Merge remote-tracking branch 'origin/main' into amethyst

_CoqProject

@@ -15,9 +15,19 @@ theories/lib/sets.v
 # STLC
 theories/type_systems/stlc/lang.v
 theories/type_systems/stlc/notation.v
+theories/type_systems/stlc/untyped.v
+theories/type_systems/stlc/types.v
+
+# Extended STLC
+theories/type_systems/stlc_extended/lang.v
+theories/type_systems/stlc_extended/notation.v
+theories/type_systems/stlc_extended/types.v
+theories/type_systems/stlc_extended/bigstep.v
+theories/type_systems/stlc_extended/ctxstep.v
 
 # By removing the # below, you can add the exercise sheets to make
 theories/type_systems/warmup/warmup.v
 #theories/type_systems/warmup/warmup_sol.v
 theories/type_systems/stlc/exercises01.v
 #theories/type_systems/stlc/exercises01_sol.v
+#theories/type_systems/stlc/exercises02.v

theories/type_systems/stlc/exercises02.v

@@ -0,0 +1,273 @@
+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.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_app_r (e1 e2 e2': expr) :
+  contextual_step e2 e2' →
+  contextual_step (e1 e2) (e1 e2').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_app_l (e1 e1' e2: expr) :
+  is_val e2 →
+  contextual_step e1 e1' →
+  contextual_step (e1 e2) (e1' e2).
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_plus_red (n1 n2 n3: Z) :
+  (n1 + n2)%Z = n3 →
+  contextual_step (n1 + n2)%E n3.
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_plus_r e1 e2 e2' :
+  contextual_step e2 e2' →
+  contextual_step (Plus e1 e2) (Plus e1 e2').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_plus_l e1 e1' e2 :
+  is_val e2 →
+  contextual_step e1 e1' →
+  contextual_step (Plus e1 e2) (Plus e1' e2).
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+(** 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.
+  (* TODO: exercise *)
+Admitted.
+
+
+(** Now the other direction. *)
+(* You may find it helpful to introduce intermediate lemmas. *)
+
+
+Lemma contextual_step_step e1 e2:
+  contextual_step e1 e2 → step e1 e2.
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+
+(** ** 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 := 0(* TODO *).
+  Definition false_scott : val := 0(* TODO *).
+
+  Lemma true_red (v1 v2 : val) :
+    closed [] v1 →
+    closed [] v2 →
+    rtc step (true_scott v1 v2) v1.
+  Proof.
+    (* TODO: exercise *)
+  Admitted.
+
+
+  Lemma false_red (v1 v2 : val) :
+    closed [] v1 →
+    closed [] v2 →
+    rtc step (false_scott v1 v2) v2.
+  Proof.
+    (* TODO: exercise *)
+  Admitted.
+
+
+  (** Scott encoding of Pairs *)
+  Definition pair_scott : val := 0(* TODO *).
+
+  Definition fst_scott : val := 0(* TODO *).
+  Definition snd_scott : val := 0(* TODO *).
+
+  Lemma fst_red (v1 v2 : val) :
+    is_closed [] v1 →
+    is_closed [] v2 →
+    rtc step (fst_scott (pair_scott v1 v2)) v1.
+  Proof.
+    (* TODO: exercise *)
+  Admitted.
+
+
+  Lemma snd_red (v1 v2 : val) :
+    is_closed [] v1 →
+    is_closed [] v2 →
+    rtc step (snd_scott (pair_scott v1 v2)) v2.
+  Proof.
+    (* TODO: exercise *)
+  Admitted.
+
+
+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 :=
+0 (* TODO *).
+
+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.
+  (* TODO: exercise *)
+Admitted.
+
+
+(** ** 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.
+  (* TODO: exercise *)
+Admitted.
+
+
+(** TODO: Is runtime typing (Curry-style) also unique? Prove it or give a counterexample. *)
+
+
+(** ** 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
+  (* TODO: complete the definition for the remaining cases *)
+  | ELitInt l => None (* TODO *)
+  | EApp E1 E2 => None (* TODO *)
+  | EPlus E1 E2 => None (* TODO *)
+  | 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.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma typing_infer_type Γ E A:
+  (Γ ⊢S E : A)%ty → infer_type Γ E = Some A.
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+
+(** ** 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 := 0 (* TODO *).
+
+  Lemma ust_safe e':
+    rtc step ust e' → is_val e' ∨ reducible e'.
+  Proof.
+    (* TODO: exercise *)
+  Admitted.
+
+
+  Lemma ust_no_type Γ A:
+    ¬ (Γ ⊢ ust : A)%ty.
+  Proof.
+    (* TODO: exercise *)
+  Admitted.
+
+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.
+    (* TODO: exercise *)
+  Admitted.
+
+End prove_non_existence.

theories/type_systems/stlc/lang.v

@@ -318,3 +318,22 @@ Qed.
 
 Lemma subst_closed_nil e x es : closed [] e → subst x es e = e.
 Proof. intros. apply subst_closed with []; set_solver. Qed.
+
+Lemma val_no_step e e':
+  step e e' → is_val e → False.
+Proof.
+  by destruct 1.
+Qed.
+
+Lemma val_no_step' (v : val) (e : expr) :
+  step (of_val v) e -> False.
+Proof.
+  intros H. eapply (val_no_step _ _ H).
+  apply is_val_val.
+Qed.
+
+Ltac val_no_step :=
+  match goal with
+  | [H: step ?e1 ?e2 |- _] =>
+    solve [exfalso; eapply (val_no_step _ _ H); done]
+  end.

