semantics-2023/theories/type_systems/stlc/exercises02_sol.v

From stdpp Require Import gmap base relations tactics.
From iris Require Import prelude.
From semantics.ts.stlc Require Import lang notation.
From semantics.ts.stlc Require untyped types.


(** README: Please also download the assigment sheet as a *.pdf from here:
            https://cms.sic.saarland/semantics_ws2324/materials/
            It contains additional explanation and excercises. **)

(** ** Exercise 1: Prove that the structural and contextual semantics are equivalent. *)
(** You will find it very helpful to separately derive the structural rules of the
  structural semantics for the contextual semantics. *)

Lemma contextual_step_beta x e e':
  is_val e' → contextual_step ((λ: x, e) e') (subst' x e' e).
Proof.
  intros Hval. eapply base_contextual_step, BetaS; done.
Qed.

Lemma contextual_step_app_r (e1 e2 e2': expr) :
  contextual_step e2 e2' →
  contextual_step (e1 e2) (e1 e2').
Proof.
  intros Hval. by eapply fill_contextual_step with (K := AppRCtx e1 HoleCtx).
Qed.

Lemma contextual_step_app_l (e1 e1' e2: expr) :
  is_val e2 →
  contextual_step e1 e1' →
  contextual_step (e1 e2) (e1' e2).
Proof.
  intros Hval. eapply is_val_spec in Hval as [v Hval].
  eapply of_to_val in Hval as <-.
  eapply fill_contextual_step with (K := (AppLCtx HoleCtx v)).
Qed.

Lemma contextual_step_plus_red (n1 n2 n3: Z) :
  (n1 + n2)%Z = n3 →
  contextual_step (n1 + n2)%E n3.
Proof.
  intros Hval. eapply base_contextual_step, PlusS; done.
Qed.

Lemma contextual_step_plus_r e1 e2 e2' :
  contextual_step e2 e2' →
  contextual_step (Plus e1 e2) (Plus e1 e2').
Proof.
  intros Hval. by eapply fill_contextual_step with (K := (PlusRCtx e1 HoleCtx)).
Qed.

Lemma contextual_step_plus_l e1 e1' e2 :
  is_val e2 →
  contextual_step e1 e1' →
  contextual_step (Plus e1 e2) (Plus e1' e2).
Proof.
  intros Hval. eapply is_val_spec in Hval as [v Hval].
  eapply of_to_val in Hval as <-.
  eapply fill_contextual_step with (K := (PlusLCtx HoleCtx v)).
Qed.

(** We register these lemmas as hints for [eauto]. *)
#[global]
Hint Resolve contextual_step_beta contextual_step_app_l contextual_step_app_r contextual_step_plus_red contextual_step_plus_l contextual_step_plus_r : core.

Lemma step_contextual_step e1 e2: step e1 e2 → contextual_step e1 e2.
Proof.
  (* The hints above make sure eauto uses the right lemmas. *)
  induction 1; eauto.
Qed.

(** Now the other direction. *)
(* You may find it helpful to introduce intermediate lemmas. *)

Lemma base_step_step e1 e2:
  base_step e1 e2 → step e1 e2.
Proof.
  destruct 1; subst.
  - eauto.
  - by econstructor.
Qed.

Lemma fill_step K e1 e2:
  step e1 e2 → step (fill K e1) (fill K e2).
Proof.
  induction K; simpl; eauto.
Qed.
Lemma contextual_step_step e1 e2:
  contextual_step e1 e2 → step e1 e2.
Proof.
  destruct 1 as [K h1 h2 -> -> Hstep].
  eapply fill_step, base_step_step, Hstep.
Qed.


(** ** Exercise 2: Scott encodings *)
Section scott.
  (* take a look at untyped.v for usefull lemmas and definitions *)
  Import semantics.ts.stlc.untyped.

  (** Scott encoding of Booleans *)
  Definition true_scott : val := λ: "t" "f", "t".
  Definition false_scott : val := λ: "t" "f", "f".

  Lemma true_red (v1 v2 : val) :
    closed [] v1 →
    closed [] v2 →
    rtc step (true_scott v1 v2) v1.
  Proof.
    intros Hcl1 Hcl2.
    eapply rtc_l.
    { econstructor; first auto.
      econstructor. auto.
    }
    simpl.
    eapply rtc_l.
    { econstructor; auto. }
    simpl.
    rewrite subst_closed_nil; done.
  Qed.

  Lemma false_red (v1 v2 : val) :
    closed [] v1 →
    closed [] v2 →
    rtc step (false_scott v1 v2) v2.
  Proof.
    intros Hcl1 Hcl2.
    eapply rtc_l.
    { econstructor; first auto.
      econstructor. auto.
    }
    simpl.
    eapply rtc_l.
    { econstructor; auto. }
    simpl. done.
  Qed.

  (** Scott encoding of Pairs *)
  Definition pair_scott : val := λ: "v1" "v2", λ: "d", "d" "v1" "v2".

  Definition fst_scott : val := λ: "p", "p" (λ: "a" "b", "a").
  Definition snd_scott : val := λ: "p", "p" (λ: "a" "b", "b").

  Lemma fst_red (v1 v2 : val) :
    is_closed [] v1 →
    is_closed [] v2 →
    rtc step (fst_scott (pair_scott v1 v2)) v1.
  Proof.
    intros Hcl1 Hcl2.
    eapply rtc_l.
    { econstructor 3. econstructor; first auto.
      econstructor. auto.
    }
    simpl. eapply rtc_l.
    { econstructor 3. econstructor; first auto. }
    simpl. eapply rtc_l.
    { econstructor. done. }
    simpl. eapply rtc_l.
    { econstructor. done. }
    simpl.
    rewrite !(subst_closed_nil v1); [ | done..].
    rewrite subst_closed_nil; last done.
    eapply rtc_l.
    { econstructor; first auto.
      econstructor. auto.
    }
    simpl. eapply rtc_l.
    { econstructor; first auto. }
    simpl. rewrite subst_closed_nil; done.
  Qed.

  Lemma snd_red (v1 v2 : val) :
    is_closed [] v1 →
    is_closed [] v2 →
    rtc step (snd_scott (pair_scott v1 v2)) v2.
  Proof.
    intros Hcl1 Hcl2.
    eapply rtc_l.
    { econstructor 3. econstructor; first auto.
      econstructor. auto.
    }
    simpl. eapply rtc_l.
    { econstructor 3. econstructor; first auto. }
    simpl. eapply rtc_l.
    { econstructor. done. }
    simpl. eapply rtc_l.
    { econstructor. done. }
    simpl.
    rewrite !(subst_closed_nil v2); [ | done..].
    rewrite !(subst_closed_nil v1); [ | done..].
    eapply rtc_l.
    { econstructor; first auto.
      econstructor. auto.
    }
    simpl. eapply rtc_l.
    { econstructor; first auto. }
    simpl. done.
  Qed.

End scott.

Import semantics.ts.stlc.types.

(** ** Exercise 3: type erasure *)
(** Source terms *)
Inductive src_expr :=
| ELitInt (n: Z)
(* Base lambda calculus *)
| EVar (x : string)
| ELam (x : binder) (A: type) (e : src_expr)
| EApp (e1 e2 : src_expr)
(* Base types and their operations *)
| EPlus (e1 e2 : src_expr).

(** The erasure function *)
Fixpoint erase (E: src_expr) : expr :=
match E with
| ELitInt n => LitInt n
| EVar x => Var x
| ELam x A E => Lam x (erase E)
| EApp E1 E2 => App (erase E1) (erase E2)
| EPlus E1 E2 => Plus (erase E1) (erase E2)
end.

Reserved Notation "Γ '⊢S' e : A" (at level 74, e, A at next level).
Inductive src_typed : typing_context → src_expr → type → Prop :=
| src_typed_var Γ x A :
    Γ !! x = Some A →
    Γ ⊢S (EVar x) : A
| src_typed_lam Γ x E A B :
    (<[ x := A]> Γ) ⊢S E : B →
    Γ ⊢S (ELam (BNamed x) A E) : (A → B)
| src_typed_int Γ z :
    Γ ⊢S (ELitInt z) : Int
| src_typed_app Γ E1 E2 A B :
    Γ ⊢S E1 : (A → B) →
    Γ ⊢S E2 : A →
    Γ ⊢S EApp E1 E2 : B
| src_typed_add Γ E1 E2 :
    Γ ⊢S E1 : Int →
    Γ ⊢S E2 : Int →
    Γ ⊢S EPlus E1 E2 : Int
where "Γ '⊢S' E : A" := (src_typed Γ E%E A%ty) : FType_scope.
#[export] Hint Constructors src_typed : core.

Lemma type_erasure_correctness Γ E A:
  (Γ ⊢S E : A)%ty → (Γ ⊢ erase E : A)%ty.
Proof.
  induction 1; simpl; eauto.
Qed.

(** ** Exercise 4: Unique Typing *)
Lemma src_typing_unique Γ E A B:
  (Γ ⊢S E : A)%ty → (Γ ⊢S E : B)%ty → A = B.
Proof.
  induction 1 as [ | ????? H1 IH1 | | ????? H1 IH1 H2 IH2 | ??? H1 IH1 H2 IH2] in B |-*; inversion 1; subst; try congruence.
  - f_equal. eauto.
  - rename select (_ ⊢S _ : (_ → _))%ty into Ha.
    eapply IH1 in Ha. by injection Ha.
Qed.

