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diff --git a/doc/sphinx/addendum/canonical-structures.rst b/doc/sphinx/addendum/canonical-structures.rst new file mode 100644 index 000000000..8961b0096 --- /dev/null +++ b/doc/sphinx/addendum/canonical-structures.rst @@ -0,0 +1,383 @@ +\achapter{Canonical Structures} +%HEVEA\cutname{canonical-structures.html} +\aauthor{Assia Mahboubi and Enrico Tassi} + +\label{CS-full} +\index{Canonical Structures!presentation} + +\noindent This chapter explains the basics of Canonical Structure and how they can be used +to overload notations and build a hierarchy of algebraic structures. +The examples are taken from~\cite{CSwcu}. We invite the interested reader +to refer to this paper for all the details that are omitted here for brevity. +The interested reader shall also find in~\cite{CSlessadhoc} a detailed +description of another, complementary, use of Canonical Structures: +advanced proof search. This latter papers also presents many techniques one +can employ to tune the inference of Canonical Structures. + +\section{Notation overloading} + +We build an infix notation $==$ for a comparison predicate. Such notation +will be overloaded, and its meaning will depend on the types of the terms +that are compared. + +\begin{coq_eval} +Require Import Arith. +\end{coq_eval} + +\begin{coq_example} +Module EQ. + Record class (T : Type) := Class { cmp : T -> T -> Prop }. + Structure type := Pack { obj : Type; class_of : class obj }. + Definition op (e : type) : obj e -> obj e -> Prop := + let 'Pack _ (Class _ the_cmp) := e in the_cmp. + Check op. + Arguments op {e} x y : simpl never. + Arguments Class {T} cmp. + Module theory. + Notation "x == y" := (op x y) (at level 70). + End theory. +End EQ. +\end{coq_example} + +We use Coq modules as name spaces. This allows us to follow the same pattern +and naming convention for the rest of the chapter. The base name space +contains the definitions of the algebraic structure. To keep the example +small, the algebraic structure \texttt{EQ.type} we are defining is very simplistic, +and characterizes terms on which a binary relation is defined, without +requiring such relation to validate any property. +The inner \texttt{theory} module contains the overloaded notation \texttt{==} and +will eventually contain lemmas holding on all the instances of the +algebraic structure (in this case there are no lemmas). + +Note that in practice the user may want to declare \texttt{EQ.obj} as a coercion, +but we will not do that here. + +The following line tests that, when we assume a type \texttt{e} that is in the +\texttt{EQ} class, then we can relates two of its objects with \texttt{==}. + +\begin{coq_example} +Import EQ.theory. +Check forall (e : EQ.type) (a b : EQ.obj e), a == b. +\end{coq_example} + +Still, no concrete type is in the \texttt{EQ} class. We amend that by equipping \texttt{nat} +with a comparison relation. + +\begin{coq_example} +Fail Check 3 == 3. +Definition nat_eq (x y : nat) := nat_compare x y = Eq. +Definition nat_EQcl : EQ.class nat := EQ.Class nat_eq. +Canonical Structure nat_EQty : EQ.type := EQ.Pack nat nat_EQcl. +Check 3 == 3. +Eval compute in 3 == 4. +\end{coq_example} + +This last test shows that Coq is now not only able to typecheck \texttt{3==3}, but +also that the infix relation was bound to the \texttt{nat\_eq} relation. This +relation is selected whenever \texttt{==} is used on terms of type \texttt{nat}. This +can be read in the line declaring the canonical structure \texttt{nat\_EQty}, +where the first argument to \texttt{Pack} is the key and its second argument +a group of canonical values associated to the key. In this case we associate +to \texttt{nat} only one canonical value (since its class, \texttt{nat\_EQcl} has just one +member). The use of the projection \texttt{op} requires its argument to be in +the class \texttt{EQ}, and uses such a member (function) to actually compare +its arguments. + +Similarly, we could equip any other type with a