\[\begin{split}\newcommand{\alors}{\textsf{then}} \newcommand{\alter}{\textsf{alter}} \newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\bool}{\textsf{bool}} \newcommand{\case}{\kw{case}} \newcommand{\conc}{\textsf{conc}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\conshl}{\textsf{cons\_hl}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\EqSt}{\textsf{EqSt}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\false}{\textsf{false}} \newcommand{\filter}{\textsf{filter}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\from}{\textsf{from}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\haslength}{\textsf{has\_length}} \newcommand{\hd}{\textsf{hd}} \newcommand{\ident}{\textsf{ident}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\lb}{\lambda} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\Nat}{\mathbb{N}} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\true}{\textsf{true}} \newcommand{\Type}{\textsf{Type}} \newcommand{\unfold}{\textsf{unfold}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \newcommand{\zeros}{\textsf{zeros}} \end{split}\]

# Extended pattern matching

Authors

  Cristina Cornes and Hugo Herbelin

This section describes the full form of pattern matching in Coq terms.

## Patterns

The full syntax of match is presented in Figures 1.1 and 1.2. 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 `(pattern as ident)` are also accepted. This pattern matches the same values as `pattern` does and `ident` is bound to the matched value. A pattern of the form `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 `Printing Matching`), then by printing the term with `Print` if the term is a constant, or using the command `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:

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.
  max is defined max is recursively defined (decreasing on 2nd argument)

## Multiple patterns

Using multiple patterns in the definition of `max` lets us write:

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.
  max is defined max is recursively defined (decreasing on 2nd argument)

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.

Check (fun x:nat => match x return nat with | y => y end).
  fun x : nat => x : nat -> nat

## Aliasing subpatterns

We can also use `as ident` to associate a name to a sub-pattern:

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.
  max is defined max is recursively defined (decreasing on 1st argument)

## Nested patterns

Here is now an example of nested patterns:

Fixpoint even (n:nat) : bool := match n with | O => true | S O => false | S (S n') => even n' end.
  even is defined even is recursively defined (decreasing on 1st argument)

This is compiled into:

Unset Printing Matching.
Print even.
  even = fix even (n : nat) : bool := match n with | 0 => true | S n0 => match n0 with | 0 => false | S n' => even n' end end : nat -> bool Argument scope is [nat_scope]

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 `lef` that given two natural numbers yields `true` if the first one is less or equal than the second one and `false` otherwise. We can write it as follows:

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.
  lef is defined lef is recursively defined (decreasing on 2nd argument)

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 `(lef O O)` yields true.

Another way to write this function is:

Fixpoint lef (n m:nat) {struct m} : bool := match n, m with | O, x => true | S n, S m => lef n m | _, _ => false end.
  lef is defined lef is recursively defined (decreasing on 2nd argument)

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:

Fail Check (fun x:nat => match x with | O => true | S _ => false | x => true end).
  The command has indeed failed with message: Pattern "x" is redundant in this clause.

## Disjunctive patterns

Multiple patterns that share the same right-hand-side can be factorized using the notation `mult_pattern+|`. For instance, `max` can be rewritten as follows:

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.
  max is defined max is recursively defined (decreasing on 2nd argument)

Similarly, factorization of (not necessarily multiple) patterns that share the same variables is possible by using the notation `pattern+|`. Here is an example:

Definition filter_2_4 (n:nat) : nat := match n with | 2 as m | 4 as m => m | _ => 0 end.
  filter_2_4 is defined

Here is another example using disjunctive subpatterns.

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.
  filter_some_square_corners is defined

## 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:

Inductive List (A:Set) : Set := | nil : List A | cons : A -> List A -> List A.
  List is defined List_rect is defined List_ind is defined List_rec is defined

We can check the function tail:

Check (fun l:List nat => match l with | nil _ => nil nat | cons _ _ l' => l' end).
  fun l : List nat => match l with | nil _ => nil nat | cons _ _ l' => l' end : List nat -> List nat

When we use parameters in patterns there is an error message:

Fail Check (fun l:List nat => match l with | nil A => nil nat | cons A _ l' => l' end).
  The command has indeed failed with message: The parameters do not bind in patterns; they must be replaced by '_'.

Flag `Asymmetric Patterns`

  This flag (off by default) removes parameters from constructors in patterns:

Set Asymmetric Patterns.
Check (fun l:List nat => match l with | nil => nil _ | cons _ l' => l' end).
  fun l : List nat => match l with | @nil _ => nil nat | @cons _ _ l' => l' end : List nat -> List nat
Unset Asymmetric Patterns.

