The SSReflect proof language¶
 Authors
Georges Gonthier, Assia Mahboubi, Enrico Tassi
Introduction¶
This chapter describes a set of tactics known as SSReflect originally designed to provide support for the socalled small scale reflection proof methodology. Despite the original purpose, this set of tactics is of general interest and is available in Coq starting from version 8.7.
SSReflect was developed independently of the tactics described in Chapter Tactics. Indeed the scope of the tactics part of SSReflect largely overlaps with the standard set of tactics. Eventually the overlap will be reduced in future releases of Coq.
Proofs written in SSReflect typically look quite different from the ones written using only tactics as per Chapter Tactics. We try to summarise here the most “visible” ones in order to help the reader already accustomed to the tactics described in Chapter Tactics to read this chapter.
The first difference between the tactics described in this chapter and the
tactics described in Chapter Tactics is the way hypotheses are managed
(we call this bookkeeping). In Chapter Tactics the most common
approach is to avoid moving explicitly hypotheses back and forth between the
context and the conclusion of the goal. On the contrary, in SSReflect all
bookkeeping is performed on the conclusion of the goal, using for that
purpose a couple of syntactic constructions behaving similar to tacticals
(and often named as such in this chapter). The :
tactical moves hypotheses
from the context to the conclusion, while =>
moves hypotheses from the
conclusion to the context, and in
moves back and forth a hypothesis from the
context to the conclusion for the time of applying an action to it.
While naming hypotheses is commonly done by means of an as
clause in the
basic model of Chapter Tactics, it is here to =>
that this task is
devoted. Tactics frequently leave new assumptions in the conclusion, and are
often followed by =>
to explicitly name them. While generalizing the
goal is normally not explicitly needed in Chapter Tactics, it is an
explicit operation performed by :
.
See also
Besides the difference of bookkeeping model, this chapter includes
specific tactics that have no explicit counterpart in Chapter Tactics
such as tactics to mix forward steps and generalizations as
generally have
or without loss
.
SSReflect adopts the point of view that rewriting, definition
expansion and partial evaluation participate all to a same concept of
rewriting a goal in a larger sense. As such, all these functionalities
are provided by the rewrite
tactic.
SSReflect includes a little language of patterns to select subterms in
tactics or tacticals where it matters. Its most notable application is
in the rewrite
tactic, where patterns are
used to specify where the rewriting step has to take place.
Finally, SSReflect supports socalled reflection steps, typically allowing to switch back and forth between the computational view and logical view of a concept.
To conclude, it is worth mentioning that SSReflect tactics can be mixed with nonSSReflect tactics in the same proof, or in the same Ltac expression. The few exceptions to this statement are described in section Compatibility issues.
Acknowledgments¶
The authors would like to thank Frédéric Blanqui, François Pottier and Laurence Rideau for their comments and suggestions.
Usage¶
Getting started¶
To be available, the tactics presented in this manual need the
following minimal set of libraries to be loaded: ssreflect.v
,
ssrfun.v
and ssrbool.v
.
Moreover, these tactics come with a methodology
specific to the authors of SSReflect and which requires a few options
to be set in a different way than in their default way. All in all,
this corresponds to working in the following context:
 From Coq Require Import ssreflect ssrfun ssrbool.
 [Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done]
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
Compatibility issues¶
Requiring the above modules creates an environment that is mostly compatible with the rest of Coq, up to a few discrepancies.
New keywords (
is
) might clash with variable, constant, tactic or tactical names, or with quasikeywords in tactic or notation commands.New tactic(al)s names (
last
,done
,have
,suffices
,suff
,without loss
,wlog
,congr
,unlock
) might clash with user tactic names.Identifiers with both leading and trailing
_
, such as_x_
, are reserved by SSReflect and cannot appear in scripts.The extensions to the
rewrite
tactic are partly incompatible with those available in current versions of Coq; in particular,rewrite .. in (type of k)
orrewrite .. in *
or any other variant ofrewrite
will not work, and the SSReflect syntax and semantics for occurrence selection and rule chaining are different. Use an explicit rewrite direction (rewrite < …
orrewrite > …
) to access the Coq rewrite tactic.New symbols (
//
,/=
,//=
) might clash with adjacent existing symbols. This can be avoided by inserting white spaces.New constant and theorem names might clash with the user theory. This can be avoided by not importing all of SSReflect:
 From Coq Require ssreflect.
 Import ssreflect.SsrSyntax.
Note that the full syntax of SSReflect’s rewrite and reserved identifiers are enabled only if the ssreflect module has been required and if
SsrSyntax
has been imported. Thus a file that requires (without importing)ssreflect
and importsSsrSyntax
can be required and imported without automatically enabling SSReflect’s extended rewrite syntax and reserved identifiers.Some user notations (in particular, defining an infix
;
) might interfere with the "open term", parenthesisfree syntax of tactics such ashave
,set (ssreflect)
andpose (ssreflect)
.The generalization of
if
statements to nonBoolean conditions is turned off by SSReflect, because it is mostly subsumed by Coercion tobool
of thesumXXX
types (declared inssrfun.v
) and theif term is pattern then term else term
construct (see Pattern conditional). To use the generalized form, turn off the SSReflect Booleanif
notation using the command:Close Scope boolean_if_scope
.The following flags can be unset to make SSReflect more compatible with parts of Coq.
 Flag SsrRewrite¶
Controls whether the incompatible rewrite syntax is enabled (the default). Disabling the flag makes the syntax compatible with other parts of Coq.
 Flag SsrIdents¶
Controls whether tactics can refer to SSReflectgenerated variables that are in the form _xxx_. Scripts with explicit references to such variables are fragile; they are prone to failure if the proof is later modified or if the details of variable name generation change in future releases of Coq.
The default is on, which gives an error message when the user tries to create such identifiers. Disabling the flag generates a warning instead, increasing compatibility with other parts of Coq.
Gallina extensions¶
Smallscale reflection makes an extensive use of the programming subset of Gallina, Coq’s logical specification language. This subset is quite suited to the description of functions on representations, because it closely follows the wellestablished design of the ML programming language. The SSReflect extension provides three additions to Gallina, for pattern assignment, pattern testing, and polymorphism; these mitigate minor but annoying discrepancies between Gallina and ML.
Pattern assignment¶
The SSReflect extension provides the following construct for irrefutable pattern matching, that is, destructuring assignment:
term+=
let: pattern := term in term
Note the colon :
after the let
keyword, which avoids any ambiguity
with a function definition or Coq’s basic destructuring let. The let:
construct differs from the latter as follows.
The pattern can be nested (deep pattern matching); in particular, this allows expression of the form:
The destructured constructor is explicitly given in the pattern, and is used for type inference.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Definition f u := let: (m, n) := u in m + n.
 f is defined
 Check f.
 f : nat * nat > nat
Using
let:
, Coq infers a type forf
, whereas with a usuallet
the same term requires an extra type annotation in order to type check. Fail Definition f u := let (m, n) := u in m + n.
 The command has indeed failed with message: Cannot infer a type for this expression.
The let:
construct is just (more legible) notation for the primitive
Gallina expression match term with pattern => term end
.
The SSReflect destructuring assignment supports all the dependent match annotations; the full syntax is
term+=
let: pattern as ident? in pattern? := term return term? in term
where the second pattern
and the second term
are types.
When the as
and return
keywords are both present, then ident
is bound
in both the second pattern
and the second term
; variables
in the optional type pattern
are bound only in the second term, and
other variables in the first pattern
are bound only in the third
term
, however.
Pattern conditional¶
The following construct can be used for a refutable pattern matching, that is, pattern testing:
term+=
if term is pattern then term else term
Although this construct is not strictly ML (it does exist in variants such as the pattern calculus or the ρcalculus), it turns out to be very convenient for writing functions on representations, because most such functions manipulate simple data types such as Peano integers, options, lists, or binary trees, and the pattern conditional above is almost always the right construct for analyzing such simple types. For example, the null and all list function(al)s can be defined as follows:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable d: Set.
 d is declared
 Definition null (s : list d) := if s is nil then true else false.
 null is defined
 Variable a : d > bool.
 a is declared
 Fixpoint all (s : list d) : bool := if s is cons x s' then a x && all s' else true.
 all is defined all is recursively defined (guarded on 1st argument)
The pattern conditional also provides a notation for destructuring
assignment with a refutable pattern, adapted to the pure functional
setting of Gallina, which lacks a Match_Failure
exception.
Like let:
above, the if…is
construct is just (more legible) notation
for the primitive Gallina expression
match term with pattern => term  _ => term end
.
Similarly, it will always be displayed as the expansion of this form in terms of primitive match expressions (where the default expression may be replicated).
Explicit pattern testing also largely subsumes the generalization of
the if
construct to all binary data types; compare
if term is inl _ then term else term
and
if term then term else term
.
The latter appears to be marginally shorter, but it is quite
ambiguous, and indeed often requires an explicit annotation
(term : {_} + {_})
to type check, which evens the character count.
Therefore, SSReflect restricts by default the condition of a plain if
construct to the standard bool
type; this avoids spurious type
annotations.
Example
 Definition orb b1 b2 := if b1 then true else b2.
 orb is defined
As pointed out in Section Compatibility issues, this restriction can be removed with the command:
Close Scope boolean_if_scope.
Like let:
above, the ifisthenelse
construct supports
the dependent match annotations:
+=
if term is pattern as ident in pattern return term then term else term
As in let:
, the variable ident
(and those in the type pattern)
are bound in the second term
; ident
is also bound in the
third term
(but not in the fourth term
), while the
variables in the first pattern
are bound only in the third
term
.
Another variant allows to treat the else
case first:
+=
if term isn't pattern then term else term
Note that pattern
eventually binds variables in the third
term
and not in the second term
.
Parametric polymorphism¶
Unlike ML, polymorphism in core Gallina is explicit: the type parameters of polymorphic functions must be declared explicitly, and supplied at each point of use. However, Coq provides two features to suppress redundant parameters.
Sections are used to provide (possibly implicit) parameters for a set of definitions.
Implicit arguments declarations are used to tell Coq to use type inference to deduce some parameters from the context at each point of call.
The combination of these features provides a fairly good emulation of MLstyle polymorphism, but unfortunately this emulation breaks down for higherorder programming. Implicit arguments are indeed not inferred at all points of use, but only at points of call, leading to expressions such as
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable T : Type.
 T is declared
 Variable null : forall T : Type, T > bool.
 null is declared
 Variable all : (T > bool) > list T > bool.
 all is declared
 Definition all_null (s : list T) := all (@null T) s.
 all_null is defined
Unfortunately, such higherorder expressions are quite frequent in
representation functions, especially those that use Coq's
Structures
to emulate Haskell typeclasses.
Therefore, SSReflect provides a variant of Coq’s implicit argument declaration, which causes Coq to fill in some implicit parameters at each point of use; e.g., the above definition can be written:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable T : Type.
 T is declared
 Variable null : forall T : Type, T > bool.
 null is declared
 Variable all : (T > bool) > list T > bool.
 all is declared
 Prenex Implicits null.
 Definition all_null (s : list T) := all null s.
 all_null is defined
Better yet, it can be omitted entirely, since all_null s
isn’t much of
an improvement over all null s
.
The syntax of the new declaration is
 Command Prenex Implicits ident_{i}+¶
This command checks that each
ident_{i}
is the name of a functional constant, whose implicit arguments are prenex, i.e., the first \(n_i > 0\) arguments ofident_{i}
are implicit; then it assignsMaximal Implicit
status to these arguments.As these prenex implicit arguments are ubiquitous and have often large display strings, it is strongly recommended to change the default display settings of Coq so that they are not printed (except after a
Set Printing All
command). All SSReflect library files thus start with the incantationSet Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive.
Anonymous arguments¶
When in a definition, the type of a certain argument is mandatory, but
not its name, one usually uses “arrow” abstractions for prenex
arguments, or the (_ : term)
syntax for inner arguments. In SSReflect,
the latter can be replaced by the open syntax of term
or
(equivalently) & term
, which are both syntactically equivalent to a
(_ : term)
expression. This feature almost behaves as the
following extension of the binder syntax:
+=
& termof term
Caveat: & T
and of T
abbreviations have to appear at the end
of a binder list. For instance, the usual twoconstructor polymorphic
type list, i.e., the one of the standard List
library, can be
defined by the following declaration:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Inductive list (A : Type) : Type := nil  cons of A & list A.
 list is defined list_rect is defined list_ind is defined list_rec is defined list_sind is defined
Wildcards¶
The terms passed as arguments to SSReflect tactics can contain
holes, materialized by wildcards _
. Since SSReflect allows a more
powerful form of type inference for these arguments, it enhances the
possibilities of using such wildcards. These holes are in particular
used as a convenient shorthand for abstractions, especially in local
definitions or type expressions.
Wildcards may be interpreted as abstractions (see for example Sections Definitions and Structure), or their content can be inferred from the whole context of the goal (see for example Section Abbreviations).
Definitions¶
 Tactic pose¶
This tactic allows to add a defined constant to a proof context. SSReflect generalizes this tactic in several ways. In particular, the SSReflect
pose (ssreflect)
tactic supports open syntax: the body of the definition does not need surrounding parentheses. For instance:
is a valid tactic expression.
The pose (ssreflect)
tactic is also improved for the local definition of higherorder terms.
Local definitions of functions can use the same syntax as
global ones.
For example, the tactic pose (ssreflect)
supports parameters:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 pose f x y := x + y.
 1 goal f := fun x y : nat => x + y : nat > nat > nat ============================ True
The SSReflect pose (ssreflect)
tactic also supports (co)fixpoints, by providing
the local counterpart of the Fixpoint f := …
and CoFixpoint f := …
constructs. For instance, the following tactic:
defines a local fixpoint f
, which mimics the standard plus operation
on natural numbers.