comparison relation, and +use the \texttt{==} notation on terms of this type. + +\subsection{Derived Canonical Structures} + +We know how to use \texttt{==} on base types, like \texttt{nat}, \texttt{bool}, \texttt{Z}. +Here we show how to deal with type constructors, i.e. how to make the +following example work: + +\begin{coq_example} +Fail Check forall (e : EQ.type) (a b : EQ.obj e), (a,b) == (a,b). +\end{coq_example} + +The error message is telling that Coq has no idea on how to compare +pairs of objects. The following construction is telling Coq exactly how to do +that. + +\begin{coq_example} +Definition pair_eq (e1 e2 : EQ.type) (x y : EQ.obj e1 * EQ.obj e2) := + fst x == fst y /\ snd x == snd y. +Definition pair_EQcl e1 e2 := EQ.Class (pair_eq e1 e2). +Canonical Structure pair_EQty (e1 e2 : EQ.type) : EQ.type := + EQ.Pack (EQ.obj e1 * EQ.obj e2) (pair_EQcl e1 e2). +Check forall (e : EQ.type) (a b : EQ.obj e), (a,b) == (a,b). +Check forall n m : nat, (3,4) == (n,m). +\end{coq_example} + +Thanks to the \texttt{pair\_EQty} declaration, Coq is able to build a comparison +relation for pairs whenever it is able to build a comparison relation +for each component of the pair. The declaration associates to the key +\texttt{*} (the type constructor of pairs) the canonical comparison relation +\texttt{pair\_eq} whenever the type constructor \texttt{*} is applied to two types +being themselves in the \texttt{EQ} class. + +\section{Hierarchy of structures} + +To get to an interesting example we need another base class to be available. +We choose the class of types that are equipped with an order relation, +to which we associate the infix \texttt{<=} notation. + +\begin{coq_example} +Module LE. + Record class T := Class { cmp : T -> T -> Prop }. + Structure type := Pack { obj : Type; class_of : class obj }. + Definition op (e : type) : obj e -> obj e -> Prop := + let 'Pack _ (Class _ f) := e in f. + Arguments op {_} x y : simpl never. + Arguments Class {T} cmp. + Module theory. + Notation "x <= y" := (op x y) (at level 70). + End theory. +End LE. +\end{coq_example} + +As before we register a canonical \texttt{LE} class for \texttt{nat}. + +\begin{coq_example} +Import LE.theory. +Definition nat_le x y := nat_compare x y <> Gt. +Definition nat_LEcl : LE.class nat := LE.Class nat_le. +Canonical Structure nat_LEty : LE.type := LE.Pack nat nat_LEcl. +\end{coq_example} + +And we enable Coq to relate pair of terms with \texttt{<=}. + +\begin{coq_example} +Definition pair_le e1 e2 (x y : LE.obj e1 * LE.obj e2) := + fst x <= fst y /\ snd x <= snd y. +Definition pair_LEcl e1 e2 := LE.Class (pair_le e1 e2). +Canonical Structure pair_LEty (e1 e2 : LE.type) : LE.type := + LE.Pack (LE.obj e1 * LE.obj e2) (pair_LEcl e1 e2). +Check (3,4,5) <= (3,4,5). +\end{coq_example} + +At the current stage we can use \texttt{==} and \texttt{<=} on concrete types, +like tuples of natural numbers, but we can't develop an algebraic +theory over the types that are equipped with both relations. + +\begin{coq_example} +Check 2 <= 3 /\ 2 == 2. +Fail Check forall (e : EQ.type) (x y : EQ.obj e), x <= y -> y <= x -> x == y. +Fail Check forall (e : LE.type) (x y : LE.obj e), x <= y -> y <= x -> x == y. +\end{coq_example} + +We need to define a new class that inherits from both \texttt{EQ} and \texttt{LE}. + +\begin{coq_example} +Module LEQ. + Record mixin (e : EQ.type) (le : EQ.obj e -> EQ.obj e -> Prop) := + Mixin { compat : forall x y : EQ.obj e, le x y /\ le y x <-> x == y }. + Record class T := Class { + EQ_class : EQ.class T; + LE_class : LE.class T; + extra : mixin (EQ.Pack T EQ_class) (LE.cmp T LE_class) }. + Structure type := _Pack { obj : Type; class_of : class obj }. + Arguments Mixin {e le} _. + Arguments Class {T} _ _ _. +\end{coq_example} + +The \texttt{mixin} component of the \texttt{LEQ} class contains all the extra content +we are adding to \texttt{EQ} and \texttt{LE}. In particular it contains the requirement +that the two relations we are combining are compatible. + +Unfortunately