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-.. _extendedpatternmatching:
-
-Extended pattern matching
-=========================
-
-:Authors: Cristina Cornes and Hugo Herbelin
-
-This section describes the full form of pattern matching in |Coq| terms.
-
-.. |rhs| replace:: right hand sides
-
-Patterns
---------
-
-The full syntax of :g:`match` is presented in section :ref:`term`.
-Identifiers in patterns are either constructor names or variables. Any
-identifier that is not the constructor of an inductive or co-inductive
-type is considered to be a variable. A variable name cannot occur more
-than once in a given pattern. It is recommended to start variable
-names by a lowercase letter.
-
-If a pattern has the form ``c x`` where ``c`` is a constructor symbol and x
-is a linear vector of (distinct) variables, it is called *simple*: it
-is the kind of pattern recognized by the basic version of match. On
-the opposite, if it is a variable ``x`` or has the form ``c p`` with ``p`` not
-only made of variables, the pattern is called *nested*.
-
-A variable pattern matches any value, and the identifier is bound to
-that value. The pattern “``_``” (called “don't care” or “wildcard” symbol)
-also matches any value, but does not bind anything. It may occur an
-arbitrary number of times in a pattern. Alias patterns written
-:n:`(@pattern as @ident)` are also accepted. This pattern matches the
-same values as :token:`pattern` does and :token:`ident` is bound to the matched
-value. A pattern of the form :n:`@pattern | @pattern` is called disjunctive. A
-list of patterns separated with commas is also considered as a pattern
-and is called *multiple pattern*. However multiple patterns can only
-occur at the root of pattern matching equations. Disjunctions of
-*multiple patterns* are allowed though.
-
-Since extended ``match`` expressions are compiled into the primitive ones,
-the expressiveness of the theory remains the same. Once parsing has finished
-only simple patterns remain. The original nesting of the ``match`` expressions
-is recovered at printing time. An easy way to see the result
-of the expansion is to toggle off the nesting performed at printing
-(use here :flag:`Printing Matching`), then by printing the term with :cmd:`Print`
-if the term is a constant, or using the command :cmd:`Check`.
-
-The extended ``match`` still accepts an optional *elimination predicate*
-given after the keyword ``return``. Given a pattern matching expression,
-if all the right-hand-sides of ``=>`` have the same
-type, then this type can be sometimes synthesized, and so we can omit
-the return part. Otherwise the predicate after return has to be
-provided, like for the basicmatch.
-
-Let us illustrate through examples the different aspects of extended
-pattern matching. Consider for example the function that computes the
-maximum of two natural numbers. We can write it in primitive syntax
-by:
-
-.. coqtop:: in
-
-   Fixpoint max (n m:nat) {struct m} : nat :=
-     match n with
-     | O => m
-     | S n' => match m with
-               | O => S n'
-               | S m' => S (max n' m')
-               end
-     end.
-
-Multiple patterns
------------------
-
-Using multiple patterns in the definition of ``max`` lets us write:
-
-.. coqtop:: in reset
-
-   Fixpoint max (n m:nat) {struct m} : nat :=
-       match n, m with
-       | O, _ => m
-       | S n', O => S n'
-       | S n', S m' => S (max n' m')
-       end.
-
-which will be compiled into the previous form.
-
-The pattern matching compilation strategy examines patterns from left
-to right. A match expression is generated **only** when there is at least
-one constructor in the column of patterns. E.g. the following example
-does not build a match expression.
-
-.. coqtop:: all
-
-   Check (fun x:nat => match x return nat with
-                       | y => y
-                       end).
-
-
-Aliasing subpatterns
---------------------
-
-We can also use :n:`as @ident` to associate a name to a sub-pattern:
-
-.. coqtop:: in reset
-
-   Fixpoint max (n m:nat) {struct n} : nat :=
-     match n, m with
-     | O, _ => m
-     | S n' as p, O => p
-     | S n', S m' => S (max n' m')
-     end.
