|
|
|
|
@ -0,0 +1,375 @@
|
|
|
|
|
From stdpp Require Import gmap base relations tactics.
|
|
|
|
|
From iris Require Import prelude.
|
|
|
|
|
From semantics.ts.systemf_mu Require Import lang notation types pure tactics logrel.
|
|
|
|
|
From Autosubst Require Import Autosubst.
|
|
|
|
|
|
|
|
|
|
(** * Exercise Sheet 6 *)
|
|
|
|
|
|
|
|
|
|
Notation sub := lang.subst.
|
|
|
|
|
|
|
|
|
|
Implicit Types
|
|
|
|
|
(e : expr)
|
|
|
|
|
(v : val)
|
|
|
|
|
(A B : type)
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
(** ** Exercise 5: Keep Rollin' *)
|
|
|
|
|
(** This exercise is slightly tricky.
|
|
|
|
|
We strongly recommend you to first work on the other exercises.
|
|
|
|
|
You may use the results from this exercise, in particular the fixpoint combinator and its typing, in other exercises, however (which is why it comes first in this Coq file).
|
|
|
|
|
*)
|
|
|
|
|
Section recursion_combinator.
|
|
|
|
|
Variable (f x: string). (* the template of the recursive function *)
|
|
|
|
|
Hypothesis (Hfx: f ≠ x).
|
|
|
|
|
|
|
|
|
|
(** Recursion Combinator *)
|
|
|
|
|
(* First, setup a definition [Rec] satisfying the reduction lemmas below. *)
|
|
|
|
|
|
|
|
|
|
(** You may find an auxiliary definition [rec_body] helpful *)
|
|
|
|
|
Definition rec_body (t: expr) : expr :=
|
|
|
|
|
roll (λ: f x, t (λ: x, (unroll f) f x) x).
|
|
|
|
|
|
|
|
|
|
Definition Rec (t: expr): val :=
|
|
|
|
|
λ: x, (unroll (rec_body t)) (rec_body t) x.
|
|
|
|
|
|
|
|
|
|
Lemma closed_rec_body t :
|
|
|
|
|
is_closed [] t → is_closed [] (rec_body t).
|
|
|
|
|
Proof. simplify_closed. Qed.
|
|
|
|
|
Lemma closed_Rec t :
|
|
|
|
|
is_closed [] t → is_closed [] (Rec t).
|
|
|
|
|
Proof. simplify_closed. Qed.
|
|
|
|
|
Lemma is_val_Rec t:
|
|
|
|
|
is_val (Rec t).
|
|
|
|
|
Proof. done. Qed.
|
|
|
|
|
|
|
|
|
|
Lemma rec_body_subst_x e t:
|
|
|
|
|
(sub x e (rec_body t))%E = rec_body t.
|
|
|
|
|
Proof.
|
|
|
|
|
simpl. destruct decide; try naive_solver.
|
|
|
|
|
destruct decide; naive_solver.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
Lemma rec_body_subst_f e t:
|
|
|
|
|
(sub f e (rec_body t))%E = rec_body t.
|
|
|
|
|
Proof.
|
|
|
|
|
simpl. destruct decide; naive_solver.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
Lemma Rec_red (t e: expr):
|
|
|
|
|
is_val e →
|
|
|
|
|
is_val t →
|
|
|
|
|
is_closed [] e →
|
|
|
|
|
is_closed [] t →
|
|
|
|
|
rtc contextual_step ((Rec t) e) (t (Rec t) e).
|
|
|
|
|
Proof.
|
|
|
|
|
intros Hval1 Hval2 Hcl1 Hcl2. do 4 try econstructor 2.
|
|
|
|
|
- rewrite /Rec. eapply app_step_beta; first eauto.
|
|
|
|
|
simplify_closed.
|
|
|
|
|
- cbn -[rec_body]. rewrite rec_body_subst_x.
|
|
|
|
|
destruct decide; try naive_solver.
|
|
|
|
|
simpl. eapply app_step_l; last eauto.
|
|
|
|
|
eapply app_step_l; last done.
|
|
|
|
|
eapply unroll_roll_step; done.
|
|
|
|
|
- eapply app_step_l; last eauto.
|
|
|
|
|
eapply app_step_beta; first done.
|
|
|
|
|
simplify_closed.
|
|
|
|
|
- simpl.
|
|
|
|
|
rewrite decide_False; last congruence.
|
|
|
|
|
rewrite decide_False; last congruence.
|
|
|
|
|
rewrite decide_True; last done.
|
|
|
|
|
rewrite !decide_False; last done.
|
|
|
|
|
eapply app_step_beta; [eauto|].
|
|
|
|
|
simplify_closed.
|
|
|
|
|
- cbn -[rec_body].
|
|
|
|
|
rewrite (subst_is_closed_nil _ f); last done.