Similarly, local cofixpoints can be defined by a tactic of the form:
The possibility to include wildcards in the body of the definitions offers a smooth way of defining local abstractions. The type of “holes” is guessed by type inference, and the holes are abstracted. For instance the tactic:
is shorthand for:
When the local definition of a function involves both arguments and holes, hole abstractions appear first. For instance, the tactic:
is shorthand for:
The interaction of the pose (ssreflect)
tactic with the interpretation of implicit
arguments results in a powerful and concise syntax for local
definitions involving dependent types. For instance, the tactic:
adds to the context the local definition:
The generalization of wildcards makes the use of the pose (ssreflect)
tactic
resemble MLlike definitions of polymorphic functions.
Abbreviations¶
 Tactic set ident : term? := occ_switch? term¶
The SSReflect
set
tactic performs abbreviations; it introduces a defined constant for a subterm appearing in the goal and/or in the context.SSReflect extends the
set
tactic by supplying:an open syntax, similarly to the
pose (ssreflect)
tactic;a more aggressive matching algorithm;
an improved interpretation of wildcards, taking advantage of the matching algorithm;
an improved occurrence selection mechanism allowing to abstract only selected occurrences of a term.
::=
{ +? natural* }
where:
ident
is a fresh identifier chosen by the user.term
1 is an optional type annotation. The type annotationterm
1 can be given in open syntax (no surrounding parentheses). If noocc_switch
(described hereafter) is present, it is also the case for the secondterm
. On the other hand, in the presence ofocc_switch
, parentheses surrounding the secondterm
are mandatory.In the occurrence switch
occ_switch
, if the first element of the list is a natural, this element should be a number, and not an Ltac variable. The empty list{}
is not interpreted as a valid occurrence switch; it is rather used as a flag to signal the intent of the user to clear the name following it (see Occurrence switches and redex switches and Introduction in the context).
The tactic:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom f : nat > nat.
 f is declared
 Lemma test x : f x + f x = f x.
 1 goal x : nat ============================ f x + f x = f x
 set t := f _.
 1 goal x : nat t := f x : nat ============================ t + t = t
 set t := {2}(f _).
 1 goal x : nat t := f x : nat ============================ f x + t = f x
The type annotation may contain wildcards, which will be filled with appropriate values by the matching process.
The tactic first tries to find a subterm of the goal matching
the second term
(and its type), and stops at the first subterm it finds. Then
the occurrences of this subterm selected by the optional occ_switch
are replaced by ident
and a definition ident := term
is added to the
context. If no occ_switch
is present, then all the occurrences are
abstracted.
Matching¶
The matching algorithm compares a pattern term
with a subterm of the
goal by comparing their heads and then pairwise unifying their
arguments (modulo conversion). Head symbols match under the following
conditions.
If the head of
term
is a constant, then it should be syntactically equal to the head symbol of the subterm.If this head is a projection of a canonical structure, then canonical structure equations are used for the matching.
If the head of
term
is not a constant, the subterm should have the same structure (λ abstraction,let…in
structure, etc.).If the head of
term
is a hole, the subterm should have at least as many arguments asterm
.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test (x y z : nat) : x + y = z.
 1 goal x, y, z : nat ============================ x + y = z
 set t := _ x.
 1 goal x, y, z : nat t := Nat.add x : nat > nat ============================ t y = z
In the special case where
term
is of the form(let f := t0 in f) t1 … tn
, then the patternterm
is treated as(_ t1 … tn)
. For each subterm in the goal having the form(A u1 … um)
with m ≥ n, the matching algorithm successively tries to find the largest partial application(A u1 … uj)
convertible to the headt0
ofterm
.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : (let f x y z := x + y + z in f 1) 2 3 = 6.
 1 goal ============================ (let f := fun x y z : nat => x + y + z in f 1) 2 3 = 6
 set t := (let g y z := S y + z in g) 2.
 1 goal t := unkeyed (fun y z : nat => S y + z) 2 : nat > nat ============================ t 3 = 6
The notation
unkeyed
defined inssreflect.v
is a shorthand for the degenerate termlet x := … in x
.
Moreover:
Multiple holes in
term
are treated as independent placeholders.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test x y z : x + y = z.
 1 goal x, y, z : nat ============================ x + y = z
 set t := _ + _.
 1 goal x, y, z : nat t := x + y : nat ============================ t = z
The type of the subterm matched should fit the type (possibly casted by some type annotations) of the pattern
term
.The replacement of the subterm found by the instantiated pattern should not capture variables. In the example above,
x
is bound and should not be captured.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : forall x : nat, x + 1 = 0.
 1 goal ============================ forall x : nat, x + 1 = 0
 Fail set t := _ + 1.
 The command has indeed failed with message: The pattern (_ + 1) did not match and has holes. Did you mean pose?
Typeclass inference should fill in any residual hole, but matching should never assign a value to a global existential variable.
Occurrence selection¶
SSReflect provides a generic syntax for the selection of occurrences by their position indexes. These occurrence switches are shared by all SSReflect tactics that require control on subterm selection like rewriting, generalization, …
An occurrence switch can be:
A list of natural numbers
{+ n1 … nm}
of occurrences affected by the tactic.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom f : nat > nat.
 f is declared
 Lemma test : f 2 + f 8 = f 2 + f 2.
 1 goal ============================ f 2 + f 8 = f 2 + f 2
 set x := {+1 3}(f 2).
 1 goal x := f 2 : nat ============================ x + f 8 = f 2 + x
Notice that some occurrences of a given term may be hidden to the user, for example because of a notation. Setting the
Printing All
flag causes these hidden occurrences to be shown when the term is displayed. This setting should be used to find the correct coding of the occurrences to be selected 11.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Notation "a < b":= (le (S a) b).
 Lemma test x y : x < y > S x < S y.
 1 goal x, y : nat ============================ x < y > S x < S y
 set t := S x.
 1 goal x, y : nat t := S x : nat ============================ t <= y > t < S y
A list of natural numbers
{n1 … nm}
. This is equivalent to the previous{+ n1 … nm}
, but the list should start with a number, and not with an Ltac variable.A list
{ n1 … nm}
of occurrences not to be affected by the tactic.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom f : nat > nat.
 f is declared
 Lemma test : f 2 + f 8 = f 2 + f 2.
 1 goal ============================ f 2 + f 8 = f 2 + f 2
 set x := {2}(f 2).
 1 goal x := f 2 : nat ============================ x + f 8 = f 2 + x
Note that, in this goal, it behaves like
set x := {1 3}(f 2).
In particular, the switch
{+}
selects all the occurrences. This switch is useful to turn off the default behavior of a tactic that automatically clears some assumptions (see Section Discharge for instance).The switch
{}
imposes that no occurrences of the term should be affected by the tactic. The tactic:set x := {}(f 2).
leaves the goal unchanged and adds the definitionx := f 2
to the context. This kind of tactic may be used to take advantage of the power of the matching algorithm in a local definition, instead of copying large terms by hand.
It is important to remember that matching precedes occurrence selection.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test x y z : x + y = x + y + z.
 1 goal x, y, z : nat ============================ x + y = x + y + z
 set a := {2}(_ + _).
 1 goal x, y, z : nat a := x + y : nat ============================ x + y = a + z
Hence, in the following goal, the same tactic fails since there is only one occurrence of the selected term.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test x y z : (x + y) + (z + z) = z + z.
 1 goal x, y, z : nat ============================ x + y + (z + z) = z + z
 Fail set a := {2}(_ + _).
 The command has indeed failed with message: Only 1 < 2 occurrence of (x + y + (z + z))
Basic localization¶
It is possible to define an abbreviation for a term appearing in the
context of a goal thanks to the in
tactical.
 Variant set ident := term in ident+
Example
 From Coq Require Import ssreflect.
 Lemma test x t (Hx : x = 3) : x + t = 4.
 1 goal x, t : nat Hx : x = 3 ============================ x + t = 4
 set z := 3 in Hx.
 1 goal x, t : nat z := 3 : nat Hx : x = z ============================ x + t = 4
 Variant set ident := term in ident+ *
Example
 From Coq Require Import ssreflect.
 Lemma test x t (Hx : x = 3) : x + t = 4.
 1 goal x, t : nat Hx : x = 3 ============================ x + t = 4
 set z := 3 in Hx * .
 1 goal x, t : nat z := 3 : nat Hx : x = z ============================ x + t = S z
Indeed, remember that 4 is just a notation for (S 3).
The use of the in
tactical is not limited to the localization of
abbreviations: for a complete description of the in
tactical, see
Section Bookkeeping and Localization.
Basic tactics¶
A sizable fraction of proof scripts consists of steps that do not "prove" anything new, but instead perform menial bookkeeping tasks such as selecting the names of constants and assumptions or splitting conjuncts. Although they are logically trivial, bookkeeping steps are extremely important because they define the structure of the dataflow of a proof script. This is especially true for reflectionbased proofs, which often involve large numbers of constants and assumptions. Good bookkeeping consists in always explicitly declaring (i.e., naming) all new constants and assumptions in the script, and systematically pruning irrelevant constants and assumptions in the context. This is essential in the context of an interactive development environment (IDE), because it facilitates navigating the proof, allowing to instantly "jump back" to the point at which a questionable assumption was added, and to find relevant assumptions by browsing the pruned context. While novice or casual Coq users may find the automatic name selection feature convenient, the usage of such a feature severely undermines the readability and maintainability of proof scripts, much like automatic variable declaration in programming languages. The SSReflect tactics are therefore designed to support precise bookkeeping and to eliminate name generation heuristics. The bookkeeping features of SSReflect are implemented as tacticals (or pseudotacticals), shared across most SSReflect tactics, and thus form the foundation of the SSReflect proof language.
Bookkeeping¶
During the course of a proof, Coq always presents the user with a sequent whose general form is:
ci : Ti
…
dj := ej : Tj
…
Fk : Pk
…
=================
forall (xl : Tl) …,
let ym := bm in … in
Pn > … > C
The goal to be proved appears below the double line; above the line
is the context of the sequent, a set of declarations of constants
ci
, defined constants dj
, and facts Fk
that can be used to
prove the goal (usually, Ti
, Tj : Type
and Pk : Prop
).
The various
kinds of declarations can come in any order. The top part of the
context consists of declarations produced by the Section
commands Variable
, Let
, and Hypothesis
.
This section context is never
affected by the SSReflect tactics: they only operate on the lower part
— the proof context. As in the figure above, the goal often
decomposes into a series of (universally) quantified variables
(xl : Tl)
, local definitions
let ym := bm in
, and assumptions
Pn >
,
and a conclusion C
(as in the context, variables, definitions, and
assumptions can appear in any order). The conclusion is what actually
needs to be proved — the rest of the goal can be seen as a part of the
proof context that happens to be “below the line”.
However, although they are logically equivalent, there are fundamental differences between constants and facts, on the one hand, and variables and assumptions, on the other. Constants and facts are unordered, but named explicitly in the proof text; variables and assumptions are ordered, but unnamed: the display names of variables may change at any time because of αconversion.
Similarly, basic deductive steps such as apply
can only operate on the
goal because the Gallina terms that control their action (e.g., the
type of the lemma used by apply
) only provide unnamed bound variables.
12 Since the proof script can only refer directly to the context, it
must constantly shift declarations from the goal to the context and
conversely in between deductive steps.
In SSReflect, these moves are performed by two tacticals, =>
and
:
, so that the bookkeeping required by a deductive step can be
directly associated with that step, and that tactics in an SSReflect
script correspond to actual logical steps in the proof rather than
merely shuffle facts. Still, some isolated bookkeeping is unavoidable,
such as naming variables and assumptions at the beginning of a
proof. SSReflect provides a specific move
tactic for this purpose.
Now, move
does essentially nothing: it is mostly a placeholder for
=>
and :
. The =>
tactical moves variables, local definitions,
and assumptions to the context, while the :
tactical moves facts and
constants to the goal.
Example
For example, the proof of 13
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma subnK : forall m n, n <= m > m  n + n = m.
 1 goal ============================ forall m n : nat, n <= m > m  n + n = m
might start with
 move=> m n le_n_m.
 1 goal m, n : nat le_n_m : n <= m ============================ m  n + n = m
where move
does nothing, but => m n le_m_n
changes
the variables and assumption of the goal in the constants
m n : nat
and the fact le_n_m : n <= m
, thus exposing the
conclusion m  n + n = m
.
The :
tactical is the converse of =>
; indeed it removes facts and
constants from the context by turning them into variables and
assumptions.
 move: m le_n_m.
 1 goal n : nat ============================ forall m : nat, n <= m > m  n + n = m
turns back m
and le_m_n
into a variable and an assumption,
removing them from the proof context, and changing the goal to
forall m, n <= m > m  n + n = m
,
which can be proved by induction on n
using elim: n
.
Because they are tacticals, :
and =>
can be combined, as in
which simultaneously renames m
and le_m_n
into p
and le_n_p
,
respectively, by first turning them into unnamed variables, then
turning these variables back into constants and facts.
Furthermore, SSReflect redefines the basic Coq tactics case
, elim
,
and apply
so that they can take better advantage of
:
and =>
. In these
SSReflect variants, these tactics operate on the first variable or
constant of the goal and they do not use or change the proof context.
The :
tactical is used to operate on an element in the context.
Example
For instance, the proof of
subnK
could continue withelim: n
. Instead ofelim n
(note, no colon), this has the advantage of removing n from the context. Better yet, thiselim
can be combined with previousmove
and with the branching version of the=>
tactical (described in Introduction in the context), to encapsulate the inductive step in a single command:
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma subnK : forall m n, n <= m > m  n + n = m.
 1 goal ============================ forall m n : nat, n <= m > m  n + n = m
 move=> m n le_n_m.
 1 goal m, n : nat le_n_m : n <= m ============================ m  n + n = m
 elim: n m le_n_m => [n IHn] m => [_  lt_n_m].
