ********************************************************************** *)
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(** * #<a name="typing"></a># Properties of typing *)
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(* ********************************************************************** *)
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(** ** Weakening (5) *)
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Lemma typing_weakening : forall E F G e T,
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typing (G ++ E) e T ->
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wf_env (G ++ F ++ E) ->
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typing (G ++ F ++ E) e T.
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Proof with simpl_env;
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eauto using wf_typ_weakening,
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wf_typ_from_wf_env_typ,
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wf_typ_from_wf_env_sub,
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sub_weakening.
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intros E F G e T Typ.
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remember (G ++ E) as H.
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generalize dependent G.
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induction Typ; intros G EQ Ok; subst...
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Case "typing_abs".
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pick fresh x and apply typing_abs.
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lapply (H x); [intros K | auto].
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rewrite <- concat_assoc.
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apply (H0 x)...
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Case "typing_tabs".
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pick fresh X and apply typing_tabs.
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lapply (H X); [intros K | auto].
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rewrite <- concat_assoc.
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apply (H0 X)...
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Qed.
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(* ********************************************************************** *)
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(** ** Strengthening (6) *)
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Lemma sub_strengthening : forall x U E F S T,
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sub (F ++ [(x, bind_typ U)] ++ E) S T ->
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sub (F ++ E) S T.
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Proof with eauto using wf_typ_strengthening, wf_env_strengthening.
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intros x U E F S T SsubT.
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remember (F ++ [(x, bind_typ U)] ++ E) as E'.
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generalize dependent F.
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induction SsubT; intros F EQ; subst...
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Case "sub_trans_tvar".
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apply (sub_trans_tvar U0)...
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binds_cases H...
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Case "sub_all".
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pick fresh X and apply sub_all...
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rewrite <- concat_assoc.
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apply H0...
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Qed.
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(************************************************************************ *)
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(** ** Narrowing for typing (7) *)
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Lemma typing_narrowing : forall Q E F X P e T,
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sub E P Q ->
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typing (F ++ [(X, bind_sub Q)] ++ E) e T ->
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typing (F ++ [(X, bind_sub P)] ++ E) e T.
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Proof with eauto 6 using wf_env_narrowing, wf_typ_narrowing, sub_narrowing.
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intros Q E F X P e T PsubQ Typ.
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remember (F ++ [(X, bind_sub Q)] ++ E) as E'.
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generalize dependent F.
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induction Typ; intros F EQ; subst...
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Case "typing_var".
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binds_cases H0...
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Case "typing_abs".
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pick fresh y and apply typing_abs.
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rewrite <- concat_assoc.
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apply H0...
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Case "typing_tabs".
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pick fresh Y and apply typing_tabs.
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rewrite <- concat_assoc.
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apply H0...
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Qed.
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(************************************************************************ *)
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(** ** Substitution preserves typing (8) *)
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Lemma typing_through_subst_ee : forall U E F x T e u,
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typing (F ++ [(x, bind_typ U)] ++ E) e T ->
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typing E u U ->
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typing (F ++ E) (subst_ee x u e) T.
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(* begin show *)
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(** We provide detailed comments for the following proof, mainly to
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point out several useful tactics and proof techniques.
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Starting a proof with "Proof with <some tactic>" allows us to
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specify a default tactic that should be used to solve goals. To
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invoke this default tactic at the end of a proof step, we signal
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the end of the step with three periods instead of a single one,
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e.g., "apply typing_weakening...". *)
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Proof with simpl_env;
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eauto 4 using wf_typ_strengthening,
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wf_env_strengthening,
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sub_strengthening.
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(** The proof proceeds by induction on the given typing derivation
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for e. We use the remember tactic, along with generalize
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dependent, to ensure that the goal is properly strengthened
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before we use induction. *)
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intros U E F x T e u TypT TypU.
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remember (F ++ [(x, bind_typ U)] ++ E) as E'.
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generalize dependent F.
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induction TypT; intros F EQ; subst; simpl subst_ee...
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(** The typing_var case involves a case analysis on whether the
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variable is the same as the one being substituted for. *)
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Case "typing_var".
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destruct (x0 == x); subst.
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(** In the case where x0=x, we first observe that hypothesis H0
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implies that T=U, since x can only be bound once in the
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environment. The conclusion then follows from hypothesis TypU
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and weakening. We can use the binds_get tactic, described in
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the Environment library, with H0 to obtain the fact that
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T=U. *)
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SCase "x0 = x".
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binds_get H0.
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inversion H2; subst.
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(** In order to apply typing_weakening, we need to rewrite the
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environment so that it has the right shape. (We could
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also prove a corollary of typing_weakening.) The
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rewrite_env tactic, described in the Environment library,
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is one way to perform this rewriting. *)
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rewrite_env (empty ++ F ++ E).
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apply typing_weakening...
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(** In the case where x0<>x, the result follows by an exhaustive
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case analysis on exactly where x0 is bound in the environment.
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We perform this case analysis by using the binds_cases tactic,
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described in the Environment library. *)
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SCase "x0 <> x".
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binds_cases H0.
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eauto using wf_env_strengthening.
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eauto using wf_env_strengthening.
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(** Informally, the typing_abs case is a straightforward application
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of the induction hypothesis, which is called H0 here. *)
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Case "typing_abs".
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(** We use the "pick fresh and apply" tactic to apply the rule
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typing_abs without having to calculate the appropriate finite
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set of atoms. *)
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pick fresh y and apply typing_abs.
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(** We cannot apply H0 directly here. The first problem is that
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the induction hypothesis has (subst_ee open_ee), whereas in
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the goal we have (open_ee subst_ee). The lemma
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subst_ee_open_ee_var lets us swap the order of these two
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operations. *)
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rewrite subst_ee_open_ee_var...
