More comments.

_CoqProject

--Q solutions tutorial_popl20
+-Q solutions solutions
 # non-canonical projections (https://github.com/coq/coq/pull/10076) do not exist yet in 8.9.
 -arg -w -arg -redundant-canonical-projection
 # change_no_check does not exist yet in 8.9.

solutions/interp.v

@@ -3,8 +3,10 @@ From tutorial_popl20 Require Export sem_typed sem_type_formers types.
 (** * Interpretation of syntactic types *)
 (** We use semantic type formers to define the interpretation [⟦ τ ⟧ : sem_ty]
 of syntactic types [τ : ty]. The interpretation is defined recursively on the
-structure of syntactic types. *)
-
+structure of syntactic types. To account for type variables (that appear freely
+in [τ]), we need to keep track of their corresponding semantic types. We
+represent these semantic types as a list, since de use De Bruijn indices for
+type variables. *)
 Reserved Notation "⟦ τ ⟧".
 Fixpoint interp `{!heapG Σ} (τ : ty) (ρ : list (sem_ty Σ)) : sem_ty Σ :=
   match τ return _ with
@@ -20,16 +22,10 @@ Fixpoint interp `{!heapG Σ} (τ : ty) (ρ : list (sem_ty Σ)) : sem_ty Σ :=
   | TRef τ => ref (⟦ τ ⟧ ρ)
   end%sem_ty
 where "⟦ τ ⟧" := (interp τ).
-
 Instance: Params (@interp) 2 := {}.
 
-(** Given a syntactic typing context [Γ : gmap string ty] (a mapping from
-variables [string] to syntactic types [ty]) together with a mapping
-[ρ : list (sem_ty Σ)] from type variables (that appear freely in [Γ]) to
-their corresponding semantic types (represented as a list, since de use De
-Bruijn indices for type variables), we define a semantic typing context
-[interp_env Γ ρ : gmap string (sem_ty Σ)], i.e., a mapping from variables
-(strings) to semantic types. *)
+(** We now lift the interpretation of types to typing contexts. This is done in
+a pointwise fashion using the [<$> : (A → B) → gmap K A → gmap K B] operator. *)
 Definition interp_env `{!heapG Σ} (Γ : gmap string ty)
   (ρ : list (sem_ty Σ)) : gmap string (sem_ty Σ) := flip interp ρ <$> Γ.
 Instance: Params (@interp_env) 3 := {}.

solutions/parametricity.v

 From tutorial_popl20 Require Export safety.
 
+(** * Parametricity *)
 Section parametricity.
   Context `{!heapG Σ}.
 
+  (** * The polymorphic identity function *)
   Lemma identity_param `{!heapPreG Σ} e (v : val) σ w es σ' :
     (∀ `{!heapG Σ}, (∅ ⊨ e : ∀ A, A → A)%I) →
     rtc erased_step ([e <_> v]%E, σ) (of_val w :: es, σ') → w = v.
@@ -20,6 +22,7 @@ Section parametricity.
     wp_apply (wp_wand with "Hu"). iIntros (w') "Hw'". by iApply "Hw'".
   Qed.
 
+  (** * Exercise (empty_type_param, easy) *)
   Lemma empty_type_param `{!heapPreG Σ} e (v : val) σ w es σ' :
     (∀ `{!heapG Σ}, (∅ ⊨ e : ∀ A, A)%I) →
     rtc erased_step ([e <_>]%E, σ) (of_val w :: es, σ') →
@@ -38,6 +41,7 @@ Section parametricity.
     iApply ("Hu" $! T).
   Qed.
 
+  (** * Exercise (boolean_param, moderate) *)
   Lemma boolean_param `{!heapPreG Σ} e (v1 v2 : val) σ w es σ' :
     (∀ `{!heapG Σ}, (∅ ⊨ e : ∀ A, A → A → A)%I) →
     rtc erased_step ([e <_> v1 v2]%E, σ) (of_val w :: es, σ') → w = v1 ∨ w = v2.
@@ -59,6 +63,7 @@ Section parametricity.
     iApply ("Hw''" $! v2). by iRight.
   Qed.
 
+  (** * Exercise (nat_param, hard) *)
   Lemma nat_param `{!heapPreG Σ} e σ w es σ' :
     (∀ `{!heapG Σ}, (∅ ⊨ e : ∀ A, (A → A) → A → A)%I) →
     rtc erased_step ([e <_> (λ: "n", "n" + #1)%V #0]%E, σ)
@@ -85,6 +90,7 @@ Section parametricity.
     by iExists 0%nat.
   Qed.
 
+  (** * Exercise (strong_nat_param, hard) *)
   Lemma strong_nat_param `{!heapPreG Σ} e σ w es σ' (vf vz : val) φ :
     (∀ `{!heapG Σ}, ∃ Φ : sem_ty Σ,
       (∅ ⊨ e : ∀ A, (A → A) → A → A)%I ∧

solutions/sem_type_formers.v

@@ -92,7 +92,7 @@ Section types.
   Definition sem_ty_ref (A : sem_ty Σ) : sem_ty Σ := SemTy (λ w,
     ∃ l : loc, ⌜w = #l⌝ ∧ inv (tyN .@ l) (∃ v, l ↦ v ∗ A v))%I.
   (** Intuitively, values of the reference type [sem_ty_ref A] should
-  be locations l that hold a value [w] in the semantic type [A] at
+  be locations [l] that hold a value [w] in the semantic type [A] at
   all times. In order to express this intuition in a formal way, we
   make use of two features of Iris: