Primitive persistent arrays

Persistent arrays expose a functional interface but are implemented
using an imperative data structure. The OCaml implementation is based on
Jean-Christophe Filliâtre's.

Co-authored-by: Benjamin Grégoire <Benjamin.Gregoire@inria.fr>
Co-authored-by: Gaëtan Gilbert <gaetan.gilbert@skyskimmer.net>

5 files changed, 101 insertions, 3 deletions

diff --git a/doc/changelog/01-kernel/11604-persistent-arrays.rst b/doc/changelog/01-kernel/11604-persistent-arrays.rst
new file mode 100644
index 0000000000..fbade033d2
--- /dev/null
+++ b/doc/changelog/01-kernel/11604-persistent-arrays.rst
@@ -0,0 +1,6 @@
+- **Added:**
+  Built-in support for persistent arrays, which expose a functional
+  interface but are implemented using an imperative data structure, for
+  better performance.
+  (`#11604 <https://github.com/coq/coq/pull/11604>`_,
+  by Maxime Dénès and Benjamin Grégoire, with help from Gaëtan Gilbert).
diff --git a/doc/sphinx/biblio.bib b/doc/sphinx/biblio.bib
index e0eec2ae2d..323da93f3e 100644
--- a/doc/sphinx/biblio.bib
+++ b/doc/sphinx/biblio.bib
@@ -617,3 +617,38 @@ the Calculus of Inductive Constructions}},
   year         = 2019,
   institution = {Chalmers and Gothenburg University},
 }
+
+@inproceedings{ConchonFilliatre07wml,
+  author = {Sylvain Conchon and Jean-Christophe Filliâtre},
+  title = {A Persistent Union-Find Data Structure},
+  booktitle = {ACM SIGPLAN Workshop on ML},
+  publisher = {ACM Press},
+  pages = {37--45},
+  year = 2007,
+  address = {Freiburg, Germany},
+  month = {October},
+  topics = {team, lri},
+  type_publi = {icolcomlec},
+  type_digiteo = {conf_isbn},
+  x-pdf = {https://www.lri.fr/~filliatr/ftp/publis/puf-wml07.pdf},
+  url = {https://www.lri.fr/~filliatr/ftp/publis/puf-wml07.pdf},
+  abstract = { The problem of disjoint sets, also known as union-find,
+  consists in maintaining a partition of a finite set within a data
+  structure. This structure provides two operations: a function find
+  returning the class of an element and a function union merging two
+  classes. An optimal and imperative solution is known since 1975.
+  However, the imperative nature of this data structure may be a
+  drawback when it is used in a backtracking algorithm. This paper
+  details the implementation of a persistent union-find data structure
+  as efficient as its imperative counterpart.  To achieve this result,
+  our solution makes heavy use of imperative features and thus it is a
+  significant example of a data structure whose side effects are
+  safely hidden behind a persistent interface.  To strengthen this
+  last claim, we also detail a formalization using the Coq proof
+  assistant which shows both the correctness of our solution and its
+  observational persistence. },
+  x-equipes = {demons PROVAL},
+  x-type = {article},
+  x-support = {actes_aux},
+  x-cle-support = {ML}
+}
diff --git a/doc/sphinx/language/core/primitive.rst b/doc/sphinx/language/core/primitive.rst
index dc8f131209..727177b23a 100644
--- a/doc/sphinx/language/core/primitive.rst
+++ b/doc/sphinx/language/core/primitive.rst
@@ -40,9 +40,8 @@ These primitive declarations are regular axioms. As such, they must be trusted a
 
    Print Assumptions one_minus_one_is_zero.
 
-The reduction machines (:tacn:`vm_compute`, :tacn:`native_compute`) implement
-dedicated, efficient, rules to reduce the applications of these primitive
-operations.
+The reduction machines implement dedicated, efficient rules to reduce the
+applications of these primitive operations.
 
 The extraction of these primitives can be customized similarly to the extraction
 of regular axioms (see :ref:`extraction`). Nonetheless, the :g:`ExtrOCamlInt63`
@@ -105,3 +104,53 @@ Literal values (of type :g:`Float64.t`) are extracted to literal OCaml
 values (of type :g:`float`) written in hexadecimal notation and
 wrapped into the :g:`Float64.of_float` constructor, e.g.:
 :g:`Float64.of_float (0x1p+0)`.
+
+.. _primitive-arrays:
+
+Primitive Arrays
+----------------
+
+The language of terms features persistent arrays as values. The type of
+such a value is *axiomatized*; it is declared through the following sentence
+(excerpt from the :g:`PArray` module):
+
+.. coqdoc::
+
+   Primitive array := #array_type.
+
+This type is equipped with a few operators, that must be similarly declared.
+For instance, elements in an array can be accessed and updated using the
+:g:`PArray.get` and :g:`PArray.set` functions, declared and specified as
+follows:
+
+.. coqdoc::
+
+   Primitive get := #array_get.
+   Primitive set := #array_set.
+   Notation "t .[ i ]" := (get t i).
+   Notation "t .[ i <- a ]" := (set t i a).
+
+   Axiom get_set_same : forall A t i (a:A), (i < length t) = true -> t.[i<-a].[i] = a.
+   Axiom get_set_other : forall A t i j (a:A), i <> j -> t.[i<-a].[j] = t.[j].
+
+The complete set of such operators can be obtained looking at the :g:`PArray` module.
+
+These primitive declarations are regular axioms. As such, they must be trusted and are listed by the
+:g:`Print Assumptions` command.
+
+The reduction machines (:tacn:`vm_compute`, :tacn:`native_compute`) implement
+dedicated, efficient rules to reduce the applications of these primitive
+operations.
+
+The extraction of these primitives can be customized similarly to the extraction
+of regular axioms (see :ref:`extraction`). Nonetheless, the :g:`ExtrOCamlPArray`
+module can be used when extracting to OCaml: it maps the Coq primitives to types
+and functions of a :g:`Parray` module. Said OCaml module is not produced by
+extraction. Instead, it has to be provided by the user (if they want to compile
+or execute the extracted code). For instance, an implementation of this module
+can be taken from the kernel of Coq (see ``kernel/parray.ml``).
+
+Primitive arrays expose a functional interface, but they are internally
+implemented using a persistent data structure :cite:`ConchonFilliatre07wml`.
+Update and access to an element in the most recent copy of an array are
+constant time operations.
diff --git a/doc/stdlib/hidden-files b/doc/stdlib/hidden-files
index 4badb20295..f39c50238a 100644
--- a/doc/stdlib/hidden-files
+++ b/doc/stdlib/hidden-files
@@ -15,6 +15,7 @@ theories/extraction/ExtrOcamlBigIntConv.v
 theories/extraction/ExtrOcamlChar.v
 theories/extraction/ExtrOCamlInt63.v
 theories/extraction/ExtrOCamlFloats.v
+theories/extraction/ExtrOCamlPArray.v
 theories/extraction/ExtrOcamlIntConv.v
 theories/extraction/ExtrOcamlNatBigInt.v
 theories/extraction/ExtrOcamlNatInt.v
diff --git a/doc/stdlib/index-list.html.template b/doc/stdlib/index-list.html.template
index ab615d5f65..7c1328916b 100644
--- a/doc/stdlib/index-list.html.template
+++ b/doc/stdlib/index-list.html.template
@@ -709,4 +709,11 @@ through the <tt>Require Import</tt> command.</p>
     theories/Compat/Coq812.v
     theories/Compat/Coq813.v
   </dd>
+
+  <dt> <b>Array</b>:
+    Persistent native arrays
+  </dt>
+  <dd>
+    theories/Array/PArray.v
+  </dd>
 </dl>