 2 goals m : nat ============================ m  0 + 0 = m goal 2 is: m  S n + S n = m
which breaks down the proof into two subgoals, the second one
having in its context
lt_n_m : S n <= m
and
IHn : forall m, n <= m > m  n + n = m
.
The :
and =>
tacticals can be explained very simply if one views
the goal as a stack of variables and assumptions piled on a conclusion:
tactic : a b c
pushes the context constantsa
,b
,c
as goal variables before performing the tactic;tactic => a b c
pops the top three goal variables as context constantsa
,b
,c
, after the tactic has been performed.
These pushes and pops do not need to balance out as in the examples
above; so move: m le_n_m => p
would rename m
into p
, but leave an extra assumption n <= p
in the goal.
Basic tactics like apply
and elim
can also be used without the ’:’
tactical: for example, we can directly start a proof of subnK
by
induction on the top variable m
with
The general form of the localization tactical in
is also best
explained in terms of the goal stack:
tactic in a H1 H2 *.
is basically equivalent to
with two differences: the in
tactical will preserve the body of a
, if a
is a defined constant, and if the *
is omitted, it will use a
temporary abbreviation to hide the statement of the goal from
tactic
.
The general form of the in
tactical can be used directly with the
move
, case
and elim
tactics, so that one can write
instead of
This is quite useful for inductive proofs that involve many facts.
See Section Localization for
the general syntax and presentation of the in
tactical.
The defective tactics¶
In this section, we briefly present the three basic tactics performing context manipulations and the main backward chaining tool.
The move tactic.¶
 Tactic move¶
This tactic, in its defective form, behaves like the
hnf
tactic.Example
 Require Import ssreflect.
 Goal not False.
 1 goal ============================ ~ False
 move.
 1 goal ============================ False > False
More precisely, the
move
tactic inspects the goal and does nothing (idtac
) if an introduction step is possible, i.e., if the goal is a product or alet … in
, and performshnf
otherwise.Of course this tactic is most often used in combination with the bookkeeping tacticals (see Sections Introduction in the context and Discharge). These combinations mostly subsume the
intros
,generalize
,revert
,rename
,clear
andpattern
tactics.
The case tactic¶
 Tactic case¶
This tactic performs primitive case analysis on (co)inductive types; specifically, it destructs the top variable or assumption of the goal, exposing its constructor(s) and its arguments, as well as setting the value of its type family indices if it belongs to a type family (see Section Type families).
The SSReflect
case
tactic has a special behavior on equalities. If the top assumption of the goal is an equality, thecase
tactic “destructs” it as a set of equalities between the constructor arguments of its left and right hand sides, as per the tactic injection. For example,case
changes the goal:(x, y) = (1, 2) > G.
into:
x = 1 > y = 2 > G.
The
case
can generate the following warning: Warning SSReflect: cannot obtain new equations out of ...¶
The tactic was run on an equation that cannot generate simpler equations, for example
x = 1
.
The warning can be silenced or made fatal by using the
Warnings
option and thespuriousssrinjection
key.Finally, the
case
tactic of SSReflect performsFalse
elimination, even if no branch is generated by this case operation. Hence the tacticcase
on a goal of the formFalse > G
will succeed and prove the goal.
The elim tactic¶
 Tactic elim¶
This tactic performs inductive elimination on inductive types. In its defective form, the tactic performs inductive elimination on a goal whose top assumption has an inductive type.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test m : forall n : nat, m <= n.
 1 goal m : nat ============================ forall n : nat, m <= n
 elim.
 2 goals m : nat ============================ m <= 0 goal 2 is: forall n : nat, m <= n > m <= S n
The apply tactic¶
 Tactic apply term?¶
This is the main backward chaining tactic of the proof system. It takes as argument any
term
and applies it to the goal. Assumptions in the type ofterm
that don’t directly match the goal may generate one or more subgoals.In its defective form, this tactic is a synonym for:
intro top; first [refine top  refine (top _)  refine (top _ _)  …]; clear top.
where
top
is a fresh name, and the sequence ofrefine
tactics tries to catch the appropriate number of wildcards to be inserted. Note that this use of therefine
tactic implies that the tactic tries to match the goal up to expansion of constants and evaluation of subterms.
apply
has a special behavior on goals containing
existential metavariables of sort Prop
.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom lt_trans : forall a b c, a < b > b < c > a < c.
 lt_trans is declared
 Lemma test : forall y, 1 < y > y < 2 > exists x : { n  n < 3 }, 0 < proj1_sig x.
 1 goal ============================ forall y : nat, 1 < y > y < 2 > exists x : {n : nat  n < 3}, 0 < proj1_sig x
 move=> y y_gt1 y_lt2; apply: (ex_intro _ (exist _ y _)).
 2 goals y : nat y_gt1 : 1 < y y_lt2 : y < 2 ============================ y < 3 goal 2 is: forall Hyp0 : y < 3, 0 < proj1_sig (exist (fun n : nat => n < 3) y Hyp0)
 by apply: lt_trans y_lt2 _.
 1 goal y : nat y_gt1 : 1 < y y_lt2 : y < 2 ============================ forall Hyp0 : y < 3, 0 < proj1_sig (exist (fun n : nat => n < 3) y Hyp0)
 by move=> y_lt3; apply: lt_trans y_gt1.
 No more goals.
Note that the last _
of the tactic
apply: (ex_intro _ (exist _ y _))
represents a proof that y < 3
. Instead of generating the goal:
0 < proj1_sig (exist (fun n : nat => n < 3) y ?Goal).
the system tries to prove y < 3
calling the trivial tactic.
If it succeeds, let’s say because the context contains
H : y < 3
, then the
system generates the following goal:
0 < proj1_sig (exist (fun n => n < 3) y H).
Otherwise the missing proof is considered to be irrelevant, and is thus discharged, generating the two goals shown above.
Last, the user can replace the trivial tactic by defining an Ltac
expression named ssrautoprop
.
Discharge¶
The general syntax of the discharging tactical :
is:
 Tactic tactic ident? : d_item+ clear_switch?¶
::=
occ_switchclear_switch? term
clear_switch::=
{ ident+ }
with the following requirements.
tactic
must be one of the four basic tactics described in The defective tactics, i.e.,move
,case
,elim
orapply
, theexact
tactic (section Terminators), thecongr
tactic (Section Congruence), or the application of the view tactical ‘/’ (Section Interpreting assumptions) to one ofmove
,case
, orelim
.The optional
ident
specifies equation generation (Section Generation of equations), and is only allowed iftactic
ismove
,case
orelim
, or the application of the view tactical ‘/’ (Section Interpreting assumptions) tocase
orelim
.An
occ_switch
selects occurrences ofterm
, as in Abbreviations;occ_switch
is not allowed iftactic
isapply
orexact
.A clear item
clear_switch
specifies facts and constants to be deleted from the proof context (as per theclear
tactic).
The :
tactical first discharges all the d_item
, right to left,
and then performs the tactic, i.e., for each d_item
, starting with the last one :
The SSReflect matching algorithm described in Section Abbreviations is used to find occurrences of
term
in the goal, after filling any holes ‘_’ in the term; however iftactic
isapply
orexact
, a different matching algorithm, described below, is used 14.These occurrences are replaced by a new variable; in particular, if the term is a fact, this adds an assumption to the goal.
If the term is exactly the name of a constant or fact in the proof context, it is deleted from the context, unless there is an
occ_switch
.
Finally, the tactic is performed just after the first d_item
has been generalized
— that is, between steps 2 and 3. The names listed in
the final clear_switch
(if it is present) are cleared first, before
d_item
n is discharged.
Switches affect the discharging of a d_item
as follows.
An
occ_switch
restricts generalization (step 2) to a specific subset of the occurrences of the term, as per Section Abbreviations, and prevents clearing (step 3).All the names specified by a
clear_switch
are deleted from the context in step 3, possibly in addition to the term.
For example, the tactic:
first generalizes
(refl_equal n : n = n)
;then generalizes the second occurrence of
n
.finally generalizes all the other occurrences of
n
, and clearsn
from the proof context (assumingn
is a proof constant).
Therefore, this tactic changes any goal G
into
where the name n0
is picked by the Coq display function, and assuming
n
appeared only in G
.
Finally, note that a discharge operation generalizes defined constants
as variables, and not as local definitions. To override this behavior,
prefix the name of the local definition with a @
, like in move: @n
.
This is in contrast with the behavior of the in
tactical (see
Section Localization), which preserves local
definitions by default.
Clear rules¶
The clear step will fail if the term is a proof constant that appears in
other facts; in that case, either the facts should be cleared
explicitly with a clear_switch
, or the clear step should be disabled.
The latter can be done by adding an occ_switch
or simply by putting
parentheses around term: both
move: (n).
and
move: {+}n.
generalize n
without clearing n
from the proof context.
The clear step will also fail if the clear_switch
contains a ident
that
is not in the proof context. Note that SSReflect never clears a
section constant.
If the tactic is move
or case
and an equation ident
is given, then clearing
(step 3) for d_item
is suppressed (see Section Generation of equations).
Intro patterns (see Section Introduction in the context)
and the rewrite
tactic (see Section Rewriting)
let one place a clear_switch
in the middle of other items
(namely identifiers, views and rewrite rules). This can trigger the
addition of proof context items to the ones being explicitly
cleared, and in turn this can result in clear
errors (e.g., if the
context item automatically added occurs in the goal). The
relevant sections describe ways to avoid the unintended clearing of
context items.
Matching for apply and exact¶
The matching algorithm for d_item
of the SSReflect
apply
and exact
tactics exploits the type of the first d_item
to interpret
wildcards in the
other d_item
and to determine which occurrences of these should be
generalized. Therefore, occur switches are not needed for apply
and
exact
.
Indeed, the SSReflect tactic apply: H x
is equivalent to
refine (@H _ … _ x); clear H x
,
with an appropriate number of wildcards between H
and x
.
Note that this means that matching for apply
and exact
has much more
context to interpret wildcards; in particular, it can accommodate the
_
d_item
, which would always be rejected after move:
.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom f : nat > nat.
 f is declared
 Axiom g : nat > nat.
 g is declared
 Lemma test (Hfg : forall x, f x = g x) a b : f a = g b.
 1 goal Hfg : forall x : nat, f x = g x a, b : nat ============================ f a = g b
 apply: trans_equal (Hfg _) _.
 1 goal Hfg : forall x : nat, f x = g x a, b : nat ============================ g a = g b
This tactic is equivalent (see Section
Bookkeeping) to:
refine (trans_equal (Hfg _) _).
and this is a common idiom for applying transitivity on the left hand
side of an equation.
The abstract tactic¶
 Tactic abstract: d_item+¶
This tactic assigns an abstract constant previously introduced with the
[: ident ]
intro pattern (see Section Introduction in the context).
In a goal like the following:
m : nat
abs : <hidden>
n : nat
=============
m < 5 + n
The tactic abstract: abs n
first generalizes the goal with respect to n
(that is not visible to the abstract constant abs
) and then assigns
abs. The resulting goal is:
m : nat
n : nat
=============
m < 5 + n
Once this subgoal is closed, all other goals having abs
in their
context see the type assigned to abs
. In this case:
m : nat
abs : forall n, m < 5 + n
=============
…
For a more detailed example, the reader should refer to Section Structure.
Introduction in the context¶
The application of a tactic to a given goal can generate (quantified) variables, assumptions, or definitions, which the user may want to introduce as new facts, constants or defined constants, respectively. If the tactic splits the goal into several subgoals, each of them may require the introduction of different constants and facts. Furthermore it is very common to immediately decompose or rewrite with an assumption instead of adding it to the context, as the goal can often be simplified and even proved after this.
All these operations are performed by the introduction tactical =>
,
whose general syntax is
::=
i_patterns_itemclear_switchi_viewi_block
s_item::=
/=////=
i_view::=
{}? /term/ltac:( tactic )
i_pattern::=
ident>_?*+occ_switch? ><[ i_item? ][: ident+ ]
i_block::=
[^ ident ][^~ identnatural ]
The =>
tactical first executes tactic
, then the i_item
s,
left to right. An s_item
specifies a
simplification operation; a clear_switch
specifies context pruning as in Discharge.
The i_pattern
s can be seen as a variant of intro patterns
(see intros
); each performs an introduction operation, i.e., pops some
variables or assumptions from the goal.
Simplification items¶
An s_item
can simplify the set of subgoals or the subgoals themselves.
//
removes all the “trivial” subgoals that can be resolved by the SSReflect tacticdone
described in Terminators, i.e., it executestry done
./=
simplifies the goal by performing partial evaluation, as per the tacticsimpl
15.//=
combines both kinds of simplification; it is equivalent to/= //
, i.e.,simpl; try done
.
When an s_item
immediately precedes a clear_switch
, then the
clear_switch
is executed
after the s_item
, e.g., {IHn}//
will solve some subgoals,
possibly using the fact IHn
, and will erase IHn
from the context
of the remaining subgoals.
Views¶
The first entry in the i_view
grammar rule, /term
,
represents a view (see Section Views and reflection).
It interprets the top of the stack with the view term
.
It is equivalent to move/term
.
A clear_switch
that immediately precedes an i_view
is complemented with the name of the view if an only if the i_view
is a simple proof context entry 20.
E.g., {}/v
is equivalent to /v{v}
.
This behavior can be avoided by separating the clear_switch
from the i_view
with the 
intro pattern or by putting
parentheses around the view.
A clear_switch
that immediately precedes an i_view
is executed after the view application.
If the next i_item
is a view, then the view is
applied to the assumption in top position once all the
previous i_item
have been performed.
The second entry in the i_view
grammar rule,
/ltac:(
tactic
)
, executes tactic
.
Notations can be used to name tactics, for example
 Tactic Notation "my" "ltac" "code" := idtac.