theories/type_systems/stlc/types.v

@@ -0,0 +1,235 @@
+From stdpp Require Import base relations tactics.
+From iris Require Import prelude.
+From semantics.ts.stlc Require Import lang notation.
+From semantics.lib Require Import sets maps.
+
+(** ** Syntactic typing *)
+(** In the Coq formalization, we exclusively define runtime typing (Curry-style). *)
+(** It will be an exercise to consider Church-style typing. *)
+
+Inductive type : Set :=
+  | Int
+  | Fun (A B : type).
+
+Definition typing_context := gmap string type.
+Implicit Types
+  (Γ : typing_context)
+  (v : val)
+  (e : expr)
+  (A : type)
+  (x: string).
+
+(** We define notation for types in a new scope with delimiter [ty].
+  See below for an example. *)
+Declare Scope FType_scope.
+Delimit Scope FType_scope with ty.
+Bind Scope FType_scope with type.
+Infix "→" := Fun : FType_scope.
+Notation "(→)" := Fun (only parsing) : FType_scope.
+
+(** Typing rules *)
+(** We need to reserve the notation beforehand to already use it when defining the
+   typing rules. *)
+Reserved Notation "Γ ⊢ e : A" (at level 74, e, A at next level).
+Inductive syn_typed : typing_context → expr → type → Prop :=
+  | typed_var Γ x A :
+      (* lookup the variable in the context *)
+      Γ !! x = Some A →
+      Γ ⊢ (Var x) : A
+  | typed_lam Γ x e A B :
+      (* add a new type assignment to the context *)
+     (<[ x := A]> Γ) ⊢ e : B →
+      Γ ⊢ (Lam (BNamed x) e) : (A → B)
+  | typed_int Γ z :
+      Γ ⊢ (LitInt z) : Int
+  | typed_app Γ e1 e2 A B :
+      Γ ⊢ e1 : (A → B) →
+      Γ ⊢ e2 : A →
+      Γ ⊢ e1 e2 : B
+  | typed_add Γ e1 e2 :
+      Γ ⊢ e1 : Int →
+      Γ ⊢ e2 : Int →
+      Γ ⊢ e1 + e2 : Int
+where "Γ ⊢ e : A" := (syn_typed Γ e%E A%ty) : FType_scope.
+(** Add constructors to [eauto]'s hint database. *)
+#[export] Hint Constructors syn_typed : core.
+Local Open Scope FType_scope.
+
+(** a small example *)
+Goal ∅ ⊢ (λ: "x", 1 + "x")%E : (Int → Int).
+Proof. eauto. Qed.
+
+(** We derive some inversion lemmas -- this is cleaner than directly
+  using the [inversion] tactic everywhere.*)
+Lemma var_inversion Γ (x: string) A: Γ ⊢ x : A → Γ !! x = Some A.
+Proof. inversion 1; subst; auto. Qed.
+
+Lemma lam_inversion Γ (x: binder) e C:
+  Γ ⊢ (λ: x, e) : C →
+  ∃ A B y, C = (A → B)%ty ∧ x = BNamed y ∧ <[y:=A]> Γ ⊢ e : B.
+Proof. inversion 1; subst; eauto 10. Qed.
+
+Lemma lit_int_inversion Γ (n: Z) A: Γ ⊢ n : A → A = Int.
+Proof. inversion 1; subst; auto. Qed.
+
+Lemma app_inversion Γ e1 e2 B:
+  Γ ⊢ e1 e2 : B →
+  ∃ A,  Γ ⊢ e1 : (A → B) ∧  Γ ⊢ e2 : A.
+Proof. inversion 1; subst; eauto. Qed.
+
+Lemma plus_inversion Γ e1 e2 B:
+  Γ ⊢ e1 + e2 : B →
+  B = Int ∧ Γ ⊢ e1 : Int ∧ Γ ⊢ e2 : Int.
+Proof. inversion 1; subst; eauto. Qed.
+
+
+Lemma syn_typed_closed Γ e A X :
+  Γ ⊢ e : A →
+  (∀ x, x ∈ dom Γ → x ∈ X) →
+  is_closed X e.
+Proof.
+  induction 1 as [ | ?????? IH | | | ] in X |-*; simpl; intros Hx; try done.
+  { (* var *) apply bool_decide_pack, Hx. apply elem_of_dom; eauto. }
+  { (* lam *) apply IH.
+    intros y. rewrite elem_of_dom lookup_insert_is_Some.
+    intros [<- | [? Hy]]; first by apply elem_of_cons; eauto.
+    apply elem_of_cons. right. eapply Hx. by apply elem_of_dom.
+  }
+  (* everything else *)
+  all: repeat match goal with
+              | |- Is_true (_ && _)  => apply andb_True; split
+              end.
+  all: naive_solver.
+Qed.
+
+Lemma typed_weakening Γ Δ e A:
+  Γ ⊢ e : A →
+  Γ ⊆ Δ →
+  Δ ⊢ e : A.
+Proof.
+  induction 1 as [| Γ x e A B Htyp IH | | | ] in Δ; intros Hsub; eauto.
+  - econstructor. by eapply lookup_weaken.
+  - econstructor. eapply IH. by eapply insert_mono.
+Qed.
+
+(** Preservation of typing under substitution *)
+Lemma typed_substitutivity e e' Γ x A B :
+  ∅ ⊢ e' : A →
+  <[x := A]> Γ ⊢ e : B →
+  Γ ⊢ subst x e' e : B.
+Proof.
+  intros He'. revert B Γ; induction e as [y | y | | |]; intros B Γ; simpl.
+  - intros Hp%var_inversion; destruct decide; subst; eauto.
+    + rewrite lookup_insert in Hp. injection Hp as ->.
+      eapply typed_weakening; first done. apply map_empty_subseteq.
+    + rewrite lookup_insert_ne in Hp; last done. auto.
+  - intros (C & D & z & -> & -> & Hty)%lam_inversion.
+    econstructor. destruct decide as [|Heq]; simplify_eq.
+    + by rewrite insert_insert in Hty.
+    + rewrite insert_commute in Hty; last naive_solver. eauto.
+  - intros (C & Hty1 & Hty2)%app_inversion; eauto.
+  - intros ->%lit_int_inversion. eauto.
+  - intros (-> & Hty1 & Hty2)%plus_inversion; eauto.
+Qed.
+
+(** Canonical value lemmas *)
+Lemma canonical_values_arr Γ e A B:
+  Γ ⊢ e : (A → B) →
+  is_val e →
+  ∃ x e', e = (λ: x, e')%E.
+Proof.
+  inversion 1; simpl; naive_solver.
+Qed.
+
+Lemma canonical_values_int Γ e:
+  Γ ⊢ e : Int →
+  is_val e →
+  ∃ n: Z, e = n.
+Proof.
+  inversion 1; simpl; naive_solver.
+Qed.
+
+
+(** Progress lemma *)
+Lemma typed_progress e A:
+  ∅ ⊢ e : A → is_val e ∨ contextual_reducible e.
+Proof.
+  remember ∅ as Γ. induction 1 as [| | | Γ e1 e2 A B Hty IH1 _ IH2 | Γ e1 e2 Hty1 IH1 Hty2 IH2].
+  - subst. naive_solver.
+  - left. done.
+  - left. done.
+  - destruct (IH2 HeqΓ) as [H2|H2]; [destruct (IH1 HeqΓ) as [H1|H1]|].
+    + eapply canonical_values_arr in Hty as (x & e & ->); last done.
+      right. eexists.
+      eapply base_contextual_step, BetaS; eauto.
+    + right. eapply is_val_spec in H2 as [v Heq].
+      replace e2 with (of_val v); last by eapply of_to_val.
+      destruct H1 as [e1' Hstep].
+      eexists. eapply (fill_contextual_step (AppLCtx HoleCtx v)). done.
+    + right. destruct H2 as [e2' H2].
+      eexists. eapply (fill_contextual_step (AppRCtx e1 HoleCtx)). done.
+  - destruct (IH2 HeqΓ) as [H2|H2]; [destruct (IH1 HeqΓ) as [H1|H1]|].
+    + right. eapply canonical_values_int in Hty1 as [n1 ->]; last done.
+      eapply canonical_values_int in Hty2 as [n2 ->]; last done.
+      eexists. eapply base_contextual_step. eapply PlusS; eauto.
+    + right. eapply is_val_spec in H2 as [v Heq].
+      replace e2 with (of_val v); last by eapply of_to_val.
+      destruct H1 as [e1' Hstep].
+      eexists. eapply (fill_contextual_step (PlusLCtx HoleCtx v)). done.
+    + right. destruct H2 as [e2' H2].
+      eexists. eapply (fill_contextual_step (PlusRCtx e1 HoleCtx)). done.
+Qed.
+
+(** Contextual typing *)
+Definition ectx_typing (K: ectx) (A B: type) :=
+  ∀ e, ∅ ⊢ e : A → ∅ ⊢ (fill K e) : B.
+
+Lemma fill_typing_decompose K e A:
+  ∅ ⊢ fill K e : A →
+  ∃ B, ∅ ⊢ e : B ∧ ectx_typing K B A.
+Proof.
+  unfold ectx_typing; induction K in e,A |-*; simpl; eauto.
+  all: inversion 1; subst; edestruct IHK as [B [Hit Hty]]; eauto.
+Qed.
+
+Lemma fill_typing_compose K e A B:
+  ∅ ⊢ e : B →
+  ectx_typing K B A →
+  ∅ ⊢ fill K e : A.
+Proof.
+  intros H1 H2; by eapply H2.
+Qed.
+
+(** Base preservation *)
+Lemma typed_preservation_base_step e e' A:
+  ∅ ⊢ e : A →
+  base_step e e' →
+  ∅ ⊢ e' : A.
+Proof.
+  intros Hty Hstep. destruct Hstep as [| e1 e2 n1 n2 n3 Heq1 Heq2 Heval]; subst.
+  - eapply app_inversion in Hty as (B & Hty1 & Hty2).
+    eapply lam_inversion in Hty1 as (B' & A' & y & Heq1 & Heq2 & Hty).
+    simplify_eq. eapply typed_substitutivity; eauto.
+  - eapply plus_inversion in Hty as (-> & Hty1 & Hty2).
+    econstructor.
+Qed.
+
+(** Preservation *)
+Lemma typed_preservation e e' A:
+  ∅ ⊢ e : A →
+  contextual_step e e' →
+  ∅ ⊢ e' : A.
+Proof.
+  intros Hty Hstep. destruct Hstep as [K e1 e2 -> -> Hstep].
+  eapply fill_typing_decompose in Hty as [B [H1 H2]].
+  eapply fill_typing_compose; last done.
+  by eapply typed_preservation_base_step.
+Qed.
+
+Lemma typed_safety e1 e2 A:
+  ∅ ⊢ e1 : A →
+  rtc contextual_step e1 e2 →
+  is_val e2 ∨ contextual_reducible e2.
+Proof.
+  induction 2; eauto using typed_progress, typed_preservation.
+Qed.

