exercises 06

theories/systemf_mu/exercises06.v

@@ -122,55 +122,50 @@ Admitted.
 
 (** ** Exercise 1: Encode arithmetic expressions *)
 
-Definition aexpr : type := μ: (Int + (#0 × #0)) + (#0 × #0).
+Definition aexpr : type := #0 (* FIXME *).
 
-Definition num_val (v : val) : val := RollV (InjLV $ InjLV v).
-Definition num_expr (e : expr) : expr := Roll (InjL $ InjL e).
+Definition num_val (v : val) : val := #0 (* FIXME *).
+Definition num_expr (e : expr) : expr := #0 (* FIXME *).
 
-Definition plus_val (v1 v2 : val) : val := RollV (InjLV $ InjRV (v1, v2)).
-Definition plus_expr (e1 e2 : expr) : expr := Roll (InjL $ InjR (e1, e2)).
+Definition plus_val (v1 v2 : val) : val := #0 (* FIXME *).
+Definition plus_expr (e1 e2 : expr) : expr := #0 (* FIXME *).
 
-Definition mul_val (v1 v2 : val) : val := RollV (InjRV (v1, v2)).
-Definition mul_expr (e1 e2 : expr) : expr := Roll (InjR (e1, e2)).
+Definition mul_val (v1 v2 : val) : val := #0 (* FIXME *).
+Definition mul_expr (e1 e2 : expr) : expr := #0 (* FIXME *).
 
 Lemma num_expr_typed n Γ e :
   TY n; Γ ⊢ e : Int →
   TY n; Γ ⊢ num_expr e : aexpr.
 Proof.
   intros. solve_typing.
-Qed.
+  (* FIXME *)
+(*Qed.*)
+Admitted.
 Lemma plus_expr_typed n Γ e1 e2 :
   TY n; Γ ⊢ e1 : aexpr →
   TY n; Γ ⊢ e2 : aexpr →
   TY n; Γ ⊢ plus_expr e1 e2 : aexpr.
 Proof.
-  intros; solve_typing.
-Qed.
+  (*intros; solve_typing.*)
+(*Qed.*)
+Admitted.
 Lemma mul_expr_typed n Γ e1 e2 :
   TY n; Γ ⊢ e1 : aexpr →
   TY n; Γ ⊢ e2 : aexpr →
   TY n; Γ ⊢ mul_expr e1 e2 : aexpr.
 Proof.
-  intros; solve_typing.
-Qed.
+  (*intros; solve_typing.*)
+(*Qed.*)
+Admitted.
 
 Definition eval_aexpr : val :=
-  fix: "rec" "e" := match: unroll "e" with
-           InjL "e'" =>
-              match: "e'" with
-                InjL "n" => "n"
-              | InjR "es" => "rec" (Fst "es") + "rec" (Snd "es")
-              end
-           | InjR "es" => "rec" (Fst "es") * "rec" (Snd "es")
-            end.
-
+  #0 (* FIXME *).
 Lemma eval_aexpr_typed Γ n :
   TY n; Γ ⊢ eval_aexpr : (aexpr → Int).
 Proof.
-  unfold eval_aexpr.
-  apply fix_typing; solve_typing.
-  done.
-Qed.
+(*Qed.*)
+(* FIXME *)
+Admitted.
 
 
 (** Exercise 3: Lists *)
@@ -182,118 +177,12 @@ Definition list_t (A : type) : type :=
   .
 
 Definition mylist_impl : val :=
-  pack ((#0, #()),        (* mynil *)
-        (λ: "a" "l", (#1 + Fst "l", ("a", Snd "l"))),   (* mycons *)
-        (Λ, λ: "l" "n" "c",
-          if: Fst "l" = #0 then "n" else "c" (Fst (Snd "l")) (Fst "l" - #1, Snd (Snd "l")))) (* mycase *)
+  #0 (* FIXME *)
   .
 