## Implicit arguments in patterns

By default, implicit arguments are omitted in patterns. So we write:

Arguments nil [A].
Arguments cons [A] _ _.
Check (fun l:List nat => match l with | nil => nil | cons _ l' => l' end).
  fun l : List nat => match l with | nil => nil | cons _ l' => l' end : List nat -> List nat

But the possibility to use all the arguments is given by “`@`” implicit explicitations (as for terms, see Explicit applications).

Check (fun l:List nat => match l with | @nil _ => @nil nat | @cons _ _ l' => l' end).
  fun l : List nat => match l with | nil => nil | cons _ l' => l' end : List nat -> List nat

## 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 `listn` of lists of a certain length:

Inductive listn : nat -> Set := | niln : listn 0 | consn : forall n:nat, nat -> listn n -> listn (S n).
  listn is defined listn_rect is defined listn_ind is defined listn_rec is defined

## Understanding dependencies in patterns

We can define the function length over `listn` by:

Definition length (n:nat) (l:listn n) := n.

Just for illustrating pattern matching, we can define it by case analysis:

Definition length (n:nat) (l:listn n) := match l with | niln => 0 | consn n _ _ => S n end.
  length is defined

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 right hand sides 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 right hand sides are different instances of the elimination predicate. The function `concat` for `listn` is an example where the branches have different types and we need to provide the elimination predicate:

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.
  concat is defined concat is recursively defined (decreasing on 2nd argument)

Reset concat.

The elimination predicate is `fun (n:nat) (l:listn n) => listn (n+m)`. In general if `m` has type `(I q1 … qr t1 … ts)` where `q1, …, qr` are parameters, the elimination predicate should be of the form `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 `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 `concat` (even though the matching on the second term is trivial) would have been:

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.
  concat is defined concat is recursively defined (decreasing on 2nd argument)

Even without real matching over the second term, this construction can be used to keep types linked. If `a` and `b` are two `listn` of the same length, by writing

Check (fun n (a b: listn n) => match a, b with | niln, b0 => tt | consn n' a y, bS => tt end).
  fun (n : nat) (a _ : listn n) => match a with | niln | _ => tt end : forall n : nat, listn n -> listn n -> unit

we have a copy of `b` in type `listn 0` resp. `listn (S n')`.

### 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:

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.
  tail is defined

and `tail n v` will be subterm of `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 `refine`.

For example, we can write the function `buildlist` that given a natural number `n` builds a list of length `n` containing zeros as follows:

Fixpoint buildlist (n:nat) : listn n := match n return listn n with | O => niln | S n => consn n 0 (buildlist n) end.
  buildlist is defined buildlist is recursively defined (decreasing on 1st argument)

We can also use multiple patterns. Consider the following definition of the predicate less-equal `Le`:

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).
  LE is defined LE_ind is defined

We can use multiple patterns to write the proof of the lemma `forall (n m:nat), (LE n m) \/ (LE m n)`:

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.
  dec is defined dec is recursively defined (decreasing on 1st argument)

In the example of `dec`, the first match is dependent while the second is not.

The user can also use match in combination with the tactic `refine` (see Section 8.2.3) to build incomplete proofs beginning with a 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

Inductive list : nat -> Set := | nil : list 0 | cons : forall n:nat, let m := (2 * n) in list m -> list (S (S m)).
  list is defined list_rect is defined list_ind is defined list_rec is defined

In the next example, the local definition is not caught.

Fixpoint length n (l:list n) {struct l} : nat := match l with | nil => 0 | cons n l0 => S (length (2 * n) l0) end.
  length is defined length is recursively defined (decreasing on 2nd argument)

But in this example, it is.

Fixpoint length' n (l:list n) {struct l} : nat := match l with | nil => 0 | @cons _ m l0 => S (length' m l0) end.
  length' is defined length' is recursively defined (decreasing on 2nd argument)

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

Inductive I : Set := | C1 : nat -> I | C2 : I -> I.
  I is defined I_rect is defined I_ind is defined I_rec is defined
Coercion C1 : nat >-> I.
  C1 is now a coercion

Check (fun x => match x with | C2 O => 0 | _ => 0 end).
  fun x : I => match x with | C1 _ | _ => 0 end : I -> nat

## 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:

Error `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).

Error `Non exhaustive pattern matching.`

  The pattern matching is not exhaustive.

Error `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.

Error `Unable to infer a match predicate`
Error `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.