-
-Nested patterns
----------------
-
-Here is now an example of nested patterns:
-
-.. coqtop:: in
-
-   Fixpoint even (n:nat) : bool :=
-     match n with
-     | O => true
-     | S O => false
-     | S (S n') => even n'
-     end.
-
-This is compiled into:
-
-.. coqtop:: all
-
-   Unset Printing Matching.
-   Print even.
-
-.. coqtop:: none
-
-   Set Printing Matching.
-
-In the previous examples patterns do not conflict with, but sometimes
-it is comfortable to write patterns that admit a non trivial
-superposition. Consider the boolean function :g:`lef` that given two
-natural numbers yields :g:`true` if the first one is less or equal than the
-second one and :g:`false` otherwise. We can write it as follows:
-
-.. coqtop:: in
-
-   Fixpoint lef (n m:nat) {struct m} : bool :=
-     match n, m with
-     | O, x => true
-     | x, O => false
-     | S n, S m => lef n m
-     end.
-
-Note that the first and the second multiple pattern overlap because
-the couple of values ``O O`` matches both. Thus, what is the result of the
-function on those values? To eliminate ambiguity we use the *textual
-priority rule:* we consider patterns to be ordered from top to bottom. A
-value is matched by the pattern at the ith row if and only if it is
-not matched by some pattern from a previous row. Thus in the example, ``O O``
-is matched by the first pattern, and so :g:`(lef O O)` yields true.
-
-Another way to write this function is:
-
-.. coqtop:: in reset
-
-   Fixpoint lef (n m:nat) {struct m} : bool :=
-     match n, m with
-     | O, x => true
-     | S n, S m => lef n m
-     | _, _ => false
-     end.
-
-Here the last pattern superposes with the first two. Because of the
-priority rule, the last pattern will be used only for values that do
-not match neither the first nor the second one.
-
-Terms with useless patterns are not accepted by the system. Here is an
-example:
-
-.. coqtop:: all
-
-   Fail Check (fun x:nat =>
-                 match x with
-                 | O => true
-                 | S _ => false
-                 | x => true
-                 end).
-
-
-Disjunctive patterns
---------------------
-
-Multiple patterns that share the same right-hand-side can be
-factorized using the notation :n:`{+| {+, @pattern } }`. For
-instance, :g:`max` can be rewritten as follows:
-
-.. coqtop:: in reset
-
-   Fixpoint max (n m:nat) {struct m} : nat :=
-     match n, m with
-     | S n', S m' => S (max n' m')
-     | 0, p | p, 0 => p
-     end.
-
-Similarly, factorization of (not necessarily multiple) patterns that
-share the same variables is possible by using the notation :n:`{+| @pattern}`.
-Here is an example:
-
-.. coqtop:: in
-
-   Definition filter_2_4 (n:nat) : nat :=
-     match n with
-     | 2 as m | 4 as m => m
-     | _ => 0
-     end.
-
-
-Nested disjunctive patterns are allowed, inside parentheses, with the
-notation :n:`({+| @pattern})`, as in:
-
-.. coqtop:: in
-
-   Definition filter_some_square_corners (p:nat*nat) : nat*nat :=
-     match p with
-     | ((2 as m | 4 as m), (3 as n | 5 as n)) => (m,n)
-     | _ => (0,0)
-     end.
-
-About patterns of parametric types
-----------------------------------
-
-Parameters in patterns
-~~~~~~~~~~~~~~~~~~~~~~
-
-When matching objects of a parametric type, parameters do not bind in
-patterns. They must be substituted by “``_``”. Consider for example the
-type of polymorphic lists:
-
-.. coqtop:: in
-
-   Inductive List (A:Set) : Set :=
-   | nil : List A
-   | cons : A -> List A -> List A.