(** TODO: Is runtime typing (Curry-style) also unique? Prove it or give a counterexample. *)
(** Runtime typing is not unique! *)
Lemma id_int: (∅ ⊢ (λ: "x", "x") : (Int → Int))%ty.
Proof. eauto. Qed.

Lemma id_int_to_int: (∅ ⊢ (λ: "x", "x") : ((Int → Int) → (Int → Int)))%ty.
Proof. eauto. Qed.

(** ** Exercise 5: Type Inference *)

Fixpoint type_eq A B :=
  match A, B with
  | Int, Int => true
  | Fun A B, Fun A' B' => type_eq A A' && type_eq B B'
  | _, _ => false
  end.

Lemma type_eq_iff A B: type_eq A B ↔ A = B.
Proof.
  induction A in B |-*; destruct B; simpl; naive_solver.
Qed.

Notation ctx := (gmap string type).
Fixpoint infer_type (Γ: ctx) E :=
  match E with
  | EVar x => Γ !! x
  | ELam (BNamed x) A E =>
    match infer_type (<[x := A]> Γ) E with
    | Some B => Some (Fun A B)
    | None => None
    end
  | ELitInt l => Some Int
  | EApp E1 E2 =>
    match infer_type Γ E1, infer_type Γ E2 with
    | Some (Fun A B), Some C => if type_eq A C then Some B else None
    | _, _ => None
    end
  | EPlus E1 E2 =>
    match infer_type Γ E1, infer_type Γ E2 with
    | Some Int, Some Int => Some Int
    | _, _ => None
    end
  | ELam BAnon A E => None
  end.

(** Prove both directions of the correctness theorem. *)
Lemma infer_type_typing Γ E A:
  infer_type Γ E = Some A → (Γ ⊢S E : A)%ty.
Proof.
  induction E as [l|x|x B E IH|E1 IH1 E2 IH2|E1 IH1 E2 IH2] in Γ, A |-*; simpl.
  - destruct l; naive_solver.
  - eauto.
  - destruct x as [|x]; first naive_solver.
    destruct infer_type eqn: Heq; last naive_solver.
    eapply IH in Heq. injection 1 as <-. eauto.
  - destruct infer_type as [B|] eqn: Heq1; last naive_solver.
    destruct B as [|B1 B2]; try naive_solver.
    destruct (infer_type Γ E2) as [C|] eqn: Heq2; last naive_solver.
    destruct type_eq eqn: Heq3; last naive_solver.
    injection 1 as <-. eapply Is_true_eq_left in Heq3.
    eapply type_eq_iff in Heq3 as ->.
    eauto.
  - destruct infer_type as [B|] eqn: Heq1; last naive_solver.
    destruct B; try naive_solver.
    destruct (infer_type Γ E2) as [C|] eqn: Heq2; last naive_solver.
    destruct C; naive_solver.
Qed.

Lemma typing_infer_type Γ E A:
  (Γ ⊢S E : A)%ty → infer_type Γ E = Some A.
Proof.
  induction 1; simpl; eauto.
  - by rewrite IHsrc_typed.
  - rewrite IHsrc_typed1 IHsrc_typed2.
    rewrite (Is_true_eq_true (type_eq A A)) //=.
    by eapply type_eq_iff.
  - by rewrite IHsrc_typed1 IHsrc_typed2.
Qed.


(** ** Exercise 6: untypable, safe term *)

(* The exercise asks you to:
  "Give one example if there is such an expression, otherwise prove their non-existence."
  So either finish one section or the other.
*)

Section give_example.
  Definition ust : expr := (λ: "f", 5) (λ: "x", 0 0)%E.

  Lemma ust_safe e':
    rtc step ust e' → is_val e' ∨ reducible e'.
  Proof.
    inversion 1 as [ | x y z H1 H2]; subst.
    - right. eexists. eapply StepBeta. done.
    - inversion H1 as [??? Ha | ??? Ha Hb | ??? Ha | | ??? Ha Hb | ??? Ha ]; subst.
      + simpl in H2. inversion H2 as [ | ??? [] _]; subst; eauto.
      + inversion Hb.
      + inversion Ha.
  Qed.

  Lemma ust_no_type Γ A:
    ¬ (Γ ⊢ ust : A)%ty.
  Proof.
    intros H. inversion H; subst.
    inversion H5; subst. inversion H6; subst.
    inversion H4.
  Qed.
End give_example.

Section prove_non_existence.
  Lemma no_ust :
  ∀ e, (∀ e', rtc step e e' → is_val e' ∨ reducible e') → ∃ A, (∅ ⊢ e : A)%ty.
  Proof.
    (* This is not possible as shown above *)
  Abort.
End prove_non_existence.