|
|
|
|
|
rewrite (subst_is_closed_nil _ x); last done.
|
|
|
|
|
destruct decide; try naive_solver.
|
|
|
|
|
destruct decide; naive_solver.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
Lemma rec_body_typing n Γ (A B: type) t :
|
|
|
|
|
Γ !! x = None →
|
|
|
|
|
Γ !! f = None →
|
|
|
|
|
type_wf n A →
|
|
|
|
|
type_wf n B →
|
|
|
|
|
TY n; Γ ⊢ t : ((A → B) → (A → B)) →
|
|
|
|
|
TY n; Γ ⊢ rec_body t : (μ: #0 → rename (+1) A → rename (+1) B).
|
|
|
|
|
Proof.
|
|
|
|
|
intros Hx Hf Hwf1 Hwf2 Hty.
|
|
|
|
|
rewrite /rec_body. econstructor; asimpl.
|
|
|
|
|
solve_typing.
|
|
|
|
|
eapply typed_weakening; eauto.
|
|
|
|
|
etrans; first eapply insert_subseteq; first done.
|
|
|
|
|
eapply insert_subseteq; rewrite lookup_insert_ne //=.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
Lemma Rec_typing n Γ A B t:
|
|
|
|
|
type_wf n A →
|
|
|
|
|
type_wf n B →
|
|
|
|
|
Γ !! x = None →
|
|
|
|
|
Γ !! f = None →
|
|
|
|
|
TY n; Γ ⊢ t : ((A → B) → (A → B)) →
|
|
|
|
|
TY n; Γ ⊢ (Rec t) : (A → B).
|
|
|
|
|
Proof.
|
|
|
|
|
intros Hwf1 Hwf2 Hx Hf Ht. econstructor; last done.
|
|
|
|
|
econstructor; last first.
|
|
|
|
|
{ econstructor. rewrite lookup_insert //=. }
|
|
|
|
|
econstructor; first eapply typed_unroll'.
|
|
|
|
|
- eapply typed_weakening with (Γ := Γ); eauto; last by eapply insert_subseteq.
|
|
|
|
|
eapply rec_body_typing with (A := A) (B := B); eauto.
|
|
|
|
|
- simpl. f_equal. f_equal; by asimpl.
|
|
|
|
|
- eapply typed_weakening with (Γ := Γ); eauto; last by eapply insert_subseteq.
|
|
|
|
|
eapply rec_body_typing; eauto.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
End recursion_combinator.
|
|
|
|
|
|
|
|
|
|
Definition Fix (f x: string) (e: expr) : val := (Rec f x (Lam f%string (Lam x%string e))).
|
|
|
|
|
(** We "seal" the definition to make it opaque to the [solve_typing] tactic.
|
|
|
|
|
We do not want [solve_typing] to unfold the definition, instead, we should manually
|
|
|
|
|
apply the typing rule below.
|
|
|
|
|
|
|
|
|
|
You do not need to understand this in detail -- the only thing of relevance is that
|
|
|
|
|
you can use the equality [Fix_eq] to unfold the definition of [Fix].
|
|
|
|
|
*)
|
|
|
|
|
Definition Fix_aux : seal (Fix). Proof. by eexists. Qed.
|
|
|
|
|
Definition Fix' := Fix_aux.(unseal).
|
|
|
|
|
Lemma Fix_eq : Fix' = Fix.
|
|
|
|
|
Proof. rewrite -Fix_aux.(seal_eq) //. Qed.
|
|
|
|
|
Arguments Fix' _ _ _ : simpl never.
|
|
|
|
|
|
|
|
|
|
(* the actual fixpoint combinator *)
|
|
|
|
|
Notation "'fix:' f x := e" := (Fix' f x e)%E
|
|
|
|
|
(at level 200, f, x at level 1, e at level 200,
|
|
|
|
|
format "'[' 'fix:' f x := '/ ' e ']'") : val_scope.
|
|
|
|
|
Notation "'fix:' f x := e" := (Fix' f x e)%E
|
|
|
|
|
(at level 200, f, x at level 1, e at level 200,
|
|
|
|
|
format "'[' 'fix:' f x := '/ ' e ']'") : expr_scope.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Lemma fix_red (f x: string) (e e': expr):
|
|
|
|
|
is_closed [x; f] e →
|
|
|
|
|
is_closed [] e' →
|
|
|
|
|
is_val e' →
|
|
|
|
|
f ≠ x →
|
|
|
|
|
rtc contextual_step ((fix: f x := e) e')%V (sub x e' (sub f (fix: f x := e)%V e)).
|
|
|
|
|
Proof.
|
|
|
|
|
intros He Hval. etransitivity; [|econstructor 2]; [| |econstructor 2].
|
|
|
|
|
- rewrite Fix_eq /Fix. eapply Rec_red; simpl; eauto.