-
-We can check the function *tail*:
-
-.. coqtop:: all
-
-   Check
-     (fun l:List nat =>
-        match l with
-        | nil _ => nil nat
-        | cons _ _ l' => l'
-        end).
-
-When we use parameters in patterns there is an error message:
-
-.. coqtop:: all
-
-   Fail Check
-     (fun l:List nat =>
-        match l with
-        | nil A => nil nat
-        | cons A _ l' => l'
-        end).
-
-.. flag:: Asymmetric Patterns
-
-   This flag (off by default) removes parameters from constructors in patterns:
-
-.. coqtop:: all
-
-   Set Asymmetric Patterns.
-   Check (fun l:List nat =>
-     match l with
-     | nil => nil _
-     | cons _ l' => l'
-     end).
-   Unset Asymmetric Patterns.
-
-Implicit arguments in patterns
-------------------------------
-
-By default, implicit arguments are omitted in patterns. So we write:
-
-.. coqtop:: all
-
-   Arguments nil {A}.
-   Arguments cons [A] _ _.
-   Check
-     (fun l:List nat =>
-        match l with
-        | nil => nil
-        | cons _ l' => l'
-        end).
-
-But the possibility to use all the arguments is given by “``@``” implicit
-explicitations (as for terms, see :ref:`explicit-applications`).
-
-.. coqtop:: all
-
-   Check
-     (fun l:List nat =>
-        match l with
-        | @nil _ => @nil nat
-        | @cons _ _ l' => l'
-        end).
-
-
-.. _matching-dependent:
-
-Matching objects of dependent types
------------------------------------
-
-The previous examples illustrate pattern matching on objects of non-
-dependent types, but we can also use the expansion strategy to
-destructure objects of dependent types. Consider the type :g:`listn` of
-lists of a certain length:
-
-.. coqtop:: in reset
-
-   Inductive listn : nat -> Set :=
-   | niln : listn 0
-   | consn : forall n:nat, nat -> listn n -> listn (S n).
-
-
-Understanding dependencies in patterns
---------------------------------------
-
-We can define the function length over :g:`listn` by:
-
-.. coqdoc::
-
-   Definition length (n:nat) (l:listn n) := n.
-
-Just for illustrating pattern matching, we can define it by case
-analysis:
-
-.. coqtop:: in
-
-   Definition length (n:nat) (l:listn n) :=
-     match l with
-     | niln => 0
-     | consn n _ _ => S n
-     end.
-
-We can understand the meaning of this definition using the same
-notions of usual pattern matching.
-
-
-When the elimination predicate must be provided
------------------------------------------------
-
-Dependent pattern matching
-~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-The examples given so far do not need an explicit elimination
-predicate because all the |rhs| have the same type and Coq
-succeeds to synthesize it. Unfortunately when dealing with dependent
-patterns it often happens that we need to write cases where the types
-of the |rhs| are different instances of the elimination predicate. The
-function :g:`concat` for :g:`listn` is an example where the branches have
-different types and we need to provide the elimination predicate:
-
-.. coqtop:: in
-
-   Fixpoint concat (n:nat) (l:listn n) (m:nat) (l':listn m) {struct l} :
-    listn (n + m) :=
-     match l in listn n return listn (n + m) with
-     | niln => l'
-     | consn n' a y => consn (n' + m) a (concat n' y m l')
-     end.
-
-.. coqtop:: none
-
-   Reset concat.
-
-The elimination predicate is :g:`fun (n:nat) (l:listn n) => listn (n+m)`.
-In general if :g:`m` has type :g:`(I q1 … qr t1 … ts)` where :g:`q1, …, qr`
-are parameters, the elimination predicate should be of the form :g:`fun y1 … ys x : (I q1 … qr y1 … ys ) => Q`.
-
-In the concrete syntax, it should be written :
-``match m as x in (I _ … _ y1 … ys) return Q with … end``.
-The variables which appear in the ``in`` and ``as`` clause are new and bounded
-in the property :g:`Q` in the return clause. The parameters of the
-inductive definitions should not be mentioned and are replaced by ``_``.
