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diff --git a/doc/sphinx/addendum/canonical-structures.rst b/doc/sphinx/addendum/canonical-structures.rst index 8961b0096..6843e9eaa 100644 --- a/doc/sphinx/addendum/canonical-structures.rst +++ b/doc/sphinx/addendum/canonical-structures.rst @@ -1,383 +1,435 @@ -\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 +.. include:: ../replaces.rst +.. _canonicalstructures: + +Canonical Structures +====================== + +:Authors: Assia Mahboubi and Enrico Tassi + +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. + + +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. + +.. coqtop:: all + + 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. + +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 ``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 theory module contains the overloaded notation ``==`` +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 +Note that in practice the user may want to declare ``EQ.obj`` as a +coercion, but we will not do that here. + +The following line tests that, when we assume a type ``e`` that is in +theEQ class, then we can relates two of its objects with ``==``. + +.. coqtop:: all + + Import EQ.theory. + Check forall (e : EQ.type) (a b : EQ.obj e), a == b. + +Still, no concrete type is in the ``EQ`` class. + +.. coqtop:: all + + Fail Check 3 == 3. + +We amend that by equipping ``nat`` with a comparison relation. + +.. coqtop:: all + + 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. + +This last test shows that |Coq| is now not only able to typecheck ``3 == 3``, +but also that the infix relation was bound to the ``nat_eq`` relation. +This relation is selected whenever ``==`` is used on terms of type nat. +This can be read in the line declaring the canonical structure +``nat_EQty``, where the first argument to ``Pack`` is the key and its second +argument a group of canonical values associated to the key. In this +case we associate to nat only one canonical value (since its class, +``nat_EQcl`` has just one member). The use of the projection ``op`` requires +its argument to be in the class ``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 ``==`` notation on terms of this type. + + +Derived Canonical Structures +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We know how to use ``== `` on base types, like ``nat``, ``bool``, ``Z``. Here we show +how to deal with type constructors, i.e. how to make the following +example work: + + +.. coqtop:: all + + Fail Check forall (e : EQ.type) (a b : EQ.obj e), (a, b) == (a, b). + +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. + +.. coqtop:: all + + 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). + +Thanks to the ``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 +for each component of the pair. The declaration associates to the key ``*`` +(the type constructor of pairs) the canonical comparison +relation ``pair_eq`` whenever the type constructor ``*`` is applied to two +types being themselves in the ``EQ`` class. + +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 ``<=`` notation. + +.. coqtop:: all + + 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. + +As before we register a canonical ``LE`` class for ``nat``. + +.. coqtop:: all + + 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. + +And we enable |Coq| to relate pair of terms with ``<=``. + +.. coqtop:: all + + 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). + +At the current stage we can use ``==`` and ``<=`` 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. + +.. coqtop:: all + + 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. + +We need to define a new class that inherits from both ``EQ`` and ``LE``. + + +.. coqtop:: all + + 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} _ _ _. + +The mixin component of the ``LEQ`` class contains all the extra content we +are adding to ``EQ`` and ``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} +Unfortunately there is still an obstacle to developing the algebraic +theory of this new class. + +.. coqtop:: all + + Module theory. + + Fail Check forall (le : type) (n m : obj le), n <= m -> n <= m -> n == m. + + +The problem is that the two classes ``LE`` and ``LEQ`` are not yet related by +a subclass relation. In other words |Coq| does not see that an object of +the ``LEQ`` class is also an object of the ``LE`` class. + +The following two constructions tell |Coq| how to canonically build the +``LE.type`` and ``EQ.type`` structure given an ``LEQ.type`` structure on the same +type. + +.. coqtop:: all + + 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. + +We can now formulate out first theorem on the objects of the ``LEQ`` +structure. + +.. coqtop:: all + + 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. 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. +.. coqtop:: all -\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. + 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. + +Again one has to tell |Coq| that the type ``nat`` is in the ``LEQ`` class, and +how the type constructor ``*`` interacts with the ``LEQ`` class. In the +following proofs are omitted for brevity. + +.. coqtop:: all + + Lemma nat_LEQ_compat (n m : nat) : n <= m /\ m <= n <-> n == m. + + Admitted. + + 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. + + Admitted. + + Definition pair_LEQmx l1 l2 := LEQ.Mixin (pair_LEQ_compat l1 l2). + +The following script registers an ``LEQ`` class for ``nat`` and for the type +constructor ``*``. 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. -\subsection{Compact declaration of Canonical Structures} +.. coqtop:: all + + 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. + +Note that no direct proof of ``n <= m -> m <= n -> n == m`` is provided by +the user for ``n`` and m of type ``nat * nat``. What the user provides is a +proof of this statement for ``n`` and ``m`` of type ``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. + +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" ] +.. coqtop:: all + + Require Import Strings.String. + + 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. + +To explain the notation ``[find v | t1 ~ t2]`` let us pick one of its +instances: ``[find e | EQ.obj e ~ 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 ``LEQ`` class. + +.. coqtop:: all + + 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}. + [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). + +The object ``Pack`` takes a type ``T`` (the key) and a mixin ``m``. It infers all +the other pieces of the class ``LEQ`` and declares them as canonical +values associated to the ``T`` key. All in all, the only new piece of +information we add in the ``LEQ`` class is the mixin, all the rest is +already canonical for ``T`` and hence can be inferred by |Coq|. + +``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 ``T`` is +going to be a concrete type. The odd arguments ``_`` and ``id`` we pass to the +packager represent respectively the classes to be inferred (like ``e``, ``o``, +etc) and a token (``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} +.. coqtop:: all + + 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). 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} +.. coqtop:: all + + Fail Canonical Structure err := Eval hnf in Pack bool nat_LEQmx. -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: |