there is still an obstacle to developing the algebraic theory +of this new class. + +\begin{coq_example} + Module theory. + Fail Check forall (le : type) (n m : obj le), n <= m -> n <= m -> n == m. +\end{coq_example} + +The problem is that the two classes \texttt{LE} and \texttt{LEQ} are not yet related by +a subclass relation. In other words Coq does not see that an object +of the \texttt{LEQ} class is also an object of the \texttt{LE} class. + +The following two constructions tell Coq how to canonically build +the \texttt{LE.type} and \texttt{EQ.type} structure given an \texttt{LEQ.type} structure +on the same type. + +\begin{coq_example} + Definition to_EQ (e : type) : EQ.type := + EQ.Pack (obj e) (EQ_class _ (class_of e)). + Canonical Structure to_EQ. + Definition to_LE (e : type) : LE.type := + LE.Pack (obj e) (LE_class _ (class_of e)). + Canonical Structure to_LE. +\end{coq_example} +We can now formulate out first theorem on the objects of the \texttt{LEQ} structure. +\begin{coq_example} + Lemma lele_eq (e : type) (x y : obj e) : x <= y -> y <= x -> x == y. + now intros; apply (compat _ _ (extra _ (class_of e)) x y); split. Qed. + Arguments lele_eq {e} x y _ _. + End theory. +End LEQ. +Import LEQ.theory. +Check lele_eq. +\end{coq_example} + +Of course one would like to apply results proved in the algebraic +setting to any concrete instate of the algebraic structure. + +\begin{coq_example} +Example test_algebraic (n m : nat) : n <= m -> m <= n -> n == m. + Fail apply (lele_eq n m). Abort. +Example test_algebraic2 (l1 l2 : LEQ.type) (n m : LEQ.obj l1 * LEQ.obj l2) : + n <= m -> m <= n -> n == m. + Fail apply (lele_eq n m). Abort. +\end{coq_example} + +Again one has to tell Coq that the type \texttt{nat} is in the \texttt{LEQ} class, and how +the type constructor \texttt{*} interacts with the \texttt{LEQ} class. In the following +proofs are omitted for brevity. + +\begin{coq_example} +Lemma nat_LEQ_compat (n m : nat) : n <= m /\ m <= n <-> n == m. +\end{coq_example} +\begin{coq_eval} + +split. + unfold EQ.op; unfold LE.op; simpl; unfold nat_le; unfold nat_eq. + case (nat_compare_spec n m); [ reflexivity | | now intros _ [H _]; case H ]. + now intro H; apply nat_compare_gt in H; rewrite -> H; intros [_ K]; case K. +unfold EQ.op; unfold LE.op; simpl; unfold nat_le; unfold nat_eq. +case (nat_compare_spec n m); [ | intros H1 H2; discriminate H2 .. ]. +intro H; rewrite H; intros _; split; [ intro H1; discriminate H1 | ]. +case (nat_compare_eq_iff m m); intros _ H1. +now rewrite H1; auto; intro H2; discriminate H2. +Qed. +\end{coq_eval} +\begin{coq_example} +Definition nat_LEQmx := LEQ.Mixin nat_LEQ_compat. +Lemma pair_LEQ_compat (l1 l2 : LEQ.type) (n m : LEQ.obj l1 * LEQ.obj l2) : +n <= m /\ m <= n <-> n == m. +\end{coq_example} +\begin{coq_eval} + +case n; case m; unfold EQ.op; unfold LE.op; simpl. +intros n1 n2 m1 m2; split; [ intros [[Le1 Le2] [Ge1 Ge2]] | intros [H1 H2] ]. + now split; apply lele_eq. +case (LEQ.compat _ _ (LEQ.extra _ (LEQ.class_of l1)) m1 n1). +case (LEQ.compat _ _ (LEQ.extra _ (LEQ.class_of l2)) m2 n2). +intros _ H3 _ H4; apply H3 in H2; apply H4 in H1; clear H3 H4. +now case H1; case H2; split; split. +Qed. +\end{coq_eval} +\begin{coq_example} +Definition pair_LEQmx l1 l2 := LEQ.Mixin (pair_LEQ_compat l1 l2). +\end{coq_example} + +The following script registers an \texttt{LEQ} class for \texttt{nat} and for the +type constructor \texttt{*}. It also tests that they work as expected. + +Unfortunately, these declarations are very verbose. In the following +subsection we show how to make these declaration more compact. + +\begin{coq_example} +Module Add_instance_attempt. + Canonical Structure nat_LEQty : LEQ.type := + LEQ._Pack nat (LEQ.Class nat_EQcl nat_LEcl nat_LEQmx). + Canonical Structure pair_LEQty (l1 l2 : LEQ.type) : LEQ.type := + LEQ._Pack (LEQ.obj l1 * LEQ.obj l2) + (LEQ.Class + (EQ.class_of (pair_EQty (to_EQ l1) (to_EQ l2))) + (LE.class_of (pair_LEty (to_LE l1) (to_LE l2))) + (pair_LEQmx l1 l2)). + Example