theories/type_systems/stlc/untyped.v

@@ -0,0 +1,221 @@
+From stdpp Require Import base relations tactics.
+From iris Require Import prelude.
+From semantics.lib Require Import sets maps.
+From semantics.ts.stlc Require Import lang notation.
+
+(** The following two lemmas will be helpful in the sequel.
+  They just lift multiple reduction steps (via [rtc]) to application.
+ *)
+Lemma steps_app_r (e1 e2 e2' : expr) :
+  rtc step e2 e2' →
+  rtc step (e1 e2) (e1 e2').
+Proof.
+  induction 1 as [ e | e e' e'' Hstep Hsteps IH].
+  - reflexivity.
+  - eapply (rtc_l _ _ (e1 e')).
+    { by eapply StepAppR. }
+    done.
+Qed.
+
+Lemma steps_app_l (e1 e1' e2 : expr) :
+  is_val e2 →
+  rtc step e1 e1' →
+  rtc step (e1 e2) (e1' e2).
+Proof.
+  intros Hv.
+  induction 1 as [ e | e e' e'' Hstep Hsteps IH].
+  - reflexivity.
+  - eapply (rtc_l _ _ (e' e2)).
+    { by eapply StepAppL. }
+    done.
+Qed.
+
+
+(** * Untyped λ calculus *)
+
+(** We do not re-define the language to remove primitive addition -- instead, we just
+  restrict our usage in this file to variables, application, and lambdas.
+ *)
+
+
+Definition I_val : val := λ: "x", "x".
+Definition F_val : val := λ: "x" "y", "x".
+Definition S_val : val := λ: "x" "y", "y".
+Definition ω : val := λ: "x", "x" "x".
+Definition Ω : expr := ω ω.
+
+
+(** Ω reduces to itself! *)
+Lemma Omega_step : step Ω Ω.
+Proof.
+  apply StepBeta. done.
+Qed.
+
+(** ** Scott encodings *)
+
+Definition zero : val := λ: "x" "y", "x".
+
+Definition succ (n : val) : val := λ: "x" "y", "y" n.
+(** [Succ] can be seen as a constructor: it takes [n] at the level of the language. *)
+Definition Succ : val := λ: "n" "x" "y", "y" "n".
+
+Fixpoint enc_nat (n : nat) : val :=
+  match n with
+  | 0 => zero
+  | S n => succ (enc_nat n)
+  end.
+
+Lemma enc_nat_closed n :
+  closed [] (enc_nat n).
+Proof.
+  induction n as [ | n IH].
+  - done.
+  - simpl. by apply closed_weaken_nil.
+Qed.
+
+Lemma enc_0_red (z s : val) :
+  is_closed [] z →
+  rtc step (enc_nat 0 z s) z.
+Proof.
+  intros Hcl.
+  eapply rtc_l.
+  { econstructor; first auto. econstructor. auto. }
+  simpl. eapply rtc_l.
+  { econstructor. auto. }
+  simpl. rewrite subst_closed_nil; done.
+Qed.
+
+Lemma enc_S_red n (z s : val) :
+  rtc step (enc_nat (S n) z s) (s (enc_nat n)).
+Proof.
+  simpl. eapply rtc_l.
+  { econstructor; first auto. econstructor. auto. }
+  simpl. eapply rtc_l.
+  { econstructor. auto. }
+  simpl. rewrite (subst_closed_nil (enc_nat n)); last apply enc_nat_closed.
+  rewrite subst_closed_nil; last apply enc_nat_closed.
+  reflexivity.
+Qed.
+
+Lemma Succ_red n : step (Succ (enc_nat n)) (enc_nat (S n)).
+Proof. econstructor. apply is_val_val. Qed.
+
+Lemma Succ_red_n n : rtc step (Nat.iter n Succ zero) (enc_nat n).
+Proof.
+  induction n as [ | n IH].
+  - reflexivity.
+  - simpl.
+    etrans.
+    { simpl. eapply steps_app_r. apply IH. }
+    eapply rtc_l.
+    { apply Succ_red. }
+    reflexivity.
+Qed.
+
+(** ** Recursion *)
+Definition Fix' : val := λ: "z" "y", "y" (λ: "x", "z" "z" "y" "x").
+Definition Fix (s : val) : val := λ: "x", Fix' Fix' s "x".
+Arguments Fix : simpl never.
+
+Local Hint Immediate is_val_val : core.
+
+(** We prove that it satisfies the recursive unfolding *)
+Lemma Fix_step (s r : val) :
+  is_closed [] s →
+  rtc step (Fix s r) (s ((Fix s))%E r).
+Proof.
+  intros Hclosed.
+  eapply rtc_l.
+  { econstructor. auto. }
+  eapply rtc_l.
+  { simpl. econstructor; first by auto.
+    econstructor. { rewrite subst_closed_nil; auto. }
+    econstructor. done.
+  }
+  simpl. rewrite subst_closed_nil; last done.
+  eapply rtc_l.
+  { econstructor; first by auto.
+    econstructor. auto.
+  }
+  simpl. reflexivity.
+Qed.
+
+(** Example usage: addition on Scott-encoded numbers *)
+Definition add_step : val := λ: "r", λ: "n" "m", "n" "m" (λ: "p", Succ ("r" "p" "m")).
+Definition add := Fix add_step.
+
+(** We are now going to prove it correct:
+ [add (enc_nat n) (enc_nat m)) ≻* (enc_nat (n + m))]
+
+ First, we prove that it satisfies the expected defining equations for Peano addition.
+ *)
+
+Lemma add_step_0 m : rtc step (add (enc_nat 0) (enc_nat m)) (enc_nat m).
+Proof.
+  (* use the unfolding equation of the fixpoint combinator *)
+  etrans.
+  { eapply steps_app_l; first by auto.
+    eapply Fix_step. done.
+  }
+  (* subst it into the [add_step] function *)
+  eapply rtc_l.
+  { econstructor; auto. econstructor; auto. econstructor. auto. }
+  simpl.
+  (* subst in the zero *)
+  eapply rtc_l.
+  { econstructor; auto. econstructor. done. }
+  simpl.
+  (* subst in the m *)
+  eapply rtc_l.
+  { econstructor; auto. }
+  simpl.
+
+  (* do a step *)
+  etrans.
+  { apply (enc_0_red (enc_nat m) (λ: "p", Succ (Fix add_step "p" (enc_nat m)))).
+    apply enc_nat_closed.
+  }
+  reflexivity.
+Qed.
+
+Lemma add_step_S n m : rtc step (add (enc_nat (S n)) (enc_nat m)) (Succ (add (enc_nat n) (enc_nat m))).
+Proof.
+  (* use the unfolding equation of the fixpoint combinator *)
+  etrans.
+  { eapply steps_app_l; first by auto.
+    eapply Fix_step. done.
+  }
+  (* subst it into the [add_step] function *)
+  eapply rtc_l.
+  { econstructor; auto. econstructor; auto. econstructor. auto. }
+  simpl.
+  (* subst in the zero *)
+  eapply rtc_l.
+  { econstructor; auto. econstructor. done. }
+  simpl.
+  (* subst in the m *)
+  eapply rtc_l.
+  { econstructor; auto. }
+  simpl.
+  (* do a step *)
+  etrans.
+  { rewrite subst_closed_nil; last apply enc_nat_closed.
+    apply (enc_S_red n (enc_nat m) (λ: "p", Succ (Fix add_step "p" (enc_nat m)))).
+  }
+
+  eapply rtc_l.
+  { econstructor. auto. }
+  simpl.
+  rewrite subst_closed_nil; last apply enc_nat_closed.
+  reflexivity.
+Qed.
+
+Lemma add_correct n m : rtc step (add (enc_nat n) (enc_nat m)) (enc_nat (n + m)).
+Proof.
+  induction n as [ | n IH].
+  - apply add_step_0.
+  - etrans. { apply add_step_S. }
+    etrans. { apply steps_app_r. apply IH. }
+    eapply rtc_l. { apply Succ_red. }
+    reflexivity.
+Qed.