|
|
|
|
|
- eapply app_step_l; last done.
|
|
|
|
|
eapply app_step_beta; last done.
|
|
|
|
|
done.
|
|
|
|
|
- simpl. rewrite decide_False; last congruence.
|
|
|
|
|
eapply app_step_beta; first done.
|
|
|
|
|
eapply is_closed_subst.
|
|
|
|
|
{ apply closed_Rec; done. }
|
|
|
|
|
eapply is_closed_weaken; eauto. set_solver.
|
|
|
|
|
- rewrite Fix_eq; done.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
Lemma fix_typing n Γ (f x: string) (A B: type) (e: expr):
|
|
|
|
|
type_wf n A →
|
|
|
|
|
type_wf n B →
|
|
|
|
|
f ≠ x →
|
|
|
|
|
TY n; <[x := A]> (<[f := (A → B)%ty]> Γ) ⊢ e : B →
|
|
|
|
|
TY n; Γ ⊢ (fix: f x := e) : (A → B).
|
|
|
|
|
Proof.
|
|
|
|
|
intros Hwf1 Hwf2 Hfx He.
|
|
|
|
|
rewrite Fix_eq /Fix.
|
|
|
|
|
eapply typed_weakening with (Γ := delete x (delete f Γ)); eauto; last first.
|
|
|
|
|
{ etrans; eapply delete_subseteq. }
|
|
|
|
|
eapply Rec_typing; eauto.
|
|
|
|
|
- eapply lookup_delete.
|
|
|
|
|
- rewrite lookup_delete_ne //=. eapply lookup_delete.
|
|
|
|
|
- econstructor; last eauto.
|
|
|
|
|
econstructor; last eauto.
|
|
|
|
|
rewrite -delete_insert_ne //=.
|
|
|
|
|
rewrite !insert_delete_insert. done.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
(** ** Exercise 1: Encode arithmetic expressions *)
|
|
|
|
|
|
|
|
|
|
Definition aexpr : type := μ: (Int + (#0 × #0)) + (#0 × #0).
|
|
|
|
|
|
|
|
|
|
Definition num_val (v : val) : val := RollV (InjLV $ InjLV v).
|
|
|
|
|
Definition num_expr (e : expr) : expr := Roll (InjL $ InjL e).
|
|
|
|
|
|
|
|
|
|
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 mul_val (v1 v2 : val) : val := RollV (InjRV (v1, v2)).
|
|
|
|
|
Definition mul_expr (e1 e2 : expr) : expr := Roll (InjR (e1, e2)).
|
|
|
|
|
|
|
|
|
|
Lemma num_expr_typed n Γ e :
|
|
|
|
|
TY n; Γ ⊢ e : Int →
|
|
|
|
|
TY n; Γ ⊢ num_expr e : aexpr.
|
|
|
|
|
Proof.
|
|
|
|
|
intros. solve_typing.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
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.
|
|
|
|
|
|
|
|
|
|
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.
|
|
|
|
|
|
|
|
|
|
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.
|
|
|
|
|
|
|
|
|
|
Lemma eval_aexpr_typed Γ n :
|
|
|
|
|
TY n; Γ ⊢ eval_aexpr : (aexpr → Int).
|
|
|
|
|
Proof.
|
|
|
|
|
unfold eval_aexpr.
|
|
|
|
|
apply fix_typing; solve_typing.
|
|
|
|
|
done.
|
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(** Exercise 3: Lists *)
|
|
|
|
|
|
|
|
|
|
Definition list_t (A : type) : type :=
|
|
|
|
|
∃: (#0 (* mynil *)
|
|
|
|
|
× (A.[ren (+1)] → #0 → #0) (* mycons *)
|
|
|
|
|
× (∀: #1 → #0 → (A.[ren (+2)] → #1 → #0) → #0)) (* mylistcase *)
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
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 *).
|
|
|
|
|
|
|
|
|
|
(** We define a recursive representation predicate for lists.
|
|
|
|
|
This is a common pattern: we specify the shape of lists in our language in terms of lists at the meta-level (i.e., in Coq).
|
|
|
|
|
*)
|
|
|
|
|
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.
|
|
|
|
|
(* A full list also needs to store the length, otherwise we'd have no way of knowing
|
|
|
|
|
when a list ends (we can't analyze whether a term is () or a pair from inside the language) *)
|
|
|
|
|
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 (fun v => 𝒱 A δ_any k' v) 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 k' Hk' 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 k'2 Hk'2 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 v' Hv'. eapply val_rel_mono; last apply Hv'. 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 k' Hk' 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 k'2 Hk'2 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 k'3 Hk'3 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 v' Hv'. eapply val_rel_mono; last apply Hv'. lia.
|
|
|
|
|
Qed.
|
mueck
11 months ago