test_algebraic (n m : nat) : n <= m -> m <= n -> n == m. + now apply (lele_eq n m). Qed. + Example test_algebraic2 (n m : nat * nat) : n <= m -> m <= n -> n == m. + now apply (lele_eq n m). Qed. +End Add_instance_attempt. +\end{coq_example} + +Note that no direct proof of \texttt{n <= m -> m <= n -> n == m} is provided by the +user for \texttt{n} and \texttt{m} of type \texttt{nat * nat}. What the user provides is a proof of +this statement for \texttt{n} and \texttt{m} of type \texttt{nat} and a proof that the pair +constructor preserves this property. The combination of these two facts is a +simple form of proof search that Coq performs automatically while inferring +canonical structures. + +\subsection{Compact declaration of Canonical Structures} + +We need some infrastructure for that. + +\begin{coq_example*} +Require Import Strings.String. +\end{coq_example*} +\begin{coq_example} +Module infrastructure. + Inductive phantom {T : Type} (t : T) : Type := Phantom. + Definition unify {T1 T2} (t1 : T1) (t2 : T2) (s : option string) := + phantom t1 -> phantom t2. + Definition id {T} {t : T} (x : phantom t) := x. + Notation "[find v | t1 ~ t2 ] p" := (fun v (_ : unify t1 t2 None) => p) + (at level 50, v ident, only parsing). + Notation "[find v | t1 ~ t2 | s ] p" := (fun v (_ : unify t1 t2 (Some s)) => p) + (at level 50, v ident, only parsing). + Notation "'Error : t : s" := (unify _ t (Some s)) + (at level 50, format "''Error' : t : s"). + Open Scope string_scope. +End infrastructure. +\end{coq_example} + +To explain the notation \texttt{[find v | t1 \textasciitilde t2]} let us pick one +of its instances: \texttt{[find e | EQ.obj e \textasciitilde T | "is not an EQ.type" ]}. +It should be read as: ``find a class e such that its objects have type T +or fail with message "T is not an EQ.type"''. + +The other utilities are used to ask Coq to solve a specific unification +problem, that will in turn require the inference of some canonical +structures. They are explained in mode details in~\cite{CSwcu}. + +We now have all we need to create a compact ``packager'' to declare +instances of the \texttt{LEQ} class. + +\begin{coq_example} +Import infrastructure. +Definition packager T e0 le0 (m0 : LEQ.mixin e0 le0) := + [find e | EQ.obj e ~ T | "is not an EQ.type" ] + [find o | LE.obj o ~ T | "is not an LE.type" ] + [find ce | EQ.class_of e ~ ce ] + [find co | LE.class_of o ~ co ] + [find m | m ~ m0 | "is not the right mixin" ] + LEQ._Pack T (LEQ.Class ce co m). +Notation Pack T m := (packager T _ _ m _ id _ id _ id _ id _ id). +\end{coq_example} + +The object \texttt{Pack} takes a type \texttt{T} (the key) and a mixin \texttt{m}. It infers all +the other pieces of the class \texttt{LEQ} and declares them as canonical values +associated to the \texttt{T} key. All in all, the only new piece of information +we add in the \texttt{LEQ} class is the mixin, all the rest is already canonical +for \texttt{T} and hence can be inferred by Coq. + +\texttt{Pack} is a notation, hence it is not type checked at the time of its +declaration. It will be type checked when it is used, an in that case +\texttt{T} is going to be a concrete type. The odd arguments \texttt{\_} and \texttt{id} we +pass to the +packager represent respectively the classes to be inferred (like \texttt{e}, \texttt{o}, etc) and a token (\texttt{id}) to force their inference. Again, for all the details the +reader can refer to~\cite{CSwcu}. + +The declaration of canonical instances can now be way more compact: + +\begin{coq_example} +Canonical Structure nat_LEQty := Eval hnf in Pack nat nat_LEQmx. +Canonical Structure pair_LEQty (l1 l2 : LEQ.type) := + Eval hnf in Pack (LEQ.obj l1 * LEQ.obj l2) (pair_LEQmx l1 l2). +\end{coq_example} + +Error messages are also quite intelligible (if one skips to the end of +the message). + +\begin{coq_example} +Fail Canonical Structure err := Eval hnf in Pack bool nat_LEQmx. +\end{coq_example} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "Reference-Manual" +%%% End: |