 Notation "'myop'" := (ltac:(my ltac code)) : ssripat_scope.
 Toplevel input, characters 059: > Notation "'myop'" := (ltac:(my ltac code)) : ssripat_scope. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: This notation contains Ltac expressions: it will not be used for printing. [nonreversiblenotation,parsing,default]
lets one write just /myop
in the intro pattern. Note the scope
annotation: views are interpreted opening the ssripat
scope. We
provide the following ltac views: /[dup]
to duplicate the top of
the stack, /[swap]
to swap the two first elements and /[apply]
to apply the top of the stack to the next.
Intro patterns¶
SSReflect supports the following i_pattern
s.
ident
pops the top variable, assumption, or local definition into a new constant, fact, or defined constant
ident
, respectively. Note that defined constants cannot be introduced when δexpansion is required to expose the top variable or assumption. Aclear_switch
(even an empty one) immediately preceding anident
is complemented with thatident
if and only if the identifier is a simple proof context entry 20. As a consequence, by prefixing theident
with{}
one can replace a context entry. This behavior can be avoided by separating theclear_switch
from theident
with the
intro pattern.Thus, trying to clear an
ident
H
with{H}H
triggers the following warning:>
pops every variable occurring in the rest of the stack. Type class instances are popped even if they don't occur in the rest of the stack. The tactic
move=> >
is equivalent tomove=> ? ?
on a goal such as:forall x y, x < y > G
A typical use if
move=>> H
to nameH
the first assumption, in the example abovex < y
.?
pops the top variable into an anonymous constant or fact, whose name is picked by the tactic interpreter. SSReflect only generates names that cannot appear later in the user script 16.
_
pops the top variable into an anonymous constant that will be deleted from the proof context of all the subgoals produced by the
=>
tactical. They should thus never be displayed, except in an error message if the constant is still actually used in the goal or context after the lasti_item
has been executed (s_item
can erase goals or terms where the constant appears).*
pops all the remaining apparent variables/assumptions as anonymous constants/facts. Unlike
?
andmove
, the*
i_item
does not expand definitions in the goal to expose quantifiers, so it may be useful to repeat amove=> *
tactic, e.g., on the goal:forall a b : bool, a <> b
a first
move=> *
adds only_a_ : bool
and_b_ : bool
to the context; it takes a secondmove=> *
to add_Hyp_ : _a_ = _b_
.+
temporarily introduces the top variable. It is discharged at the end of the intro pattern. For example
move=> + y
on a goal:forall x y, P
is equivalent to
move=> _x_ y; move: _x_
that results in the goal:forall x, P
occ_switch? >
(resp.
occ_switch
<
) pops the top assumption (which should be a rewritable proposition) into an anonymous fact, rewrites (resp. rewrites right to left) the goal with this fact (using the SSReflectrewrite
tactic described in Section Rewriting, and honoring the optional occurrence selector), and finally deletes the anonymous fact from the context.[
i_item
* … 
i_item
*]
when it is the very first
i_pattern
after tactic=>
tactical and the tactic is not a move, is a branchingi_pattern
. It executes the sequencei_item_{i}
on the ith subgoal produced by the tactic. The execution of the tactic should thus generate exactly m subgoals, unless the[…]
i_pattern
comes after an initial//
or//=
s_item
that closes some of the goals produced by the tactic, in which case exactly m subgoals should remain after thes_item
, or we have the trivial branchingi_pattern
[], which always does nothing, regardless of the number of remaining subgoals.[
i_item
* … 
i_item
*]
when it is not the first
i_pattern
or when the tactic is amove
, is a destructingi_pattern
. It starts by destructing the top variable, using the SSReflectcase
tactic described in The defective tactics. It then behaves as the corresponding branchingi_pattern
, executing the sequencei_item_{i}
in the ith subgoal generated by the case analysis; unless we have the trivial destructingi_pattern
[]
, the latter should generate exactly m subgoals, i.e., the top variable should have an inductive type with exactly m constructors 17. While it is good style to use thei_item
i * to pop the variables and assumptions corresponding to each constructor, this is not enforced by SSReflect.
does nothing, but counts as an intro pattern. It can also be used to force the interpretation of
[
i_item
* … 
i_item
*]
as a case analysis like inmove=> [H1 H2]
. It can also be used to indicate explicitly the link between a view and a name like inmove=> /eqPH1
. Last, it can serve as a separator between views. Section Views and reflection 19 explains in which respect the tacticmove=> /v1/v2
differs from the tacticmove=> /v1/v2
.[:
ident
…]
introduces in the context an abstract constant for each
ident
. Its type has to be fixed later on by using theabstract
tactic. Before then the type displayed is<hidden>
.
Note that SSReflect does not support the syntax (ipat, …, ipat)
for
destructing intro patterns.
Clear switch¶
Clears are deferred until the end of the intro pattern.
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test x y : Nat.leb 0 x = true > (Nat.leb 0 x) && (Nat.leb y 2) = true.
 1 goal x, y : nat ============================ Nat.leb 0 x = true > Nat.leb 0 x && Nat.leb y 2 = true
 move=> {x} >.
 1 goal y : nat ============================ true && Nat.leb y 2 = true
If the cleared names are reused in the same intro pattern, a renaming is performed behind the scenes.
Facts mentioned in a clear switch must be valid names in the proof context (excluding the section context).
Branching and destructuring¶
The rules for interpreting branching and destructing i_pattern
are
motivated by the fact that it would be pointless to have a branching
pattern if the tactic is a move
, and in most of the remaining cases
the tactic is case
or elim
, which implies destructuring.
The rules above imply that:
move=> [a b].
case=> [a b].
case=> a b.
are all equivalent, so which one to use is a matter of style; move
should
be used for casual decomposition, such as splitting a pair, and case
should be used for actual decompositions, in particular for type families
(see Type families) and proof by contradiction.
The trivial branching i_pattern
can be used to force the branching
interpretation, e.g.:
case=> [] [a b] c.
move=> [[a b] c].
case; case=> a b c.
are all equivalent.
Block introduction¶
SSReflect supports the following i_block
s.
[^ ident ]
block destructing
i_pattern
. It performs a case analysis on the top variable and introduces, in one go, all the variables coming from the case analysis. The names of these variables are obtained by taking the names used in the inductive type declaration and prefixing them withident
. If the intro pattern immediately follows a call toelim
with a custom eliminator (see Interpreting eliminations), then the names are taken from the ones used in the type of the eliminator.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Record r := { a : nat; b := (a, 3); _ : bool; }.
 r is defined a is defined b is defined
 Lemma test : r > True.
 1 goal ============================ r > True
 Proof. move => [^ x ].
 1 goal xa : nat xb := (xa, 3) : nat * nat _x?_ : bool ============================ True
[^~ ident ]
block destructing using
ident
as a suffix.[^~ natural ]
block destructing using
natural
as a suffix.Only a
s_item
is allowed between the elimination tactic and the block destructing.
Generation of equations¶
The generation of named equations option stores the definition of a new constant as an equation. The tactic:
where l
is a list, replaces size l
by n
in the goal and
adds the fact En : size l = n
to the context.
This is quite different from:
which generates a definition n := (size l)
. It is not possible to
generalize or rewrite such a definition; on the other hand, it is
automatically expanded during computation, whereas expanding the
equation En
requires explicit rewriting.
The use of this equation name generation option with a case
or an
elim
tactic changes the status of the first i_item
, in order to
deal with the possible parameters of the constants introduced.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test (a b :nat) : a <> b.
 1 goal a, b : nat ============================ a <> b
 case E : a => [n].
 2 goals a, b : nat E : a = 0 ============================ 0 <> b goal 2 is: S n <> b
If the user does not provide a branching i_item
as first
i_item
, or if the i_item
does not provide enough names for
the arguments of a constructor, then the constants generated are introduced
under fresh SSReflect names.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test (a b :nat) : a <> b.
 1 goal a, b : nat ============================ a <> b
 case E : a => H.
 2 goals a, b : nat E : a = 0 H : 0 = b ============================ False goal 2 is: False
 Show 2.
 goal 2 is: a, b, _n_ : nat E : a = S _n_ H : S _n_ = b ============================ False
Combining the generation of named equations mechanism with the case
tactic strengthens the power of a case analysis. On the other hand,
when combined with the elim
tactic, this feature is mostly useful for
debug purposes, to trace the values of decomposed parameters and
pinpoint failing branches.
Type families¶
When the top assumption of a goal has an inductive type, two specific
operations are possible: the case analysis performed by the case
tactic, and the application of an induction principle, performed by
the elim
tactic. When this top assumption has an inductive type, which
is moreover an instance of a type family, Coq may need help from the
user to specify which occurrences of the parameters of the type should
be substituted.
 Variant case: d_item+ / d_item+
 Variant elim: d_item+ / d_item+
A specific
/
switch indicates the type family parameters of the type of ad_item
immediately following this/
switch. Thed_item
on the right side of the/
switch are discharged as described in Section Discharge. The case analysis or elimination will be done on the type of the top assumption after these discharge operations.Every
d_item
preceding the/
is interpreted as an argument of this type, which should be an instance of an inductive type family. These terms are not actually generalized, but rather selected for substitution. Occurrence switches can be used to restrict the substitution. If a term is left completely implicit (e.g., writing just_
), then a pattern is inferred by looking at the type of the top assumption. This allows for the compact syntax:case: {2}_ / eqP.where
_
is interpreted as(_ == _)
, sinceeqP T a b : reflect (a = b) (a == b)
andreflect
is a type family with one index.Moreover, if the
d_item
list is too short, it is padded with an initial sequence of_
of the right length.Example
Here is a small example on lists. We define first a function that adds an element at the end of a given list.
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Require Import List.
 Section LastCases.
 Variable A : Type.
 A is declared
 Implicit Type l : list A.
 Fixpoint add_last a l : list A := match l with  nil => a :: nil  hd :: tl => hd :: (add_last a tl) end.
 add_last is defined add_last is recursively defined (guarded on 2nd argument)
Then we define an inductive predicate for case analysis on lists according to their last element:
 Inductive last_spec : list A > Type :=  LastSeq0 : last_spec nil  LastAdd s x : last_spec (add_last x s).
 last_spec is defined last_spec_rect is defined last_spec_ind is defined last_spec_rec is defined last_spec_sind is defined
 Theorem lastP : forall l : list A, last_spec l.
 1 goal A : Type ============================ forall l, last_spec l
 Admitted.
 lastP is declared
We are now ready to use
lastP
in conjunction withcase
. Lemma test l : (length l) * 2 = length (l ++ l).
 1 goal A : Type l : list A ============================ length l * 2 = length (l ++ l)
 case: (lastP l).
 2 goals A : Type l : list A ============================ length nil * 2 = length (nil ++ nil) goal 2 is: forall (s : list A) (x : A), length (add_last x s) * 2 = length (add_last x s ++ add_last x s)
Applied to the same goal, the tactic
case: l / (lastP l)
generates the same subgoals, butl
has been cleared from both contexts: case: l / (lastP l).
 2 goals A : Type ============================ length nil * 2 = length (nil ++ nil) goal 2 is: forall (s : list A) (x : A), length (add_last x s) * 2 = length (add_last x s ++ add_last x s)
Again applied to the same goal:
 case: {1 3}l / (lastP l).
 2 goals A : Type l : list A ============================ length nil * 2 = length (l ++ nil) goal 2 is: forall (s : list A) (x : A), length (add_last x s) * 2 = length (l ++ add_last x s)
Note that the selected occurrences on the left of the
/
switch have been substituted withl
instead of being affected by the case analysis.The equation name generation feature combined with a type family
/
switch generates an equation for the first dependentd_item
specified by the user. Again starting with the above goal, the command:Example
 Lemma test l : (length l) * 2 = length (l ++ l).
 1 goal A : Type l : list A ============================ length l * 2 = length (l ++ l)
 case E: {1 3}l / (lastP l) => [s x].
 2 goals A : Type l : list A E : l = nil ============================ length nil * 2 = length (l ++ nil) goal 2 is: length (add_last x s) * 2 = length (l ++ add_last x s)
 Show 2.
 goal 2 is: A : Type l, s : list A x : A E : l = add_last x s ============================ length (add_last x s) * 2 = length (l ++ add_last x s)
There must be at least one
d_item
to the left of the/
switch; this prevents any confusion with the view feature. However, thed_item
to the right of the/
are optional, and if they are omitted, the first assumption provides the instance of the type family.The equation always refers to the first
d_item
in the actual tactic call, before any padding with initial_
. Thus, if an inductive type has two family parameters, it is possible to have SSReflect generate an equation for the second one by omitting the pattern for the first; note however that this will fail if the type of the second parameter depends on the value of the first parameter.
Control flow¶
Indentation and bullets¶
A linear development of Coq scripts gives little information on the structure of the proof. In addition, replaying a proof after some changes in the statement to be proved will usually not display information to distinguish between the various branches of case analysis for instance.
To help the user in this organization of the proof script at development
time, SSReflect provides some bullets to highlight the structure of branching
proofs. The available bullets are 
, +
and *
. Combined with
tabulation, this lets us highlight four nested levels of branching; the most
we have ever needed is three. Indeed, the use of “simpl and closing”
switches, of terminators (see Section Terminators) and
selectors (see Section Selectors) is powerful enough to avoid most
of the time more than two levels of indentation.
Here is a fragment of such a structured script:
case E1: (abezoutn _ _) => [[ k1] [ k2]].
 rewrite !muln0 !gexpn0 mulg1 => H1.
move/eqP: (sym_equal F0); rewrite H1 orderg1 eqn_mul1.
by case/andP; move/eqP.
 rewrite muln0 gexpn0 mulg1 => H1.
have F1: t % t * S k2.+1  1.
apply: (@dvdn_trans (orderg x)); first by rewrite F0; exact: dvdn_mull.
rewrite orderg_dvd; apply/eqP; apply: (mulgI x).
rewrite {1}(gexpn1 x) mulg1 gexpn_add leq_add_sub //.
by move: P1; case t.
rewrite dvdn_subr in F1; last by exact: dvdn_mulr.