-
-Multiple dependent pattern matching
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-Recall that a list of patterns is also a pattern. So, when we
-destructure several terms at the same time and the branches have
-different types we need to provide the elimination predicate for this
-multiple pattern. It is done using the same scheme: each term may be
-associated to an ``as`` clause and an ``in`` clause in order to introduce
-a dependent product.
-
-For example, an equivalent definition for :g:`concat` (even though the
-matching on the second term is trivial) would have been:
-
-.. coqtop:: in
-
-   Fixpoint concat (n:nat) (l:listn n) (m:nat) (l':listn m) {struct l} :
-    listn (n + m) :=
-     match l in listn n, l' return listn (n + m) with
-     | niln, x => x
-     | consn n' a y, x => consn (n' + m) a (concat n' y m x)
-     end.
-
-Even without real matching over the second term, this construction can
-be used to keep types linked. If :g:`a` and :g:`b` are two :g:`listn` of the same
-length, by writing
-
-.. coqtop:: in
-
-   Check (fun n (a b: listn n) =>
-    match a, b with
-    | niln, b0 => tt
-    | consn n' a y, bS => tt
-    end).
-
-we have a copy of :g:`b` in type :g:`listn 0` resp. :g:`listn (S n')`.
-
-.. _match-in-patterns:
-
-Patterns in ``in``
-~~~~~~~~~~~~~~~~~~
-
-If the type of the matched term is more precise than an inductive
-applied to variables, arguments of the inductive in the ``in`` branch can
-be more complicated patterns than a variable.
-
-Moreover, constructors whose types do not follow the same pattern will
-become impossible branches. In an impossible branch, you can answer
-anything but False_rect unit has the advantage to be subterm of
-anything.
-
-To be concrete: the ``tail`` function can be written:
-
-.. coqtop:: in
-
-   Definition tail n (v: listn (S n)) :=
-     match v in listn (S m) return listn m with
-     | niln => False_rect unit
-     | consn n' a y => y
-     end.
-
-and :g:`tail n v` will be subterm of :g:`v`.
-
-Using pattern matching to write proofs
---------------------------------------
-
-In all the previous examples the elimination predicate does not depend
-on the object(s) matched. But it may depend and the typical case is
-when we write a proof by induction or a function that yields an object
-of a dependent type. An example of a proof written using ``match`` is given
-in the description of the tactic :tacn:`refine`.
-
-For example, we can write the function :g:`buildlist` that given a natural
-number :g:`n` builds a list of length :g:`n` containing zeros as follows:
-
-.. coqtop:: in
-
-   Fixpoint buildlist (n:nat) : listn n :=
-     match n return listn n with
-     | O => niln
-     | S n => consn n 0 (buildlist n)
-     end.
-
-We can also use multiple patterns. Consider the following definition
-of the predicate less-equal :g:`Le`:
-
-.. coqtop:: in
-
-   Inductive LE : nat -> nat -> Prop :=
-     | LEO : forall n:nat, LE 0 n
-     | LES : forall n m:nat, LE n m -> LE (S n) (S m).
-
-We can use multiple patterns to write the proof of the lemma
-:g:`forall (n m:nat), (LE n m) \/ (LE m n)`:
-
-.. coqtop:: in
-
-   Fixpoint dec (n m:nat) {struct n} : LE n m \/ LE m n :=
-     match n, m return LE n m \/ LE m n with
-     | O, x => or_introl (LE x 0) (LEO x)
-     | x, O => or_intror (LE x 0) (LEO x)
-     | S n as n', S m as m' =>
-         match dec n m with
-         | or_introl h => or_introl (LE m' n') (LES n m h)
-         | or_intror h => or_intror (LE n' m') (LES m n h)
-         end
-     end.
-
-In the example of :g:`dec`, the first match is dependent while the second
-is not.