theories/type_systems/stlc_extended/bigstep.v

@@ -0,0 +1,47 @@
+From stdpp Require Import gmap base relations.
+From iris Require Import prelude.
+From semantics.ts.stlc_extended Require Import lang notation.
+
+(** * Big-step semantics *)
+
+Implicit Types
+  (v : val)
+  (e : expr).
+
+Inductive big_step : expr → val → Prop :=
+  | bs_lit (n : Z) :
+      big_step (LitInt n) (LitIntV n)
+  | bs_lam (x : binder) (e : expr) :
+      big_step (λ: x, e)%E (λ: x, e)%V
+  | bs_add e1 e2 (z1 z2 : Z) :
+      big_step e1 (LitIntV z1) →
+      big_step e2 (LitIntV z2) →
+      big_step (Plus e1 e2) (LitIntV (z1 + z2))%Z
+  | bs_app e1 e2 x e v2 v :
+      big_step e1 (LamV x e) →
+      big_step e2 v2 →
+      big_step (subst' x (of_val v2) e) v →
+      big_step (App e1 e2) v
+
+(* TODO : extend the big-step semantics *)
+    .
+#[export] Hint Constructors big_step : core.
+
+Lemma big_step_of_val e v :
+  e = of_val v →
+  big_step e v.
+Proof.
+  intros ->.
+  induction v; simpl; eauto.
+
+(* TODO : this should be fixed once you have added the right semantics *)
+Admitted.
+
+
+Lemma big_step_val v v' :
+  big_step (of_val v) v' → v' = v.
+Proof.
+  enough (∀ e, big_step e v' → e = of_val v → v' = v) by naive_solver.
+  intros e Hb.
+  induction Hb in v |-*; intros Heq; subst; destruct v; inversion Heq; subst; naive_solver.
+Qed.