+ rewrite H1 F0 {2}(muln1 (p ^ l)); congr (_ * _).
by apply/eqP; rewrite dvdn1.
+ by move: P1; case: (t) => [ [ s1]].
 rewrite muln0 gexpn0 mul1g => H1.
...
Terminators¶
To further structure scripts, SSReflect supplies terminating tacticals to explicitly close off tactics. When replaying scripts, we then have the nice property that an error immediately occurs when a closed tactic fails to prove its subgoal.
It is hence recommended practice that the proof of any subgoal should
end with a tactic that fails if it does not solve the current goal,
like discriminate
, contradiction
or assumption
.
In fact, SSReflect provides a generic tactical that turns any tactic
into a closing one (similar to now
). Its general syntax is:
The Ltac expression by [tactic  tactic  …]
is equivalent to
do [done  by tactic  by tactic  …]
, which corresponds to the
standard Ltac expression first [done  tactic; done  tactic; done  …]
.
In the script provided as example in Section Indentation and bullets, the
paragraph corresponding to each subcase ends with a tactic line prefixed
with a by
, like in:
 Tactic done¶
The
by
tactical is implemented using the userdefined, and extensible,done
tactic. Thisdone
tactic tries to solve the current goal by some trivial means and fails if it doesn’t succeed. Indeed, the tactic expressionby tactic
is equivalent totactic; done
.Conversely, the tactic
by [ ]
is equivalent todone
.The default implementation of the
done
tactic, in thessreflect.v
file, is:Ltac done := trivial; hnf; intros; solve [ do ![solve [trivial  apply: sym_equal; trivial]  discriminate  contradiction  split]  case not_locked_false_eq_true; assumption  match goal with H : ~ _  _ => solve [case H; trivial] end ].The lemma
not_locked_false_eq_true
is needed to discriminate locked boolean predicates (see Section Locking, unlocking). The iterator tacticaldo
is presented in Section Iteration. This tactic can be customized by the user, for instance to include anauto
tactic.
A natural and common way of closing a goal is to apply a lemma that is the exact one needed for the goal to be solved. The defective form of the tactic:
is equivalent to:
where top
is a fresh name assigned to the top assumption of the goal.
This applied form is supported by the :
discharge tactical, and the
tactic:
is equivalent to:
(see Section Discharge for the documentation of the apply: combination).
Warning
The list of tactics (possibly chained by semicolons) that
follows the by
keyword is considered to be a parenthesized block applied to
the current goal. Hence for example if the tactic:
succeeds, then the tactic:
usually fails since it is equivalent to:
Selectors¶
 Tactic last¶
 Tactic first¶
When composing tactics, the two tacticals
first
andlast
let the user restrict the application of a tactic to only one of the subgoals generated by the previous tactic. This covers the frequent cases where a tactic generates two subgoals one of which can be easily disposed of.This is another powerful way of linearization of scripts, since it happens very often that a trivial subgoal can be solved in a less than one line tactic. For instance,
tactic ; last by tactic
tries to solve the last subgoal generated by the first tactic using the given second tactic, and fails if it does not succeed. Its analoguetactic ; first by tactic
tries to solve the first subgoal generated by the first tactic using the second given tactic, and fails if it does not succeed.
SSReflect also offers an extension of this facility, by supplying tactics to permute the subgoals generated by a tactic.
 Variant last first¶
 Variant first last¶
These two equivalent tactics invert the order of the subgoals in focus.
Finally, the tactics last
and first
combine with the branching syntax
of Ltac: if the tactic generates n subgoals on a given goal,
then the tactic
applies tactic1
to the
\(n−k+1\)th goal, … tacticm
to the \(n−k+m\)th goal and tacticn
to the others.
Example
Here is a small example on lists. We define first a function that adds an element at the end of a given list.
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Inductive test : nat > Prop :=  C1 n of n = 1 : test n  C2 n of n = 2 : test n  C3 n of n = 3 : test n  C4 n of n = 4 : test n.
 test is defined test_ind is defined test_sind is defined
 Lemma example n (t : test n) : True.
 1 goal n : nat t : test n ============================ True
 case: t; last 2 [move=> k move=> l]; idtac.
 4 goals n : nat ============================ forall n0 : nat, n0 = 1 > True goal 2 is: k = 2 > True goal 3 is: l = 3 > True goal 4 is: forall n0 : nat, n0 = 4 > True
Iteration¶
 Tactic do mult? tactic[ tactic+ ]¶
This tactical offers an accurate control on the repetition of tactics.
mult
is a multiplier.Brackets can only be omitted if a single tactic is given and a multiplier is present.
A tactic of the form:
is equivalent to the standard Ltac expression:
The optional multiplier mult
specifies how many times the action of
tactic
should be repeated on the current subgoal.
There are four kinds of multipliers:
mult::=
natural !!natural ??
Their meaning is as follows.
With
n!
, the step tactic is repeated exactlyn
times (wheren
is a positive integer argument).With
!
, the step tactic is repeated as many times as possible, and done at least once.With
?
, the step tactic is repeated as many times as possible, optionally.Finally, with
n?
, the step tactic is repeated up ton
times, optionally.
For instance, the tactic:
rewrites at most one time the lemma mult_comm
in all the subgoals
generated by tactic, whereas the tactic:
rewrites exactly two times the lemma mult_comm
in all the subgoals
generated by tactic
, and fails if this rewrite is not possible in some
subgoal.
Note that the combination of multipliers and rewrite is so often used that multipliers are in fact integrated to the syntax of the SSReflect rewrite tactic, see Section Rewriting.
Localization¶
In Sections Basic localization and Bookkeeping, we have
already presented the localization tactical in
, whose general syntax is:
where ident
is a name in the
context. On the left side of in
,
tactic
can be move
, case
, elim
, rewrite
, set
,
or any tactic formed with the general iteration tactical do
(see Section
Iteration).
The operation described by the tactic is performed in the facts listed after
in
and in the goal if a *
ends the list of names.
The in
tactical successively:
generalizes the selected hypotheses, possibly “protecting” the goal if
*
is not present;performs
tactic
, on the obtained goal;reintroduces the generalized facts, under the same names.
This defective form of the do
tactical is useful to avoid clashes
between standard Ltac in
and the SSReflect tactical in.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Ltac mytac H := rewrite H.
 mytac is defined
 Lemma test x y (H1 : x = y) (H2 : y = 3) : x + y = 6.
 1 goal x, y : nat H1 : x = y H2 : y = 3 ============================ x + y = 6
 do [mytac H2] in H1 *.
 1 goal x, y : nat H2 : y = 3 H1 : x = 3 ============================ x + 3 = 6
the last tactic rewrites the hypothesis H2 : y = 3
both in
H1 : x = y
and in the goal x + y = 6
.
By default, in
keeps the body of local definitions. To erase the body
of a local definition during the generalization phase, the name of the
local definition must be written between parentheses, like in
rewrite H in H1 (def_n) H2.
 Variant tactic in clear_switch@?ident( ident )( @?ident := c_pattern )+ *?
This is the most general form of the
in
tactical. In its simplest form, the last option lets one rename hypotheses that can’t be cleared (like section variables). For example,(y := x)
generalizes overx
and reintroduces the generalized variable under the namey
(and does not clearx
). For a more precise description of this form of localization, refer to Advanced generalization.
Structure¶
Forward reasoning structures the script by explicitly specifying some assumptions to be added to the proof context. It is closely associated with the declarative style of proof, since an extensive use of these highlighted statements makes the script closer to a (very detailed) textbook proof.
Forward chaining tactics allow to state an intermediate lemma and start a piece of script dedicated to the proof of this statement. The use of closing tactics (see Section Terminators) and of indentation makes syntactically explicit the portion of the script building the proof of the intermediate statement.
The have tactic.¶
 Tactic have : term¶
This is the main SSReflect forward reasoning tactic. It can be used in two modes: one starts a new (sub)proof for an intermediate result in the main proof, and the other provides explicitly a proof term for this intermediate step.
This tactic supports open syntax for
term
. Applied to a goalG
, it generates a first subgoal requiring a proof ofterm
in the context ofG
. The second generated subgoal is of the formterm > G
, where term becomes the new top assumption, instead of being introduced with a fresh name. At the proofterm level, thehave
tactic creates a β redex, and introduces the lemma under a fresh name, automatically chosen.
Like in the case of the pose (ssreflect)
tactic (see Section Definitions), the types of
the holes are abstracted in term.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 have: _ * 0 = 0.
 2 goals ============================ forall n : nat, n * 0 = 0 goal 2 is: (forall n : nat, n * 0 = 0) > True
The invocation of have
is equivalent to:
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 have: forall n : nat, n * 0 = 0.
 2 goals ============================ forall n : nat, n * 0 = 0 goal 2 is: (forall n : nat, n * 0 = 0) > True
The have
tactic also enjoys the same abstraction mechanism as the pose (ssreflect)
tactic for the noninferred implicit arguments. For instance, the
tactic:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 have: forall x y, (x, y) = (x, y + 0).
 2 goals ============================ forall (T : Type) (x : T) (y : nat), (x, y) = (x, y + 0) goal 2 is: (forall (T : Type) (x : T) (y : nat), (x, y) = (x, y + 0)) > True
opens a new subgoal where the type of x
is quantified.
The behavior of the defective have tactic makes it possible to generalize it in the following general construction:
Open syntax is supported for both term
. For the description
of i_item
and s_item
, see Section
Introduction in the context. The first mode of the
have tactic, which opens a subproof for an intermediate result, uses
tactics of the form:
 Variant have clear_switch i_item : term by tactic
which behaves like:
Note that the clear_switch
precedes the i_item
, which
allows to reuse
a name of the context, possibly used by the proof of the assumption,
to introduce the new assumption itself.
The by
feature is especially convenient when the proof script of the
statement is very short, basically when it fits in one line like in:
The possibility of using i_item
supplies a very concise syntax for
the further use of the intermediate step. For instance,
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test a : 3 * a  1 = a.
 1 goal a : nat ============================ 3 * a  1 = a
 have > : forall x, x * a = a.
 2 goals a : nat ============================ forall x : nat, x * a = a goal 2 is: a  1 = a
Note how the second goal was rewritten using the stated equality. Also note that in this last subgoal, the intermediate result does not appear in the context.
Thanks to the deferred execution of clears, the following idiom is also supported (assuming x occurs in the goal only):
Another frequent use of the intro patterns combined with have
is the
destruction of existential assumptions like in the tactic:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 have [x Px]: exists x : nat, x > 0; last first.
 2 goals x : nat Px : x > 0 ============================ True goal 2 is: exists x : nat, x > 0
An alternative use of the have
tactic is to provide the explicit proof
term for the intermediate lemma, using tactics of the form:
 Variant have ident? := term
This tactic creates a new assumption of type the type of
term
. If the optionalident
is present, this assumption is introduced under the nameident
. Note that the body of the constant is lost for the user.Again, noninferred implicit arguments and explicit holes are abstracted.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 have H := forall x, (x, x) = (x, x).
 1 goal H : Type > Prop ============================ True
adds to the context H : Type > Prop.
This is a schematic example, but
the feature is specially useful when the proof term to give involves
for instance a lemma with some hidden implicit arguments.
After the i_pattern
, a list of binders is allowed.
Example
 From Coq Require Import ssreflect.
 From Coq Require Import ZArith Lia.
 [Loading ML file ring_plugin.cmxs (using legacy method) ... done] [Loading ML file zify_plugin.cmxs (using legacy method) ... done] [Loading ML file micromega_core_plugin.cmxs (using legacy method) ... done] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done]
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test : True.
 1 goal ============================ True
 have H x (y : nat) : 2 * x + y = x + x + y by lia.
 1 goal H : forall x y : nat, 2 * x + y = x + x + y ============================ True
A proof term provided after :=
can mention these bound variables
(that are automatically introduced with the given names).
Since the i_pattern
can be omitted, to avoid ambiguity,
bound variables can be surrounded
with parentheses even if no type is specified:
 have (x) : 2 * x = x + x by lia.
 1 goal ============================ (forall x : nat, 2 * x = x + x) > True
The i_item
and s_item
can be used to interpret the asserted
hypothesis with views (see Section Views and reflection) or simplify the resulting
goals.
The have
tactic also supports a suff
modifier that allows for
asserting that a given statement implies the current goal without
copying the goal itself.
Example
 have suff H : 2 + 2 = 3; last first.
 2 goals H : 2 + 2 = 3 > True ============================ True goal 2 is: 2 + 2 = 3 > True
Note that H is introduced in the second goal.
The suff
modifier is not
compatible with the presence of a list of binders.
Generating let in context entries with have¶
Since SSReflect 1.5, the have
tactic supports a “transparent” modifier
to generate let in
context entries: the @
symbol in front of the
context entry name.
Example
 Set Printing Depth 15.
 Inductive Ord n := Sub x of x < n.
 Ord is defined Ord_rect is defined Ord_ind is defined Ord_rec is defined Ord_sind is defined
 Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
 Arguments Sub {_} _ _.
 Lemma test n m (H : m + 1 < n) : True.
 1 goal n, m : nat H : m + 1 < n ============================ True
 have @i : 'I_n by apply: (Sub m); lia.
 1 goal n, m : nat H : m + 1 < n i := Sub m (ZifyClasses.rew_iff_rev (m < n) (Z.of_nat m < Z.of_nat n)%Z (ZifyClasses.mkrel nat Z lt Z.of_nat Z.lt Nat2Z.inj_lt m (...) eq_refl n (...) eq_refl) (let H0 : ...%Z := ... in ... ...)) : 'I_n ============================ True
Note that the subterm produced by lia
is in general huge and
uninteresting, and hence one may want to hide it.