-
-The user can also use match in combination with the tactic :tacn:`refine`
-to build incomplete proofs beginning with a :g:`match` construction.
-
-
-Pattern-matching on inductive objects involving local definitions
------------------------------------------------------------------
-
-If local definitions occur in the type of a constructor, then there
-are two ways to match on this constructor. Either the local
-definitions are skipped and matching is done only on the true
-arguments of the constructors, or the bindings for local definitions
-can also be caught in the matching.
-
-.. example::
-
-   .. coqtop:: in reset
-
-      Inductive list : nat -> Set :=
-      | nil : list 0
-      | cons : forall n:nat, let m := (2 * n) in list m -> list (S (S m)).
-
-   In the next example, the local definition is not caught.
-
-   .. coqtop:: in
-
-      Fixpoint length n (l:list n) {struct l} : nat :=
-        match l with
-        | nil => 0
-        | cons n l0 => S (length (2 * n) l0)
-        end.
-
-   But in this example, it is.
-
-   .. coqtop:: in
-
-      Fixpoint length' n (l:list n) {struct l} : nat :=
-        match l with
-        | nil => 0
-        | @cons _ m l0 => S (length' m l0)
-        end.
-
-.. note:: For a given matching clause, either none of the local
-          definitions or all of them can be caught.
-
-.. note:: You can only catch let bindings in mode where you bind all
-          variables and so you have to use ``@`` syntax.
-
-.. note:: this feature is incoherent with the fact that parameters
-          cannot be caught and consequently is somehow hidden. For example,
-          there is no mention of it in error messages.
-
-Pattern-matching and coercions
-------------------------------
-
-If a mismatch occurs between the expected type of a pattern and its
-actual type, a coercion made from constructors is sought. If such a
-coercion can be found, it is automatically inserted around the
-pattern.
-
-.. example::
-
-   .. coqtop:: in
-
-      Inductive I : Set :=
-        | C1 : nat -> I
-        | C2 : I -> I.
-
-      Coercion C1 : nat >-> I.
-
-   .. coqtop:: all
-
-      Check (fun x => match x with
-                      | C2 O => 0
-                      | _ => 0
-                      end).
-
-
-When does the expansion strategy fail?
---------------------------------------
-
-The strategy works very like in ML languages when treating patterns of
-non-dependent types. But there are new cases of failure that are due to
-the presence of dependencies.
-
-The error messages of the current implementation may be sometimes
-confusing. When the tactic fails because patterns are somehow
-incorrect then error messages refer to the initial expression. But the
-strategy may succeed to build an expression whose sub-expressions are
-well typed when the whole expression is not. In this situation the
-message makes reference to the expanded expression. We encourage
-users, when they have patterns with the same outer constructor in
-different equations, to name the variable patterns in the same
-positions with the same name. E.g. to write ``(cons n O x) => e1`` and
-``(cons n _ x) => e2`` instead of ``(cons n O x) => e1`` and
-``(cons n' _ x') => e2``. This helps to maintain certain name correspondence between the
-generated expression and the original.
-
-Here is a summary of the error messages corresponding to each
-situation:
-
-.. exn:: The constructor @ident expects @num arguments.
-
-   The variable ident is bound several times in pattern termFound a constructor
-   of inductive type term while a constructor of term is expectedPatterns are
-   incorrect (because constructors are not applied to the correct number of the
-   arguments, because they are not linear or they are wrongly typed).
-
-.. exn:: Non exhaustive pattern matching.
-
-   The pattern matching is not exhaustive.
-
-.. exn:: The elimination predicate term should be of arity @num (for non \
-         dependent case) or @num (for dependent case).
-
-   The elimination predicate provided to match has not the expected arity.
-
-.. exn:: Unable to infer a match predicate
-         Either there is a type incompatibility or the problem involves dependencies.
-
-   There is a type mismatch between the different branches. The user should
-   provide an elimination predicate.