-Fixpoint represents_list_rec (l : list val) (v : val) :=
-  match l with
-  | [] => v = #()
-  | h :: l' => ∃ v' : val, v = (h, v')%V ∧ represents_list_rec l' v'
-  end.
-Definition represents_list (l : list val) (v : val) :=
-  ∃ (n : Z) (v' : val), n = length l ∧ v = (#n, v')%V ∧
-  represents_list_rec l v'.
-
 Lemma mylist_impl_sem_typed A :
   type_wf 0 A →
   ∀ k, 𝒱 (list_t A) δ_any k mylist_impl.
 Proof.
-  intros Hwf k.
-  unfold list_t.
-  simp type_interp.
-  eexists _. split; first done.
-  pose_sem_type (λ k' (v : val), ∃ l : list val, represents_list l v ∧ Forall (𝒱 A δ_any k') l) as τ.
-  {
-    intros k' v (l & Hrep & Hl).
-    destruct Hrep as (n & v' & -> & -> & Hrep). simplify_closed.
-    induction l as [ | h l IH] in v', Hrep, Hl |-*; simpl in Hrep.
-    - rewrite Hrep. done.
-    - destruct Hrep as (v'' & -> & Hrep).
-      simplify_closed.
-      + eapply Forall_inv in Hl. eapply val_rel_is_closed in Hl. done.
-      + eapply IH; first done. eapply Forall_inv_tail in Hl. done.
-  }
-  {
-    intros k' k'' v (l & Hrep & Hl) Hle.
-    exists l. split; first done.
-    eapply Forall_impl; first done.
-    intros v'. by eapply val_rel_mono.
-  }
-  exists τ.
-  simp type_interp. eexists _, _. split; first done. split;
-    [ simp type_interp; eexists _, _; split; first done; split | ].
-  - simp type_interp. simpl. exists []. split; last done.
-    eexists _, _; simpl; split; first done. done.
-  - simp type_interp. eexists _, _. split; first done. split; first done.
-    intros v2 kd Hv2. simpl. eapply (sem_val_expr_rel _ _ _ (LamV _ _)).
-    simp type_interp. eexists _, _. split; first done.
-    specialize (val_rel_is_closed _ _ _ _ Hv2) as ?.
-    split; first by simplify_closed.
-
-    intros v3 kd2 Hv3.
-    simpl. rewrite subst_is_closed_nil; last done.
-    simp type_interp in Hv3. destruct Hv3 as (l & (len & hv & -> & -> & Hv3) & Hl).
-    simpl.
-
-    eapply expr_det_steps_closure.
-    { repeat do_det_step. constructor. }
-    eapply (sem_val_expr_rel _ _ _ (PairV _ (PairV _ _))).
-    simp type_interp. simpl.
-    exists (v2 :: l). split.
-    { simpl. eexists _, (v2, hv)%V. split; first done.
-      simpl. split. { f_equal. f_equal. f_equal. lia. }
-      eexists _; done.
-    }
-    econstructor.
-    { eapply sem_val_rel_cons; eapply val_rel_mono; last done. lia. }
-    eapply Forall_impl; first done.
-    intros. eapply val_rel_mono; last done. lia.
-  - simp type_interp. eexists; split; first done. split; first done.
-    intros τ'. eapply (sem_val_expr_rel _ _ _ (LamV _ _)).
-    simp type_interp. eexists _, _. split; first done. split; first done.
-    intros v2 kd Hv2. simpl.
-    eapply (sem_val_expr_rel _ _ _ (LamV _ _)).
-    simp type_interp. eexists _, _. split; first done.
-    specialize (val_rel_is_closed _ _ _ _ Hv2) as ?.
-    split; first by simplify_closed.
-    intros v3 kd2 Hv3. simpl.
-    rewrite subst_is_closed_nil; last done.
-    eapply (sem_val_expr_rel _ _ _ (LamV _ _)).
-    simp type_interp. eexists _, _. split; first done.
-    specialize (val_rel_is_closed _ _ _ _ Hv3) as ?.
-    split; first by simplify_closed.
-    intros v4 kd3 Hv4. simpl.
-    rewrite !subst_is_closed_nil; [ | done..].
-
-    simp type_interp in Hv2. simpl in Hv2. destruct Hv2 as (l & (len & vl & -> & -> & Hl) & Hlf).
-    simpl.
-
-    destruct l as [ | h l].
-    + eapply expr_det_steps_closure.
-      { repeat do_det_step. econstructor. }
-      eapply sem_val_expr_rel.
-      eapply val_rel_mono; last done. lia.
-    + simpl in Hl. destruct Hl as (vl' & -> & Hl).
-      eapply expr_det_steps_closure.
-      { repeat do_det_step. econstructor. }
-      replace (S (length l) - 1)%Z with (Z.of_nat $ length l) by lia.
-      eapply semantic_app.
-      { eapply semantic_app.
-        { eapply sem_val_expr_rel.
-          eapply val_rel_mono; last done. lia.
-        }
-        apply Forall_inv in Hlf.
-        eapply sem_val_expr_rel, val_rel_mono.
-        2: { eapply sem_val_rel_cons in Hlf. erewrite sem_val_rel_cons in Hlf. asimpl in Hlf. apply Hlf. }
-        lia.
-      }
-
-      eapply (sem_val_expr_rel _ _ _ (PairV (LitV _) _)). simp type_interp. simpl. exists l. split.
-      { eexists _, _. done. }
-      eapply Forall_inv_tail.
-      eapply Forall_impl; first done.
-      intros. eapply val_rel_mono; last done. lia.
-Qed.
+  (* FIXME *)
+Admitted.