For this purpose the [: name]
intro pattern and the tactic
abstract
(see The abstract tactic) are provided.
Example
 Lemma test n m (H : m + 1 < n) : True.
 1 goal n, m : nat H : m + 1 < n ============================ True
 have [:pm] @i : 'I_n by apply: (Sub m); abstract: pm; lia.
 1 goal n, m : nat H : m + 1 < n pm : m < n (*1*) i := Sub m pm : 'I_n ============================ True
The type of pm
can be cleaned up by its annotation (*1*)
by just
simplifying it. The annotations are there for technical reasons only.
When intro patterns for abstract constants are used in conjunction with`` have`` and an explicit term, they must be used as follows:
Example
 Lemma test n m (H : m + 1 < n) : True.
 1 goal n, m : nat H : m + 1 < n ============================ True
 have [:pm] @i : 'I_n := Sub m pm.
 2 goals n, m : nat H : m + 1 < n ============================ S m <= n goal 2 is: True
 by lia.
 1 goal n, m : nat H : m + 1 < n pm : S m <= n (*1*) i := Sub m pm : 'I_n : 'I_n ============================ True
In this case, the abstract constant pm
is assigned by using it in
the term that follows :=
and its corresponding goal is left to be
solved. Goals corresponding to intro patterns for abstract constants
are opened in the order in which the abstract constants are declared
(not in the “order” in which they are used in the term).
Note that abstract constants do respect scopes. Hence, if a variable is declared after their introduction, it has to be properly generalized (i.e., explicitly passed to the abstract constant when one makes use of it).
Example
 Lemma test n m (H : m + 1 < n) : True.
 1 goal n, m : nat H : m + 1 < n ============================ True
 have [:pm] @i k : 'I_(n+k) by apply: (Sub m); abstract: pm k; lia.
 1 goal n, m : nat H : m + 1 < n pm : (forall k : nat, m < n + k) (*1*) i := fun k : nat => Sub m (pm k) : forall k : nat, 'I_(n + k) ============================ True
Last, notice that the use of intro patterns for abstract constants is
orthogonal to the transparent flag @
for have
.
The have tactic and typeclass resolution¶
Since SSReflect 1.5, the have
tactic behaves as follows with respect to
typeclass inference.
 Axiom ty : Type.
 ty is declared
 Axiom t : ty.
 t is declared
 Goal True.
 1 goal ============================ True
 have foo : ty.
 2 goals ============================ ty goal 2 is: True
Full inference for
ty
. The first subgoal demands a proof of such instantiated statement.have foo : ty := .No inference for
ty
. Unresolved instances are quantified inty
. The first subgoal demands a proof of such quantified statement. Note that no proof term follows:=
; hence two subgoals are generated.
 have foo : ty := t.
 1 goal foo : ty ============================ True
No inference for
ty
andt
.
 have foo := t.
 1 goal foo : ty ============================ True
No inference for
t
. Unresolved instances are quantified in the (inferred) type oft
and abstracted int
.
Variants: the suff and wlog tactics¶
As is often the case in mathematical textbooks, forward reasoning may be used in slightly different variants. One of these variants is to show that the intermediate step L easily implies the initial goal G. By easily we mean here that the proof of L ⇒ G is shorter than the one of L itself. This kind of reasoning step usually starts with: “It suffices to show that …”.
This is such a frequent way of reasoning that SSReflect has a variant
of the have
tactic called suffices
(whose abridged name is suff
).
The
have
and suff
tactics are equivalent and have the same syntax but:
the order of the generated subgoals is inverted;
the optional clear item is still performed in the second branch, which means that the tactic:
suff {H} H : forall x : nat, x >= 0.fails if the context of the current goal indeed contains an assumption named
H
.
The rationale of this clearing policy is to make possible “trivial” refinements of an assumption, without changing its name in the main branch of the reasoning.
The have
modifier can follow the suff
tactic.
Example
 Axioms G P : Prop.
 G is declared P is declared
 Lemma test : G.
 1 goal ============================ G
 suff have H : P.
 2 goals H : P ============================ G goal 2 is: (P > G) > G
Note that, in contrast with have suff
, the name H has been introduced
in the first goal.
Another useful construct is reduction, showing that a particular case
is in fact general enough to prove a general property. This kind of
reasoning step usually starts with: “Without loss of generality, we
can suppose that …”. Formally, this corresponds to the proof of a goal
G
by introducing a cut: wlog_statement > G
. Hence the user shall
provide a proof for both (wlog_statement > G) > G
and
wlog_statement > G
. However, such cuts are usually rather
painful to perform by
hand, because the statement wlog_statement
is tedious to write by hand,
and sometimes even to read.
SSReflect implements this kind of reasoning step through the without loss
tactic, whose short name is wlog
. It offers support to describe
the shape of the cut statements, by providing the simplifying
hypothesis and by pointing at the elements of the initial goals that
should be generalized. The general syntax of without loss is:
 Tactic wlog suff? clear_switch? i_item? : ident* / term¶
 Tactic without loss suff? clear_switch? i_item? : ident* / term¶
where each ident
is a constant in the context
of the goal. Open syntax is supported for term
.
In its defective form:
on a goal G, it creates two subgoals: a first one to prove the formula (term > G) > G and a second one to prove the formula term > G.
If the optional list of ident
is present
on the left side of /
, these constants are generalized in the
premise (term > G) of the first subgoal. By default bodies of local
definitions are erased. This behavior can be inhibited by prefixing the
name of the local definition with the @
character.
In the second subgoal, the tactic:
is performed if at least one of these optional switches is present in
the wlog
tactic.
The wlog
tactic is specially useful when a symmetry argument
simplifies a proof. Here is an example showing the beginning of the
proof that quotient and reminder of natural number euclidean division
are unique.
Example
 Lemma quo_rem_unicity d q1 q2 r1 r2 : q1*d + r1 = q2*d + r2 > r1 < d > r2 < d > (q1, r1) = (q2, r2).
 1 goal d, q1, q2, r1, r2 : nat ============================ q1 * d + r1 = q2 * d + r2 > r1 < d > r2 < d > (q1, r1) = (q2, r2)
 wlog: q1 q2 r1 r2 / q1 <= q2.
 2 goals d, q1, q2, r1, r2 : nat ============================ (forall q3 q4 r3 r4 : nat, q3 <= q4 > q3 * d + r3 = q4 * d + r4 > r3 < d > r4 < d > (q3, r3) = (q4, r4)) > q1 * d + r1 = q2 * d + r2 > r1 < d > r2 < d > (q1, r1) = (q2, r2) goal 2 is: q1 <= q2 > q1 * d + r1 = q2 * d + r2 > r1 < d > r2 < d > (q1, r1) = (q2, r2)
 by case (le_gt_dec q1 q2)=> H; last symmetry; eauto with arith.
 1 goal d, q1, q2, r1, r2 : nat ============================ q1 <= q2 > q1 * d + r1 = q2 * d + r2 > r1 < d > r2 < d > (q1, r1) = (q2, r2)
The wlog suff
variant is simpler, since it cuts wlog_statement
instead
of wlog_statement > G
. It thus opens the goals
wlog_statement > G
and wlog_statement
.
In its simplest form, the generally have : …
tactic is equivalent to
wlog suff : …
followed by last first
. When the have
tactic is used
with the generally
(or gen
) modifier, it accepts an extra identifier
followed by a comma before the usual intro pattern. The identifier
will name the new hypothesis in its more general form, while the intro
pattern will be used to process its instance.
Example
 From Coq Require Import ssreflect ssrfun ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom P : nat > Prop.
 P is declared
 Axioms eqn leqn : nat > nat > bool.
 eqn is declared leqn is declared
 Declare Scope this_scope.
 Notation "a != b" := (eqn a b) (at level 70) : this_scope.
 Notation "a <= b" := (leqn a b) (at level 70) : this_scope.
 Open Scope this_scope.
 Lemma simple n (ngt0 : 0 < n ) : P n.
 1 goal n : nat ngt0 : 0 < n ============================ P n
 gen have ltnV, /andP[nge0 neq0] : n ngt0 / (0 <= n) && (n != 0); last first.
 2 goals n : nat ngt0 : 0 < n ltnV : forall n : nat, 0 < n > (0 <= n) && (n != 0) nge0 : 0 <= n neq0 : n != 0 ============================ P n goal 2 is: (0 <= n) && (n != 0)
Advanced generalization¶
The complete syntax for the items on the left hand side of the /
separator is the following one:
 Variant wlog … : clear_switch@?ident( @?ident := c_pattern)? / term
Clear operations are intertwined with generalization operations. This helps in particular avoiding dependency issues while generalizing some facts.
If an ident
is prefixed with the @
mark, then a letin redex is
created, which keeps track of its body (if any). The syntax
(ident := c_pattern)
allows to generalize an arbitrary term using a
given name. Note that its simplest form (x := y)
is just a renaming of
y
into x
. In particular, this can be useful in order to simulate the
generalization of a section variable, otherwise not allowed. Indeed,
renaming does not require the original variable to be cleared.
The syntax (@x := y)
generates a letin abstraction but with the
following caveat: x
will not bind y
, but its body, whenever y
can be
unfolded. This covers the case of both local and global definitions, as
illustrated in the following example.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable x : nat.
 x is declared
 Definition addx z := z + x.
 addx is defined
 Lemma test : x <= addx x.
 1 goal x : nat ============================ x <= addx x
 wlog H : (y := x) (@twoy := addx x) / twoy = 2 * y.
 2 goals x : nat ============================ (forall y : nat, let twoy := y + y in twoy = 2 * y > y <= twoy) > x <= addx x goal 2 is: y <= twoy
To avoid unfolding the term captured by the pattern add x
, one can use
the pattern id (addx x)
, which would produce the following first
subgoal
 From Coq Require Import ssreflect Lia.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable x : nat.
 x is declared
 Definition addx z := z + x.
 addx is defined
 Lemma test : x <= addx x.
 1 goal x : nat ============================ x <= addx x
 wlog H : (y := x) (@twoy := id (addx x)) / twoy = 2 * y.
 2 goals x : nat ============================ (forall y : nat, let twoy := addx y in twoy = 2 * y > y <= addx y) > x <= addx x goal 2 is: y <= addx y
Rewriting¶
The generalized use of reflection implies that most of the
intermediate results handled are properties of effectively computable
functions. The most efficient means of establishing such results are
computation and simplification of expressions involving such
functions, i.e., rewriting. SSReflect therefore includes an
extended rewrite
tactic that unifies and combines most of the
rewriting functionalities.
An extended rewrite tactic¶
The main features of the rewrite tactic are:
it can perform an entire series of such operations in any subset of the goal and/or context;
it allows to perform rewriting, simplifications, folding/unfolding of definitions, closing of goals;
several rewriting operations can be chained in a single tactic;
control over the occurrence at which rewriting is to be performed is significantly enhanced.
The general form of an SSReflect rewrite tactic is:
The combination of a rewrite tactic with the in
tactical (see Section
Localization) performs rewriting in both the context and the goal.
A rewrite step rstep
has the general form:
::=
r_prefix? r_item
r_prefix::=
? mult? occ_switchclear_switch? [ r_pattern ]?
r_pattern::=
termin ident in? termterm interm as ident in term
r_item::=
/? terms_item
An r_prefix
contains annotations to qualify where and how the rewrite
operation should be performed.
The optional initial

indicates the direction of the rewriting ofr_item
: if present, the direction is righttoleft and it is lefttoright otherwise.The multiplier
mult
(see Section Iteration) specifies if and how the rewrite operation should be repeated.A rewrite operation matches the occurrences of a rewrite pattern, and replaces these occurrences by another term, according to the given
r_item
. The optional redex switch[r_pattern]
, which should always be surrounded by brackets, gives explicitly this rewrite pattern. In its simplest form, it is a regular term. If no explicit redex switch is present, the rewrite pattern to be matched is inferred from ther_item
.This optional term, or the
r_item
, may be preceded by anocc_switch
(see Section Selectors) or aclear_switch
(see Section Discharge), these two possibilities being exclusive.An occurrence switch selects the occurrences of the rewrite pattern that should be affected by the rewrite operation.
A clear switch, even an empty one, is performed after the
r_item
is actually processed and is complemented with the name of the rewrite rule if and only if it is a simple proof context entry 20. As a consequence, one can writerewrite {}H
to rewrite withH
and disposeH
immediately afterwards. This behavior can be avoided by putting parentheses around the rewrite rule.
A r_item
can be one of the following.
A simplification
r_item
, represented by as_item
(see Section Introduction in the context). Simplification operations are intertwined with the possible other rewrite operations specified by the list ofr_item
.A folding/unfolding
r_item
. The tacticrewrite /term
unfolds the head constant ofterm
in every occurrence of the first matching ofterm
in the goal. In particular, ifmy_def
is a (local or global) defined constant, the tacticrewrite /my_def.
is analogous to:unfold my_def
. Conversely,rewrite /my_def.
is equivalent tofold my_def
. When an unfoldr_item
is combined with a redex pattern, a conversion operation is performed. A tactic of the formrewrite [term1]/term2.
is equivalent tochange term1 with term2.
Ifterm2
is a single constant andterm1
head symbol is notterm2
, then the head symbol ofterm1
is repeatedly unfolded untilterm2
appears. A
term
can be: a term whose type has the form:
forall (x1 : A1 )…(xn : An ), eq term1 term2
, whereeq
is the Leibniz equality or a registered setoid equality;a list of terms
(t1 ,…,tn)
, eachti
having a type as above, and the tacticrewrite r_prefix (t1 ,…,tn ).
is equivalent todo [rewrite r_prefix t1  …  rewrite r_prefix tn ].