theories/type_systems/stlc_extended/ctxstep.v

@@ -0,0 +1,180 @@
+From semantics.ts.stlc_extended Require Export lang.
+
+(** The stepping relation *)
+
+Inductive base_step : expr → expr → Prop :=
+  | BetaS x e1 e2 e' :
+     is_val e2 →
+     e' = subst' x e2 e1 →
+     base_step (App (Lam x e1) e2) e'
+  | PlusS e1 e2 (n1 n2 n3 : Z):
+     e1 = (LitInt n1) →
+     e2 = (LitInt n2) →
+     (n1 + n2)%Z = n3 →
+     base_step (Plus e1 e2) (LitInt n3)
+  (* TODO: extend the definition *)
+.
+
+#[export] Hint Constructors base_step : core.
+
+(** We define evaluation contexts *)
+Inductive ectx :=
+  | HoleCtx
+  | AppLCtx (K: ectx) (v2 : val)
+  | AppRCtx (e1 : expr) (K: ectx)
+  | PlusLCtx (K: ectx) (v2 : val)
+  | PlusRCtx (e1 : expr) (K: ectx)
+  (* TODO: extend the definition *)
+.
+
+Fixpoint fill (K : ectx) (e : expr) : expr :=
+  match K with
+  | HoleCtx => e
+  | AppLCtx K v2 => App (fill K e) (of_val v2)
+  | AppRCtx e1 K => App e1 (fill K e)
+  | PlusLCtx K v2 => Plus (fill K e) (of_val v2)
+  | PlusRCtx e1 K => Plus e1 (fill K e)
+  (* TODO: extend the definition *)
+  end.
+
+Fixpoint comp_ectx (K: ectx) (K' : ectx) : ectx :=
+  match K with
+  | HoleCtx => K'
+  | AppLCtx K v2 => AppLCtx (comp_ectx K K') v2
+  | AppRCtx e1 K => AppRCtx e1 (comp_ectx K K')
+  | PlusLCtx K v2 => PlusLCtx (comp_ectx K K') v2
+  | PlusRCtx e1 K => PlusRCtx e1 (comp_ectx K K')
+  (* TODO: extend the definition *)
+  end.
+
+(** Contextual steps *)
+Inductive contextual_step (e1 : expr) (e2 : expr) : Prop :=
+  Ectx_step K e1' e2' :
+    e1 = fill K e1' → e2 = fill K e2' →
+    base_step e1' e2' → contextual_step e1 e2.
+
+#[export] Hint Constructors contextual_step : core.
+
+Definition reducible (e : expr) :=
+  ∃ e', contextual_step e e'.
+
+Definition empty_ectx := HoleCtx.
+
+(** Basic properties about the language *)
+Lemma fill_empty e : fill empty_ectx e = e.
+Proof. done. Qed.
+
+Lemma fill_comp (K1 K2 : ectx) e : fill K1 (fill K2 e) = fill (comp_ectx K1 K2) e.
+Proof. induction K1; simpl; congruence. Qed.
+
+Lemma base_contextual_step e1 e2 :
+  base_step e1 e2 → contextual_step e1 e2.
+Proof. apply Ectx_step with empty_ectx; by rewrite ?fill_empty. Qed.
+
+Lemma fill_contextual_step K e1 e2 :
+  contextual_step e1 e2 → contextual_step (fill K e1) (fill K e2).
+Proof.
+  destruct 1 as [K' e1' e2' -> ->].
+  rewrite !fill_comp. by econstructor.
+Qed.
+
+(** We derive a few lemmas about contextual steps:
+  these essentially provide rules for structural lifting
+   akin to the structural semantics.
+ *)
+Lemma contextual_step_app_l e1 e1' e2:
+  is_val e2 →
+  contextual_step e1 e1' →
+  contextual_step (App e1 e2) (App e1' e2).
+Proof.
+  intros [v <-%of_to_val]%is_val_spec Hcontextual.
+  by eapply (fill_contextual_step (AppLCtx HoleCtx v)).
+Qed.
+
+Lemma contextual_step_app_r e1 e2 e2':
+  contextual_step e2 e2' →
+  contextual_step (App e1 e2) (App e1 e2').
+Proof.
+  intros Hcontextual.
+  by eapply (fill_contextual_step (AppRCtx 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 [v <-%of_to_val]%is_val_spec Hcontextual.
+  by eapply (fill_contextual_step (PlusLCtx HoleCtx v)).
+Qed.
+
+Lemma contextual_step_plus_r e1 e2 e2':
+  contextual_step e2 e2' →
+  contextual_step (Plus e1 e2) (Plus e1 e2').
+Proof.
+  intros Hcontextual.
+  by eapply (fill_contextual_step (PlusRCtx e1 HoleCtx)).
+Qed.
+
+Lemma contextual_step_pair_l e1 e1' e2:
+  is_val e2 →
+  contextual_step e1 e1' →
+  contextual_step (Pair e1 e2) (Pair e1' e2).
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_pair_r e1 e2 e2':
+  contextual_step e2 e2' →
+  contextual_step (Pair e1 e2) (Pair e1 e2').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_fst e e':
+  contextual_step e e' →
+  contextual_step (Fst e) (Fst e').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_snd e e':
+  contextual_step e e' →
+  contextual_step (Snd e) (Snd e').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_injl e e':
+  contextual_step e e' →
+  contextual_step (InjL e) (InjL e').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_injr e e':
+  contextual_step e e' →
+  contextual_step (InjR e) (InjR e').
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+Lemma contextual_step_case e e' e1 e2:
+  contextual_step e e' →
+  contextual_step (Case e e1 e2) (Case e' e1 e2).
+Proof.
+  (* TODO: exercise *)
+Admitted.
+
+
+#[global]
+Hint Resolve
+  contextual_step_app_l contextual_step_app_r contextual_step_plus_l contextual_step_plus_r
+  contextual_step_case contextual_step_fst contextual_step_injl contextual_step_injr
+  contextual_step_pair_l contextual_step_pair_r contextual_step_snd : core.