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(** The second problem is how the concatenations are associated in
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the environments. In the goal, we currently have
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<< ([(y, bind_typ V)] ++ F ++ E),
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>>
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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
|
||
|
|
Environment library. *)
|
||
|
|
|
||
|
|
rewrite <- concat_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 <- concat_assoc.
|
||
|
|
apply H0...
|
||
|
|
Qed.
|
||
|
|
(* end show *)
|
||
|
|
|
||
|
|
|
||
|
|
(************************************************************************ *)
|
||
|
|
(** ** Type substitution preserves typing (11) *)
|
||
|
|
|
||
|
|
Lemma typing_through_subst_te : forall Q E F Z e T P,
|
||
|
|
typing (F ++ [(Z, bind_sub Q)] ++ E) e T ->
|
||
|
|
sub E P Q ->
|
||
|
|
typing (map (subst_tb Z P) F ++ E) (subst_te Z P e) (subst_tt Z P T).
|
||
|
|
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 ].
|
||
|
|
binds_cases 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...
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** * #<a name="preservation"></a># Preservation *)
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** ** Inversion of typing (13) *)
|
||
|
|
|
||
|
|
Lemma typing_inv_abs : forall E S1 e1 T,
|
||
|
|
typing E (exp_abs S1 e1) T ->
|
||
|
|
forall U1 U2, sub E T (typ_arrow U1 U2) ->
|
||
|
|
sub E U1 S1
|
||
|
|
/\ exists S2, exists L, forall x, x `notin` L ->
|
||
|
|
typing ([(x, bind_typ S1)] ++ E) (open_ee e1 x) S2 /\ sub E S2 U2.
|
||
|
|
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.
|
||
|
|
|
||
|
|
Lemma typing_inv_tabs : forall E S1 e1 T,
|
||
|
|
typing E (exp_tabs S1 e1) T ->
|
||
|
|
forall U1 U2, sub E T (typ_all U1 U2) ->
|
||
|
|
sub E U1 S1
|
||
|
|
/\ exists S2, exists L, forall X, X `notin` L ->
|
||
|
|
typing ([(X, bind_sub U1)] ++ E) (open_te e1 X) (open_tt S2 X)
|
||
|
|
/\ sub ([(X, bind_sub U1)] ++ E) (open_tt S2 X) (open_tt U2 X).
|
||
|
|
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.
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** ** Preservation (20) *)
|
||
|
|
|
||
|
|
Lemma preservation : forall E e e' T,
|
||
|
|
typing E e T ->
|
||
|
|
red e e' ->
|
||
|
|
typing E e' T.
|
||
|
|
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)...
|
||
|
|
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.
|
||
|
|
Proof.
|
||
|
|
intros e U1 U2 Val Typ.
|
||
|
|
remember empty as E.
|
||
|
|
remember (typ_arrow U1 U2) as T.
|
||
|
|
revert U1 U2 HeqT HeqE.
|
||
|
|
induction Typ; intros U1 U2 EQT EQE; subst;
|
||
|
|
try solve [ inversion Val | inversion EQT | eauto ].
|
||
|
|
Case "typing_sub".
|
||
|
|
inversion H; subst; eauto.
|
||
|
|
inversion H0.
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma canonical_form_tabs : forall e U1 U2,
|
||
|
|
value e ->
|
||
|
|
typing empty e (typ_all U1 U2) ->
|
||
|
|
exists V, exists e1, e = exp_tabs V e1.
|
||
|
|
Proof.
|
||
|
|
intros e U1 U2 Val Typ.
|
||
|
|
remember empty as E.
|
||
|
|
remember (typ_all U1 U2) as T.
|
||
|
|
revert U1 U2 HeqT HeqT.
|
||
|
|
induction Typ; intros U1 U2 EQT EQE; subst;
|
||
|
|
try solve [ inversion Val | inversion EQT | eauto ].
|
||
|
|
Case "typing_sub".
|
||
|
|
inversion H; subst; eauto.
|
||
|
|
inversion H0.
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
(* ********************************************************************** *)
|
||
|
|
(** ** Progress (16) *)
|
||
|
|
|
||
|
|
Lemma progress : forall e T,
|
||
|
|
typing empty e T ->
|
||
|
|
value e \/ exists e', red e e'.
|
||
|
|
Proof with eauto.
|
||
|
|
intros e T Typ.
|
||
|
|
remember empty as E. generalize dependent HeqE.
|
||
|
|
assert (Typ' : typing E e T)...
|
||
|
|
induction Typ; intros EQ; subst...
|
||
|
|
Case "typing_var".
|
||
|
|
inversion H0.
|
||
|
|
Case "typing_app".
|
||
|
|
right.
|
||
|
|
destruct IHTyp1 as [Val1 | [e1' Rede1']]...
|
||
|
|
SCase "Val1".
|
||
|
|
destruct IHTyp2 as [Val2 | [e2' Rede2']]...
|
||
|
|
SSCase "Val2".
|
||
|
|
destruct (canonical_form_abs _ _ _ Val1 Typ1) as [S [e3 EQ]].
|
||
|
|
subst.
|
||
|
|
exists (open_ee e3 e2)...
|
||
|
|
Case "typing_tapp".
|
||
|
|
right.
|
||
|
|
destruct IHTyp as [Val1 | [e1' Rede1']]...
|
||
|
|
SCase "Val1".
|
||
|
|
destruct (canonical_form_tabs _ _ _ Val1 Typ) as [S [e3 EQ]].
|
||
|
|
subst.
|
||
|
|
exists (open_te e3 T)...
|
||
|
|
SCase "e1' Rede1'".
|
||
|
|
exists (exp_tapp e1' T)...
|
||
|
|
Qed.
|