;an anonymous rewrite lemma
(_ : term)
, whereterm
has a type as above.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Definition double x := x + x.
 double is defined
 Definition ddouble x := double (double x).
 ddouble is defined
 Lemma test x : ddouble x = 4 * x.
 1 goal x : nat ============================ ddouble x = 4 * x
 rewrite [ddouble _]/double.
 1 goal x : nat ============================ double x + double x = 4 * x
Warning
The SSReflect terms containing holes are not typed as abstractions in this context. Hence the following script fails.
 Definition f := fun x y => x + y.
 f is defined
 Lemma test x y : x + y = f y x.
 1 goal x, y : nat ============================ x + y = f y x
 rewrite [f y]/(y + _).
 Toplevel input, characters 022: > rewrite [f y]/(y + _). > ^^^^^^^^^^^^^^^^^^^^^^ Error: fold pattern (y + _) does not match redex (f y)
but the following script succeeds
 rewrite [f y x]/(y + _).
 1 goal x, y : nat ============================ x + y = y + x
 A
 Flag SsrOldRewriteGoalsOrder¶
Controls the order in which generated subgoals (side conditions) are added to the proof context. The flag is off by default, which puts subgoals generated by conditional rules first, followed by the main goal. When it is on, the main goal appears first. If your proofs are organized to complete proving the main goal before side conditions, turning the flag on will save you from having to add
last first
tactics that would be needed to keep the main goal as the currently focused goal.
Remarks and examples¶
Rewrite redex selection¶
The general strategy of SSReflect is to grasp as many redexes as possible and to let the user select the ones to be rewritten thanks to the improved syntax for the control of rewriting.
This may be a source of incompatibilities between the two rewrite tactics.
In a rewrite tactic of the form:
term1
is the explicit rewrite redex and term2
is the rewrite rule.
This execution of this tactic unfolds as follows.
First
term1
andterm2
are βι normalized. Thenterm2
is put in head normal form if the Leibniz equality constructoreq
is not the head symbol. This may involve ζ reductions.Then, the matching algorithm (see Section Abbreviations) determines the first subterm of the goal matching the rewrite pattern. The rewrite pattern is given by
term1
, if an explicit redex pattern switch is provided, or by the type ofterm2
otherwise. However, matching skips over matches that would lead to trivial rewrites. All the occurrences of this subterm in the goal are candidates for rewriting.Then only the occurrences coded by
occ_switch
(see again Section Abbreviations) are finally selected for rewriting.The lefthand side of
term2
is unified with the subterm found by the matching algorithm, and if this succeeds, all the selected occurrences in the goal are replaced by the righthand side ofterm2
.Finally the goal is βι normalized.
In the case term2
is a list of terms, the first topdown (in the
goal) lefttoright (in the list) matching rule gets selected.
Chained rewrite steps¶
The possibility to chain rewrite operations in a single tactic makes scripts more compact and gathers in a single command line a bunch of surgical operations that would be described by a one sentence in a pen and paper proof.
Performing rewrite and simplification operations in a single tactic enhances significantly the concision of scripts. For instance the tactic:
unfolds my_def
in the goal, simplifies the second occurrence of the
first subterm matching pattern [f _]
, rewrites my_eq
, simplifies the
goals and closes trivial goals.
Here are some concrete examples of chained rewrite operations, in the proof of basic results on natural numbers arithmetic.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom addn0 : forall m, m + 0 = m.
 addn0 is declared
 Axiom addnS : forall m n, m + S n = S (m + n).
 addnS is declared
 Axiom addSnnS : forall m n, S m + n = m + S n.
 addSnnS is declared
 Lemma addnCA m n p : m + (n + p) = n + (m + p).
 1 goal m, n, p : nat ============================ m + (n + p) = n + (m + p)
 by elim: m p => [  m Hrec] p; rewrite ?addSnnS ?addnS.
 No more goals.
 Qed.
 Lemma addnC n m : m + n = n + m.
 1 goal n, m : nat ============================ m + n = n + m
 by rewrite {1}[n]addn0 addnCA addn0.
 No more goals.
 Qed.
Note the use of the ?
switch for parallel rewrite operations in the
proof of addnCA
.
Explicit redex switches are matched first¶
If an r_prefix
involves a redex switch, the first step is to find a
subterm matching this redex pattern, independently from the lefthand
side of the equality the user wants to rewrite.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test (H : forall t u, t + u = u + t) x y : x + y = y + x.
 1 goal H : forall t u : nat, t + u = u + t x, y : nat ============================ x + y = y + x
 rewrite [y + _]H.
 1 goal H : forall t u : nat, t + u = u + t x, y : nat ============================ x + y = x + y
Note that if this first pattern matching is not compatible with the
r_item
, the rewrite fails, even if the goal contains a
correct redex matching both the redex switch and the lefthand side of
the equality.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test (H : forall t u, t + u * 0 = t) x y : x + y * 4 + 2 * 0 = x + 2 * 0.
 1 goal H : forall t u : nat, t + u * 0 = t x, y : nat ============================ x + y * 4 + 2 * 0 = x + 2 * 0
 Fail rewrite [x + _]H.
 The command has indeed failed with message: pattern (x + y * 4) does not match LHS of H
Indeed, the lefthand side of H
does not match
the redex identified by the pattern x + y * 4
.
Occurrence switches and redex switches¶
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test x y : x + y + 0 = x + y + y + 0 + 0 + (x + y + 0).
 1 goal x, y : nat ============================ x + y + 0 = x + y + y + 0 + 0 + (x + y + 0)
 rewrite {2}[_ + y + 0](_: forall z, z + 0 = z).
 2 goals x, y : nat ============================ forall z : nat, z + 0 = z goal 2 is: x + y + 0 = x + y + y + 0 + 0 + (x + y)
The second subgoal is generated by the use of an anonymous lemma in
the rewrite tactic. The effect of the tactic on the initial goal is to
rewrite this lemma at the second occurrence of the first matching
x + y + 0
of the explicit rewrite redex _ + y + 0
.
Occurrence selection and repetition¶
Occurrence selection has priority over repetition switches. This means the repetition of a rewrite tactic specified by a multiplier will perform matching each time an elementary rewrite operation is performed. Repeated rewrite tactics apply to every subgoal generated by the previous tactic, including the previous instances of the repetition.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Lemma test x y (z : nat) : x + 1 = x + y + 1.
 1 goal x, y, z : nat ============================ x + 1 = x + y + 1
 rewrite 2!(_ : _ + 1 = z).
 4 goals x, y, z : nat ============================ x + 1 = z goal 2 is: z = z goal 3 is: x + y + 1 = z goal 4 is: z = z
This last tactic generates three subgoals because
the second rewrite operation specified with the 2!
multiplier
applies to the two subgoals generated by the first rewrite.
Multirule rewriting¶
The rewrite tactic can be provided a tuple of rewrite rules, or more generally a tree of such rules, since this tuple can feature arbitrary inner parentheses. We call multirule such a generalized rewrite rule. This feature is of special interest when it is combined with multiplier switches, which makes the rewrite tactic iterate the rewrite operations prescribed by the rules on the current goal.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables (a b c : nat).
 a is declared b is declared c is declared
 Hypothesis eqab : a = b.
 eqab is declared
 Hypothesis eqac : a = c.
 eqac is declared
 Lemma test : a = a.
 1 goal a, b, c : nat eqab : a = b eqac : a = c ============================ a = a
 rewrite (eqab, eqac).
 1 goal a, b, c : nat eqab : a = b eqac : a = c ============================ b = b
Indeed, rule eqab
is the first to apply among the ones
gathered in the tuple passed to the rewrite tactic. This multirule
(eqab, eqac)
is actually a Coq term and we can name it with a
definition:
 Definition multi1 := (eqab, eqac).
 multi1 is defined
In this case, the tactic rewrite multi1
is a synonym for
rewrite (eqab, eqac)
.
More precisely, a multirule rewrites the first subterm to which one of the rules applies in a lefttoright traversal of the goal, with the first rule from the multirule tree in lefttoright order. Matching is performed according to the algorithm described in Section Abbreviations, but literal matches have priority.
Example
 Definition d := a.
 d is defined
 Hypotheses eqd0 : d = 0.
 eqd0 is declared
 Definition multi2 := (eqab, eqd0).
 multi2 is defined
 Lemma test : d = b.
 1 goal a, b, c : nat eqab : a = b eqac : a = c eqd0 : d = 0 ============================ d = b
 rewrite multi2.
 1 goal a, b, c : nat eqab : a = b eqac : a = c eqd0 : d = 0 ============================ 0 = b
Indeed, rule eqd0
applies without unfolding the
definition of d
.
For repeated rewrites, the selection process is repeated anew.
Example
 Hypothesis eq_adda_b : forall x, x + a = b.
 eq_adda_b is declared
 Hypothesis eq_adda_c : forall x, x + a = c.
 eq_adda_c is declared
 Hypothesis eqb0 : b = 0.
 eqb0 is declared
 Definition multi3 := (eq_adda_b, eq_adda_c, eqb0).
 multi3 is defined
 Lemma test : 1 + a = 12 + a.
 1 goal a, b, c : nat eqab : a = b eqac : a = c eqd0 : d = 0 eq_adda_b : forall x : nat, x + a = b eq_adda_c : forall x : nat, x + a = c eqb0 : b = 0 ============================ 1 + a = 12 + a
 rewrite 2!multi3.
 1 goal a, b, c : nat eqab : a = b eqac : a = c eqd0 : d = 0 eq_adda_b : forall x : nat, x + a = b eq_adda_c : forall x : nat, x + a = c eqb0 : b = 0 ============================ 0 = 12 + a
It uses eq_adda_b
then eqb0
on the lefthand
side only. Without the bound 2
, one would obtain 0 = 0
.
The grouping of rules inside a multirule does not affect the selection
strategy, but can make it easier to include one rule set in another or
to (universally) quantify over the parameters of a subset of rules (as
there is special code that will omit unnecessary quantifiers for rules
that can be syntactically extracted). It is also possible to reverse
the direction of a rule subset, using a special dedicated syntax: the
tactic rewrite (=^~ multi1)
is equivalent to rewrite multi1_rev
.
Example
 Hypothesis eqba : b = a.
 eqba is declared
 Hypothesis eqca : c = a.
 eqca is declared
 Definition multi1_rev := (eqba, eqca).
 multi1_rev is defined
except that the constants eqba
, eqab
and mult1_rev
have not been created.
Rewriting with multirules is useful to implement simplification or
transformation procedures, to be applied on terms of small to medium
size. For instance, the library ssrnat
(Mathematical Components library)
provides two implementations
for arithmetic operations on natural numbers: an elementary one and a
tail recursive version, less inefficient but also less convenient for
reasoning purposes. The library also provides one lemma per such
operation, stating that both versions return the same values when
applied to the same arguments:
The operation on the lefthand side of each lemma is the efficient
version, and the corresponding naive implementation is on the righthand side.
In order to reason conveniently on expressions involving
the efficient operations, we gather all these rules in the definition
trecE
:
The tactic rewrite !trecE.
restores the naive version of each operation in a goal involving the
efficient ones, e.g., for the purpose of a correctness proof.
Wildcards vs abstractions¶
The rewrite tactic supports r_item
s containing holes. For example, in
the tactic rewrite (_ : _ * 0 = 0).
,
the term _ * 0 = 0
is interpreted as forall n : nat, n * 0 = 0.
Anyway this tactic is not equivalent to
rewrite (_ : forall x, x * 0 = 0).
.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test y z : y * 0 + y * (z * 0) = 0.
 1 goal y, z : nat ============================ y * 0 + y * (z * 0) = 0
 rewrite (_ : _ * 0 = 0).
 2 goals y, z : nat ============================ y * 0 = 0 goal 2 is: 0 + y * (z * 0) = 0
while the other tactic results in
 rewrite (_ : forall x, x * 0 = 0).
 2 goals y, z : nat ============================ forall x : nat, x * 0 = 0 goal 2 is: 0 + y * (z * 0) = 0
The first tactic requires you to prove the instance of the (missing) lemma that was used, while the latter requires you prove the quantified form.
When SSReflect rewrite fails on standard Coq licit rewrite¶
In a few cases, the SSReflect rewrite tactic fails rewriting some redexes that standard Coq successfully rewrites. There are two main cases.
SSReflect never accepts to rewrite indeterminate patterns like:
Lemma foo (x : unit) : x = tt.SSReflect will however accept the ηζ expansion of this rule:
Lemma fubar (x : unit) : (let u := x in u) = tt.The standard rewrite tactic provided by Coq uses a different algorithm to find instances of the rewrite rule.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable g : nat > nat.
 g is declared
 Definition f := g.
 f is defined
 Axiom H : forall x, g x = 0.
 H is declared
 Lemma test : f 3 + f 3 = f 6.
 1 goal g : nat > nat ============================ f 3 + f 3 = f 6
 (* we call the standard rewrite tactic here *)
 rewrite > H.
 1 goal g : nat > nat ============================ 0 + 0 = f 6
This rewriting is not possible in SSReflect, because there is no occurrence of the head symbol
f
of the rewrite rule in the goal. rewrite H.
 Toplevel input, characters 09: > rewrite H. > ^^^^^^^^^ Error: The LHS of H (g _) does not match any subterm of the goal
Rewriting with
H
first requires unfolding the occurrences off
where the substitution is to be performed (here there is a single such occurrence), using tacticrewrite /f
(for a global replacement off
byg
) orrewrite pattern/f
, for a finer selection. rewrite /f H.
 1 goal g : nat > nat ============================ 0 + 0 = g 6
Alternatively, one can override the pattern inferred from
H
 rewrite [f _]H.
 1 goal g : nat > nat ============================ 0 + 0 = f 6
Existential metavariables and rewriting¶
The rewrite tactic will not instantiate existing existential metavariables when matching a redex pattern.
If a rewrite rule generates a goal with new existential metavariables
in the Prop
sort, these will be generalized as for apply
(see The apply tactic) and
corresponding new goals will be generated.