theories/type_systems/stlc_extended/lang.v

@@ -0,0 +1,175 @@
+From stdpp Require Export binders strings.
+From iris.prelude Require Import options.
+From semantics.lib Require Export maps.
+
+Declare Scope expr_scope.
+Declare Scope val_scope.
+Delimit Scope expr_scope with E.
+Delimit Scope val_scope with V.
+
+Inductive expr :=
+  (* Base lambda calculus *)
+  | Var (x : string)
+  | Lam (x : binder) (e : expr)
+  | App (e1 e2 : expr)
+  (* Base types and their operations *)
+  | LitInt (n: Z)
+  | Plus (e1 e2 : expr)
+  (* Products *)
+  | Pair (e1 e2 : expr)
+  | Fst (e : expr)
+  | Snd (e : expr)
+  (* Sums *)
+  | InjL (e : expr)
+  | InjR (e : expr)
+  | Case (e0 : expr) (e1 : expr) (e2 : expr).
+
+Bind Scope expr_scope with expr.
+
+Inductive val :=
+  | LitIntV (n: Z)
+  | LamV (x : binder) (e : expr)
+  | PairV (v1 v2 : val)
+  | InjLV (v : val)
+  | InjRV (v : val)
+.
+
+Bind Scope val_scope with val.
+
+Fixpoint of_val (v : val) : expr :=
+  match v with
+  | LitIntV n => LitInt n
+  | LamV x e => Lam x e
+  | PairV v1 v2 => Pair (of_val v1) (of_val v2)
+  | InjLV v => InjL (of_val v)
+  | InjRV v => InjR (of_val v)
+  end.
+
+Fixpoint to_val (e : expr) : option val :=
+  match e with
+  | LitInt n => Some $ LitIntV n
+  | Lam x e => Some (LamV x e)
+  | Pair e1 e2 =>
+    to_val e1 ≫= (λ v1, to_val e2 ≫= (λ v2, Some $ PairV v1 v2))
+  | InjL e => to_val e ≫= (λ v, Some $ InjLV v)
+  | InjR e => to_val e ≫= (λ v, Some $ InjRV v)
+  | _ => None
+  end.
+
+(** Equality and other typeclass stuff *)
+Lemma to_of_val v : to_val (of_val v) = Some v.
+Proof.
+  by induction v; simplify_option_eq; repeat f_equal; try apply (proof_irrel _).
+Qed.
+
+Lemma of_to_val e v : to_val e = Some v → of_val v = e.
+Proof.
+  revert v; induction e; intros v ?; simplify_option_eq; auto with f_equal.
+Qed.
+
+#[export] Instance of_val_inj : Inj (=) (=) of_val.
+Proof. by intros ?? Hv; apply (inj Some); rewrite <-!to_of_val, Hv. Qed.
+
+(** structural computational version *)
+Fixpoint is_val (e : expr) : Prop :=
+  match e with
+  | LitInt l => True
+  | Lam x e => True
+  | Pair e1 e2 => is_val e1 ∧ is_val e2
+  | InjL e => is_val e
+  | InjR e => is_val e
+  | _ => False
+  end.
+Lemma is_val_spec e : is_val e ↔ ∃ v, to_val e = Some v.
+Proof.
+  induction e as [ | ? e IH | e1 IH1 e2 IH2 | | e1 IH1 e2 IH2 | e1 IH1 e2 IH2 | e IH | e IH | e IH | e IH | e1 IH1 e2 IH2 e3 IH3];
+    simpl; (split; [ | intros (v & Heq)]); simplify_option_eq; try done; eauto.
+  - rewrite IH1, IH2. intros [(v1 & ->) (v2 & ->)]. eauto.
+  - rewrite IH1, IH2. eauto.
+  - rewrite IH. intros (v & ->). eauto.
+  - apply IH. eauto.
+  - rewrite IH. intros (v & ->); eauto.
+  - apply IH. eauto.
+Qed.
+
+Ltac simplify_val :=
+  repeat match goal with
+  | H: to_val (of_val ?v) = ?o |- _ => rewrite to_of_val in H
+  | H: is_val ?e |- _ => destruct (proj1 (is_val_spec e) H) as (? & ?); clear H
+  end.
+
+(* Misc *)
+Lemma is_val_of_val v : is_val (of_val v).
+Proof. apply is_val_spec. rewrite to_of_val. eauto. Qed.
+
+(** Substitution *)
+Fixpoint subst (x : string) (es : expr) (e : expr)  : expr :=
+  match e with
+  | LitInt _ => e
+  | Var y => if decide (x = y) then es else Var y
+  | Lam y e =>
+      Lam y $ if decide (BNamed x = y) then e else subst x es e
+  | App e1 e2 => App (subst x es e1) (subst x es e2)
+  | Plus e1 e2 => Plus (subst x es e1) (subst x es e2)
+  | Pair e1 e2 => Pair (subst x es e1) (subst x es e2)
+  | Fst e => Fst (subst x es e)
+  | Snd e => Snd (subst x es e)
+  | InjL e => InjL (subst x es e)
+  | InjR e => InjR (subst x es e)
+  | Case e0 e1 e2 => Case (subst x es e0) (subst x es e1) (subst x es e2)
+  end.
+
+Definition subst' (mx : binder) (es : expr) : expr → expr :=
+  match mx with BNamed x => subst x es | BAnon => id end.
+
+(** Closed terms **)
+
+Fixpoint is_closed (X : list string) (e : expr) : bool :=
+  match e with
+  | Var x => bool_decide (x ∈ X)
+  | Lam x e => is_closed (x :b: X) e
+  | LitInt _ => true
+  | Fst e | Snd e | InjL e | InjR e => is_closed X e
+  | App e1 e2 | Plus e1 e2 | Pair e1 e2 => is_closed X e1 && is_closed X e2
+  | Case e0 e1 e2 =>
+   is_closed X e0 && is_closed X e1 && is_closed X e2
+  end.
+
+(** [closed] states closedness as a Coq proposition, through the [Is_true] transformer. *)
+Definition closed (X : list string) (e : expr) : Prop := Is_true (is_closed X e).
+#[export] Instance closed_proof_irrel X e : ProofIrrel (closed X e).
+Proof. unfold closed. apply _. Qed.
+#[export] Instance closed_dec X e : Decision (closed X e).
+Proof. unfold closed. apply _. Defined.
+
+(** closed expressions *)
+Lemma is_closed_weaken X Y e : is_closed X e → X ⊆ Y → is_closed Y e.
+Proof. revert X Y; induction e; naive_solver (eauto; set_solver). Qed.
+
+Lemma is_closed_weaken_nil X e : is_closed [] e → is_closed X e.
+Proof. intros. by apply is_closed_weaken with [], list_subseteq_nil. Qed.
+
+Lemma is_closed_subst X e x es :
+  is_closed [] es → is_closed (x :: X) e → is_closed X (subst x es e).
+Proof.
+  intros ?.
+  induction e in X |-*; simpl; intros ?; destruct_and?; split_and?; simplify_option_eq;
+    try match goal with
+    | H : ¬(_ ∧ _) |- _ => apply not_and_l in H as [?%dec_stable|?%dec_stable]
+    end; eauto using is_closed_weaken with set_solver.
+Qed.
+Lemma is_closed_do_subst' X e x es :
+  is_closed [] es → is_closed (x :b: X) e → is_closed X (subst' x es e).
+Proof. destruct x; eauto using is_closed_subst. Qed.
+
+(** Substitution lemmas *)
+Lemma subst_is_closed X e x es : is_closed X e → x ∉ X → subst x es e = e.
+Proof.
+  induction e in X |-*; simpl; rewrite ?bool_decide_spec, ?andb_True; intros ??;
+    repeat case_decide; simplify_eq; simpl; f_equal; intuition eauto with set_solver.
+Qed.
+
+Lemma subst_is_closed_nil e x es : is_closed [] e → subst x es e = e.
+Proof. intros. apply subst_is_closed with []; set_solver. Qed.
+Lemma subst'_is_closed_nil e x es : is_closed [] e → subst' x es e = e.
+Proof. intros. destruct x as [ | x]. { done. } by apply subst_is_closed_nil. Qed.

