Commit af43752f authored by Amin Timany's avatar Amin Timany
Browse files

coqdoc for interp.v and sem_type_formers.v

parent aae79b93
From tutorial_popl20 Require Export sem_type_formers types.
(** * Here we use semantic type formers to define semantics of syntactic types.
This is done by a straightforward induction on the syntactic type. *)
Reserved Notation "⟦ τ ⟧".
Fixpoint interp `{heapG Σ} (τ : ty) (ρ : list (sem_ty Σ)) : sem_ty Σ :=
match τ return _ with
From tutorial_popl20 Require Export sem_types.
(** The definitions of the type formers *)
(** * The definitions of the semantic type formers
These type formers correspond to those of syntactic types, e.g., a
product former that given two semantic types [A] and [B], gives
the semantic type of the product [A * B], i.e., values that are
pairs where the first component belongs to [A] and the second
component to [B]. *)
Section types.
Context `{heapG Σ}.
......@@ -25,17 +31,76 @@ Section types.
(* REMOVE *) Definition sem_ty_sum (A1 A2 : sem_ty Σ) : sem_ty Σ := SemTy (λ w,
( w1, w = InjLV w1 A1 w1) ( w2, w = InjRV w2 A2 w2))%I.
(** The value interpretation for function types is as follows: *)
Definition sem_ty_arr (A1 A2 : sem_ty Σ) : sem_ty Σ := SemTy (λ w,
v, A1 v - WP App w v {{ A2 }})%I.
(** This definition is very close to the usual way of defining the
value interpretation of the function type [A1 → A2] in ordinary
logical relations: it expresses that arguments of semantic type
[A1] are mapped to results of semantic type [A2] . The only Iris
specific feature one has to take into account is the [□]
modality. Recall that the value interpretation should be a
persistent predicate; however, even if both [P] and [Q] are
persistent propositions, the magic wand [P -∗ Q] is not
necessarily persistent. Hence, we use the [□] modality to make
the magic wand persistent. *)
(** The value interpretation for type variables, universal types,
and existential types is: *)
Definition sem_ty_forall (C : sem_ty Σ sem_ty Σ) : sem_ty Σ := SemTy (λ w,
A : sem_ty Σ, WP w #() {{ w, C A w }})%I.
Definition sem_ty_exist (C : sem_ty Σ sem_ty Σ) : sem_ty Σ := SemTy (λ w,
A : sem_ty Σ, C A w)%I.
(** The interpretations of these types are fairly straightforward.
Given a higher-order type former [C] that maps semantic types to
semantic types, we define the universal type [sem_ty_forall A]
using the universal quantification in Iris. That is, a value [w]
is considered a polymorphic value if for any semantic type [A],
when [w] is specialized to the type [A] (written as [w #()] as
(semantic) types never appear in terms in our untyped syntax)
the _resulting expression_ is an expression in the semantics of
the type [C A] (defined using WP).
Similarly, given a higher-order type former [C] that maps
semantic types to semantic types, we define the existential type
[sem_ty_exist A] using the existential quantification in Iris.
Notice how the impredicative nature of Iris propositions and
predicates allows us to quantify over Iris predicates to define
an Iris predicate. This is crucial for giving semantics to
parametric polymorphism, i.e., universal and existential types.
Remark: notice that for technical reasons (related to the value
restriction problem in ML-like languages) universally quantified
expressions are not evaluated until they are applied to a
specific type. *)
(** The value interpretation of reference types is as follows: *)
Definition tyN := nroot .@ "ty".
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
all times. In order to express this intuition in a formal way, we
make use of two features of Iris:
- The points-to connective l ↦ v (from vanilla separation logic)
provides exclusive ownership of the location l with value
v. The points-to connective is an ephemeral proposition, and
necessarily not a persistent proposition.
- The invariant assertion [inv N P] expresses that a (typically
ephemeral) proposition [P] holds at all times -- i.e., [P] is
invariant. The invariant assertion is persistent. *)
(** Remark: Iris is also capable giving semantics to recursive
types. However, for the sake of simplicity we did not consider
recursive types for this tutorial. In particular, to give the
semantics of recursive types one needs to use Iris's guarded
fixpoints which in turn requires us to enforce that semantics types,
in addition to being persistent, are also non-expansive. *)
End types.
(** We introduce nicely looking notations for our semantic types. This allows
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