theories/type_systems/stlc_extended/notation.v

@@ -0,0 +1,56 @@
+From semantics.ts.stlc_extended Require Export lang.
+Set Default Proof Using "Type".
+
+
+(** Coercions to make programs easier to type. *)
+Coercion of_val : val >-> expr.
+Coercion App : expr >-> Funclass.
+Coercion Var : string >-> expr.
+
+(** Define some derived forms. *)
+Notation Let x e1 e2 := (App (Lam x e2) e1) (only parsing).
+Notation Seq e1 e2 := (Let BAnon e1 e2) (only parsing).
+Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)) (only parsing).
+
+(* No scope for the values, does not conflict and scope is often not inferred
+properly. *)
+Notation "# l" := (LitIntV l%Z%V%stdpp) (at level 8, format "# l").
+Notation "# l" := (LitInt l%Z%E%stdpp) (at level 8, format "# l") : expr_scope.
+
+(** Syntax inspired by Coq/Ocaml. Constructions with higher precedence come
+    first. *)
+Notation "( e1 , e2 , .. , en )" := (Pair .. (Pair e1 e2) .. en) : expr_scope.
+Notation "( e1 , e2 , .. , en )" := (PairV .. (PairV e1 e2) .. en) : val_scope.
+
+Notation "'match:' e0 'with' 'InjL' x1 => e1 | 'InjR' x2 => e2 'end'" :=
+  (Match e0 x1%binder e1 x2%binder e2)
+  (e0, x1, e1, x2, e2 at level 200,
+   format "'[hv' 'match:'  e0  'with'  '/  ' '[' 'InjL'  x1  =>  '/  ' e1 ']'  '/' '[' |  'InjR'  x2  =>  '/  ' e2 ']'  '/' 'end' ']'") : expr_scope.
+Notation "'match:' e0 'with' 'InjR' x1 => e1 | 'InjL' x2 => e2 'end'" :=
+  (Match e0 x2%binder e2 x1%binder e1)
+  (e0, x1, e1, x2, e2 at level 200, only parsing) : expr_scope.
+
+Notation "e1 + e2" := (Plus e1%E e2%E) : expr_scope.
+
+(*Notation "~ e" := (UnOp NegOp e%E) (at level 75, right associativity) : expr_scope.*)
+
+Notation "λ: x , e" := (Lam x%binder e%E)
+  (at level 200, x at level 1, e at level 200,
+   format "'[' 'λ:'  x ,  '/  ' e ']'") : expr_scope.
+Notation "λ: x y .. z , e" := (Lam x%binder (Lam y%binder .. (Lam z%binder e%E) ..))
+  (at level 200, x, y, z at level 1, e at level 200,
+   format "'[' 'λ:'  x  y  ..  z ,  '/  ' e ']'") : expr_scope.
+
+Notation "λ: x , e" := (LamV x%binder e%E)
+  (at level 200, x at level 1, e at level 200,
+   format "'[' 'λ:'  x ,  '/  ' e ']'") : val_scope.
+Notation "λ: x y .. z , e" := (LamV x%binder (Lam y%binder .. (Lam z%binder e%E) .. ))
+  (at level 200, x, y, z at level 1, e at level 200,
+   format "'[' 'λ:'  x  y  ..  z ,  '/  ' e ']'") : val_scope.
+
+Notation "'let:' x := e1 'in' e2" := (Lam x%binder e2%E e1%E)
+  (at level 200, x at level 1, e1, e2 at level 200,
+   format "'[' 'let:'  x  :=  '[' e1 ']'  'in'  '/' e2 ']'") : expr_scope.
+Notation "e1 ;; e2" := (Lam BAnon e2%E e1%E)
+  (at level 100, e2 at level 200,
+   format "'[' '[hv' '[' e1 ']' ;;  ']' '/' e2 ']'") : expr_scope.

theories/type_systems/stlc_extended/types.v

@@ -0,0 +1,255 @@
+From stdpp Require Import base relations.
+From iris Require Import prelude.
+From semantics.lib Require Import maps.
+From semantics.ts.stlc_extended Require Import lang notation ctxstep.
+
+(** ** Syntactic typing *)
+Inductive type : Type :=
+  | Int
+  | Fun (A B : type)
+  | Prod (A B : type)
+  | Sum (A B : type).
+
+Definition typing_context := gmap string type.
+Implicit Types
+  (Γ : typing_context)
+  (v : val)
+  (e : expr).
+
+Declare Scope FType_scope.
+Delimit Scope FType_scope with ty.
+Bind Scope FType_scope with type.
+Infix "→" := Fun : FType_scope.
+Notation "(→)" := Fun (only parsing) : FType_scope.
+Infix "×" := Prod (at level 70) : FType_scope.
+Notation "(×)" := Prod (only parsing) : FType_scope.
+Infix "+" := Sum : FType_scope.
+Notation "(+)" := Sum (only parsing) : FType_scope.
+
+Reserved Notation "Γ ⊢ e : A" (at level 74, e, A at next level).
+
+Inductive syn_typed : typing_context → expr → type → Prop :=
+  | typed_var Γ x A :
+      Γ !! x = Some A →
+      Γ ⊢ (Var x) : A
+  | typed_lam Γ x e A B :
+      (<[ x := A]> Γ) ⊢ e : B →
+      Γ ⊢ (Lam (BNamed x) e) : (A → B)
+  | typed_lam_anon Γ e A B :
+      Γ ⊢ e : B →
+      Γ ⊢ (Lam BAnon e) : (A → B)
+  | typed_int Γ z : Γ ⊢ (LitInt z) : Int
+  | typed_app Γ e1 e2 A B :
+      Γ ⊢ e1 : (A → B) →
+      Γ ⊢ e2 : A →
+      Γ ⊢ (e1 e2)%E : B
+  | typed_add Γ e1 e2 :
+      Γ ⊢ e1 : Int →
+      Γ ⊢ e2 : Int →
+      Γ ⊢ e1 + e2 : Int
+  (* TODO: provide the new typing rules *)
+where "Γ ⊢ e : A" := (syn_typed Γ e%E A%ty).
+#[export] Hint Constructors syn_typed : core.
+
+(** Examples *)
+Goal ∅ ⊢ (λ: "x", "x")%E : (Int → Int).
+Proof. eauto. Qed.
+
+Lemma syn_typed_closed Γ e A X :
+  Γ ⊢ e : A →
+  (∀ x, x ∈ dom Γ → x ∈ X) →
+  is_closed X e.
+Proof.
+  (* TODO: you will need to add the new cases, i.e. "|"'s to the intro pattern. The proof then should go through *)
+  induction 1 as [ | ?????? IH | | | | ] in X |-*; simpl; intros Hx; try done.
+  { (* var *) apply bool_decide_pack, Hx. apply elem_of_dom; eauto. }
+  { (* lam *) apply IH.
+    intros y. rewrite elem_of_dom lookup_insert_is_Some.
+    intros [<- | [? Hy]]; first by apply elem_of_cons; eauto.
+    apply elem_of_cons. right. eapply Hx. by apply elem_of_dom.
+  }
+  { (* anon lam *) naive_solver. }
+  (* everything else *)
+  all: repeat match goal with
+              | |- Is_true (_ && _)  => apply andb_True; split
+              end.
+  all: try naive_solver.
+Qed.
+
+Lemma typed_weakening Γ Δ e A:
+  Γ ⊢ e : A →
+  Γ ⊆ Δ →
+  Δ ⊢ e : A.
+Proof.
+  (* TODO: here you will need to add the new cases to the intro pattern as well. The proof then should go through *)
+  induction 1 as [| Γ x e A B Htyp IH | | | | ] in Δ |-*; intros Hsub; eauto.
+  - (* var *) econstructor. by eapply lookup_weaken.
+  - (* lam *) econstructor. eapply IH; eauto. by eapply insert_mono.
+Qed.
+
+(** Typing inversion lemmas *)
+Lemma var_inversion Γ (x: string) A: Γ ⊢ x : A → Γ !! x = Some A.
+Proof. inversion 1; subst; auto. Qed.
+
+Lemma lam_inversion Γ (x: string) e C:
+  Γ ⊢ (λ: x, e) : C →
+  ∃ A B, C = (A → B)%ty ∧ <[x:=A]> Γ ⊢ e : B.
+Proof. inversion 1; subst; eauto 10. Qed.
+
+Lemma lam_anon_inversion Γ e C:
+  Γ ⊢ (λ: <>, e) : C →
+  ∃ A B, C = (A → B)%ty ∧ Γ ⊢ e : B.
+Proof. inversion 1; subst; eauto 10. Qed.
+
+Lemma app_inversion Γ e1 e2 B:
+  Γ ⊢ e1 e2 : B →
+  ∃ A, Γ ⊢ e1 : (A → B) ∧ Γ ⊢ e2 : A.
+Proof. inversion 1; subst; eauto. Qed.
+
+Lemma plus_inversion Γ e1 e2 B:
+  Γ ⊢ e1 + e2 : B →
+  B = Int ∧ Γ ⊢ e1 : Int ∧ Γ ⊢ e2 : Int.
+Proof. inversion 1; subst; eauto. Qed.
+
+(* TODO: add inversion lemmas for the new typing rules.
+  They will be very useful for the proofs below!
+*)
+
+
+Lemma typed_substitutivity e e' Γ (x: string) A B :
+  ∅ ⊢ e' : A →
+  (<[x := A]> Γ) ⊢ e : B →
+  Γ ⊢ lang.subst x e' e : B.
+Proof.
+  intros He'. revert B Γ; induction e as [y | y | | | | | | |  | | ]; intros B Γ; simpl.
+  - intros Hp % var_inversion.
+    destruct (decide (x = y)).
+    + subst. rewrite lookup_insert in Hp. injection Hp as ->.
+      eapply typed_weakening; [done| ]. apply map_empty_subseteq.
+    + rewrite lookup_insert_ne in Hp; last done. auto.
+  - destruct y as [ | y].
+    { intros (A' & C & -> & Hty) % lam_anon_inversion.
+      econstructor. destruct decide as [Heq|].
+      + congruence.
+      + eauto.
+    }
+    intros (A' & C & -> & Hty) % lam_inversion.
+    econstructor. destruct decide as [Heq|].
+    + injection Heq as [= ->]. by rewrite insert_insert in Hty.
+    + rewrite insert_commute in Hty; last naive_solver. eauto.
+  - intros (C & Hty1 & Hty2) % app_inversion. eauto.
+  - inversion 1; subst; auto.
+  - intros (-> & Hty1 & Hty2)%plus_inversion; eauto.
+  - (* TODO *) admit.
+  - (* TODO *) admit.
+  - (* TODO *) admit.
+  - (* TODO *) admit.
+  - (* TODO *) admit.
+  - (* TODO *) admit.
+Admitted.
+
+(** Canonical values *)
+Lemma canonical_values_arr Γ e A B:
+  Γ ⊢ e : (A → B) →
+  is_val e →
+  ∃ x e', e = (λ: x, e')%E.
+Proof.
+  inversion 1; simpl; naive_solver.
+Qed.
+
+Lemma canonical_values_int Γ e:
+  Γ ⊢ e : Int →
+  is_val e →
+  ∃ n: Z, e = (#n)%E.
+Proof.
+  inversion 1; simpl; naive_solver.
+Qed.
+
+(* TODO: add canonical forms lemmas for the new types *)
+
+(** Progress *)
+Lemma typed_progress e A:
+  ∅ ⊢ e : A → is_val e ∨ reducible e.
+Proof.
+  remember ∅ as Γ.
+  (* TODO: you will need to extend the intro pattern *)
+  induction 1 as [| | | | Γ e1 e2 A B Hty IH1 _ IH2 | Γ e1 e2 Hty1 IH1 Hty2 IH2].
+  - subst. naive_solver.
+  - left. done.
+  - left. done.
+  - (* int *)left. done.
+  - (* app *)
+    destruct (IH2 HeqΓ) as [H2|H2]; [destruct (IH1 HeqΓ) as [H1|H1]|].
+    + eapply canonical_values_arr in Hty as (x & e & ->); last done.
+      right. eexists.
+      eapply base_contextual_step, BetaS; eauto.
+    + right. destruct H1 as [e1' Hstep].
+      eexists. eauto.
+    + right. destruct H2 as [e2' H2].
+      eexists. eauto.
+  - (* plus *)
+    destruct (IH2 HeqΓ) as [H2|H2]; [destruct (IH1 HeqΓ) as [H1|H1]|].
+    + right. eapply canonical_values_int in Hty1 as [n1 ->]; last done.
+      eapply canonical_values_int in Hty2 as [n2 ->]; last done.
+      subst. eexists; eapply base_contextual_step. eauto.
+    + right. destruct H1 as [e1' Hstep]. eexists. eauto.
+    + right. destruct H2 as [e2' H2]. eexists. eauto.
+
+(* FIXME: prove the new cases *)
+Admitted.
+
+Definition ectx_typing (K: ectx) (A B: type) :=
+  ∀ e, ∅ ⊢ e : A → ∅ ⊢ (fill K e) : B.
+
+Lemma fill_typing_decompose K e A:
+  ∅ ⊢ fill K e : A →
+  ∃ B, ∅ ⊢ e : B ∧ ectx_typing K B A.
+Proof.
+  unfold ectx_typing; induction K in e,A |-*; simpl; eauto.
+  all: inversion 1; subst; edestruct IHK as [? [Hit Hty]]; eauto.
+Qed.
+
+Lemma fill_typing_compose K e A B:
+  ∅ ⊢ e : B →
+  ectx_typing K B A →
+  ∅ ⊢ fill K e : A.
+Proof.
+  intros H1 H2; by eapply H2.
+Qed.
+
+Lemma typed_preservation_base_step e e' A:
+  ∅ ⊢ e : A →
+  base_step e e' →
+  ∅ ⊢ e' : A.
+Proof.
+  intros Hty Hstep.
+  destruct Hstep as [  ]; subst.
+  - eapply app_inversion in Hty as (B & H1 & H2).
+    destruct x as [|x].
+    { eapply lam_anon_inversion in H1 as (C & D & [= -> ->] & Hty). done. }
+    eapply lam_inversion in H1 as (C & D & Heq & Hty).
+    injection Heq as -> ->.
+    eapply typed_substitutivity; eauto.
+  - eapply plus_inversion in Hty as (-> & Hty1 & Hty2). constructor.
+
+(* TODO: extend this for the new cases *)
+Admitted.
+
+Lemma typed_preservation e e' A:
+  ∅ ⊢ e : A →
+  contextual_step e e' →
+  ∅ ⊢ e' : A.
+Proof.
+  intros Hty Hstep. destruct Hstep as [K e1 e2 -> -> Hstep].
+  eapply fill_typing_decompose in Hty as [B [H1 H2]].
+  eapply fill_typing_compose; last done.
+  by eapply typed_preservation_base_step.
+Qed.
+
+Lemma type_safety e1 e2 A:
+  ∅ ⊢ e1 : A →
+  rtc contextual_step e1 e2 →
+  is_val e2 ∨ reducible e2.
+Proof.
+  induction 2; eauto using typed_progress, typed_preservation.
+Qed.