Example
 From Coq Require Import ssreflect ssrfun ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Set Warnings "notationoverridden".
 Axiom leq : nat > nat > bool.
 leq is declared
 Notation "m <= n" := (leq m n) : nat_scope.
 Notation "m < n" := (S m <= n) : nat_scope.
 Inductive Ord n := Sub x of x < n.
 Ord is defined Ord_rect is defined Ord_ind is defined Ord_rec is defined Ord_sind is defined
 Notation "'I_ n" := (Ord n) (at level 8, n at level 2, format "''I_' n").
 Arguments Sub {_} _ _.
 Definition val n (i : 'I_n) := let: Sub a _ := i in a.
 val is defined
 Definition insub n x := if @idP (x < n) is ReflectT _ Px then Some (Sub x Px) else None.
 insub is defined
 Axiom insubT : forall n x Px, insub n x = Some (Sub x Px).
 insubT is declared
 Lemma test (x : 'I_2) y : Some x = insub 2 y.
 1 goal x : 'I_2 y : nat ============================ Some x = insub 2 y
 rewrite insubT.
 2 goals x : 'I_2 y : nat ============================ forall Hyp0 : y < 2, Some x = Some (Sub y Hyp0) goal 2 is: y < 2
Since the argument corresponding to Px
is not supplied by the user, the
resulting goal should be Some x = Some (Sub y ?Goal).
Instead, SSReflect rewrite
tactic hides the existential variable.
As in The apply tactic, the ssrautoprop
tactic is used to try to
solve the existential variable.
 Lemma test (x : 'I_2) y (H : y < 2) : Some x = insub 2 y.
 1 goal x : 'I_2 y : nat H : y < 2 ============================ Some x = insub 2 y
 rewrite insubT.
 1 goal x : 'I_2 y : nat H : y < 2 ============================ Some x = Some (Sub y H)
As a temporary limitation, this behavior is available only if the rewriting rule is stated using Leibniz equality (as opposed to setoid relations). It will be extended to other rewriting relations in the future.
Rewriting under binders¶
Goals involving objects defined with higherorder functions often require "rewriting under binders". While setoid rewriting is a possible approach in this case, it is common to use regular rewriting along with dedicated extensionality lemmas. This may cause some practical issues during the development of the corresponding scripts, notably as we might be forced to provide the rewrite tactic with complete terms, as shown by the simple example below.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Axiom subnn : forall n : nat, n  n = 0.
 subnn is declared
 Parameter map : (nat > nat) > list nat > list nat.
 map is declared
 Parameter sumlist : list nat > nat.
 sumlist is declared
 Axiom eq_map : forall F1 F2 : nat > nat, (forall n : nat, F1 n = F2 n) > forall l : list nat, map F1 l = map F2 l.
 eq_map is declared
 Lemma example_map l : sumlist (map (fun m => m  m) l) = 0.
 1 goal l : list nat ============================ sumlist (map (fun m : nat => m  m) l) = 0
In this context, one cannot directly use eq_map
:
 rewrite eq_map.
 Toplevel input, characters 014: > rewrite eq_map. > ^^^^^^^^^^^^^^ Error: Unable to find an instance for the variable F2. Rule's type: (forall F1 F2 : nat > nat, (forall n : nat, F1 n = F2 n) > forall l : list nat, map F1 l = map F2 l)
as we need to explicitly provide the noninferable argument F2
,
which corresponds here to the term we want to obtain after the
rewriting step. In order to perform the rewrite step, one has to
provide the term by hand as follows:
 rewrite (@eq_map _ (fun _ : nat => 0)).
 2 goals l : list nat ============================ forall n : nat, n  n = 0 goal 2 is: sumlist (map (fun _ : nat => 0) l) = 0
 by move=> m; rewrite subnn.
 1 goal l : list nat ============================ sumlist (map (fun _ : nat => 0) l) = 0
The under
tactic lets one perform the same operation in a more
convenient way:
 Lemma example_map l : sumlist (map (fun m => m  m) l) = 0.
 1 goal l : list nat ============================ sumlist (map (fun m : nat => m  m) l) = 0
 under eq_map => m do rewrite subnn.
 1 goal l : list nat ============================ sumlist (map (fun _ : nat => 0) l) = 0
The under tactic¶
The convenience under
tactic supports the following syntax:
 Tactic under r_prefix? term => i_item+? do tactic[ tactic* ]?¶
It operates under the context proved to be extensional by lemma
term
. Error Incorrect number of tactics (expected N tactics, was given M).¶
This error can occur when using the version with a
do
clause.
The multiplier part of
r_prefix
is not supported.
We distinguish two modes:
interactive mode, without a do
clause, and
oneliner mode, with a do
clause,
which are explained in more detail below.
Interactive mode¶
Let us redo the running example in interactive mode.
Example
 Lemma example_map l : sumlist (map (fun m => m  m) l) = 0.
 1 goal l : list nat ============================ sumlist (map (fun m : nat => m  m) l) = 0
 under eq_map => m.
 2 focused goals (shelved: 1) l : list nat m : nat ============================ 'Under[ m  m ] goal 2 is: sumlist (map ?Goal l) = 0
 rewrite subnn.
 2 focused goals (shelved: 1) l : list nat m : nat ============================ 'Under[ 0 ] goal 2 is: sumlist (map ?Goal l) = 0
 over.
 1 goal l : list nat ============================ sumlist (map (fun _ : nat => 0) l) = 0
The execution of the Ltac expression:
under term => [ i_item_{1}  …  i_item_{n} ].
involves the following steps.
It performs a
rewrite term
without failing like in the first example withrewrite eq_map.
, but creating evars (seeevar
). Ifterm
is prefixed by a pattern or an occurrence selector, then the modifiers are honoured.As an nbranch intro pattern is provided,
under
checks that n+1 subgoals have been created. The last one is the main subgoal, while the other ones correspond to premises of the rewrite rule (such asforall n, F1 n = F2 n
foreq_map
).If so,
under
puts these n goals in head normal form (using the defective form of the tacticmove
), then executes the corresponding intro patterni_pattern_{i}
in each goal.Then,
under
checks that the first n subgoals are (quantified) Leibniz equalities, double implications or registered relations (w.r.t. ClassRewriteRelation
) between a term and an evar, e.g.,m  m = ?F2 m
in the running example. (This support for setoidlike relations is enabled as soon as one does bothRequire Import ssreflect.
andRequire Setoid.
)If so
under
protects these n goals against an accidental instantiation of the evar. These protected goals are displayed using the'Under[ … ]
notation (e.g.'Under[ m  m ]
in the running example).The expression inside the
'Under[ … ]
notation can be proved equivalent to the desired expression by using a regularrewrite
tactic.Interactive editing of the first n goals has to be signalled by using the
over
tactic or rewrite rule (see below), which requires that the underlying relation is reflexive. (The running example deals with Leibniz equality, butPreOrder
relations are also supported, for example.)Finally, a postprocessing step is performed in the main goal to keep the name(s) for the bound variables chosen by the user in the intro pattern for the first branch.
The over tactic¶
Two equivalent facilities (a terminator and a lemma) are provided to
close intermediate subgoals generated by under
(i.e., goals
displayed as 'Under[ … ]
):
 Tactic over¶
This terminator tactic allows one to close goals of the form
'Under[ … ]
.
 Variant by rewrite over
This is a variant of
over
in order to close'Under[ … ]
goals, relying on theover
rewrite rule.
Note that a rewrite rule UnderE
is available as well, if one wants
to "unprotect" the evar, without closing the goal automatically (e.g.,
to instantiate it manually with another rule than reflexivity).
Oneliner mode¶
The Ltac expression:
under term => [ i_item_{1}  …  i_item_{n} ] do [ tactic_{1}  …  tactic_{n} ].
can be seen as a shorter form for the following expression:
(under term) => [ i_item_{1}  …  i_item_{n}  ]; [ tactic_{1}; over  …  tactic_{n}; over  cbv beta iota ].
Notes:
The
betaiota
reduction here is useful to get rid of the beta redexes that could be introduced after the substitution of the evars by theunder
tactic.Note that the provided tactics can as well involve other
under
tactics. See below for a typical example involving thebigop
theory from the Mathematical Components library.If there is only one tactic, the brackets can be omitted, e.g.:
under term => i do tactic.
and that shorter form should be preferred.If the
do
clause is provided and the intro pattern is omitted, then the defaulti_item
*
is applied to each branch. E.g., the Ltac expressionunder term do [ tactic_{1}  …  tactic_{n} ]
is equivalent tounder term => [ *  …  * ] do [ tactic_{1}  …  tactic_{n} ]
(and it can be noted here that theunder
tactic performs amove.
before processing the intro patterns=> [ *  …  * ]
).
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Coercion is_true : bool >> Sortclass.
 is_true is now a coercion
 Reserved Notation "\big [ op / idx ]_ ( m <= i < n  P ) F" (at level 36, F at level 36, op, idx at level 10, m, i, n at level 50, format "'[' \big [ op / idx ]_ ( m <= i < n  P ) F ']'").
 Variant bigbody (R I : Type) : Type := BigBody : forall (_ : I) (_ : forall (_ : R) (_ : R), R) (_ : bool) (_ : R), bigbody R I.
 bigbody is defined
 Parameter bigop : forall (R I : Type) (_ : R) (_ : list I) (_ : forall _ : I, bigbody R I), R.
 bigop is declared
 Axiom eq_bigr_ : forall (R : Type) (idx : R) (op : forall (_ : R) (_ : R), R) (I : Type) (r : list I) (P : I > bool) (F1 F2 : I > R), (forall x : I, is_true (P x) > F1 x = F2 x) > bigop idx r (fun i : I => BigBody i op (P i) (F1 i)) = bigop idx r (fun i : I => BigBody i op (P i) (F2 i)).
 eq_bigr_ is declared
 Axiom eq_big_ : forall (R : Type) (idx : R) (op : R > R > R) (I : Type) (r : list I) (P1 P2 : I > bool) (F1 F2 : I > R), (forall x : I, P1 x = P2 x) > (forall i : I, is_true (P1 i) > F1 i = F2 i) > bigop idx r (fun i : I => BigBody i op (P1 i) (F1 i)) = bigop idx r (fun i : I => BigBody i op (P2 i) (F2 i)).
 eq_big_ is declared
 Reserved Notation "\sum_ ( m <= i < n  P ) F" (at level 41, F at level 41, i, m, n at level 50, format "'[' \sum_ ( m <= i < n  P ) '/ ' F ']'").
 Parameter index_iota : nat > nat > list nat.
 index_iota is declared
 Notation "\big [ op / idx ]_ ( m <= i < n  P ) F" := (bigop idx (index_iota m n) (fun i : nat => BigBody i op P%bool F)).
 Notation "\sum_ ( m <= i < n  P ) F" := (\big[plus/O]_(m <= i < n  P%bool) F%nat).
 Notation eq_bigr := (fun n m => eq_bigr_ 0 plus (index_iota n m)).
 Notation eq_big := (fun n m => eq_big_ 0 plus (index_iota n m)).
 Parameter odd : nat > bool.
 odd is declared
 Parameter prime : nat > bool.
 prime is declared
 Parameter addnC : forall m n : nat, m + n = n + m.
 addnC is declared
 Parameter muln1 : forall n : nat, n * 1 = n.
 muln1 is declared
 Check eq_bigr.
 eq_bigr : forall (n m : nat) (P : nat > bool) (F1 F2 : nat > nat), (forall x : nat, P x > F1 x = F2 x) > \sum_(n <= i < m  P i) F1 i = \sum_(n <= i < m  P i) F2 i
 Check eq_big.
 eq_big : forall (n m : nat) (P1 P2 : nat > bool) (F1 F2 : nat > nat), (forall x : nat, P1 x = P2 x) > (forall i : nat, P1 i > F1 i = F2 i) > \sum_(n <= i < m  P1 i) F1 i = \sum_(n <= i < m  P2 i) F2 i
 Lemma test_big_nested (m n : nat) : \sum_(0 <= a < m  prime a) \sum_(0 <= j < n  odd (j * 1)) (a + j) = \sum_(0 <= i < m  prime i) \sum_(0 <= j < n  odd j) (j + i).
 1 goal m, n : nat ============================ \sum_(0 <= a < m  prime a) \sum_(0 <= j < n  odd (j * 1)) (a + j) = \sum_(0 <= i < m  prime i) \sum_(0 <= j < n  odd j) (j + i)
 under eq_bigr => i prime_i do under eq_big => [ j  j odd_j ] do [ rewrite (muln1 j)  rewrite (addnC i j) ].
 1 goal m, n : nat ============================ \sum_(0 <= i < m  prime i) \sum_(0 <= j < n  odd j) (j + i) = \sum_(0 <= i < m  prime i) \sum_(0 <= j < n  odd j) (j + i)
Remark how the final goal uses the name i
(the name given in the
intro pattern) rather than a
in the binder of the first summation.
Locking, unlocking¶
As program proofs tend to generate large goals, it is important to be
able to control the partial evaluation performed by the simplification
operations that are performed by the tactics. These evaluations can,
for example, come from a /=
simplification switch, or from rewrite
steps, which may expand large terms while performing conversion. We
definitely want to avoid repeating large subterms of the goal in the
proof script. We do this by “clamping down” selected function symbols
in the goal, which prevents them from being considered in
simplification or rewriting steps. This clamping is accomplished by
using the occurrence switches (see Section Abbreviations)
together with “term tagging” operations.
SSReflect provides two levels of tagging.
The first one uses auxiliary definitions to introduce a provably equal
copy of any term t
. However this copy is (on purpose) not
convertible to t
in the Coq system 18. The job is done by the
following construction:
Note that the definition of master_key is explicitly opaque. The
equation t = locked t
given by the lock
lemma can be used for
selective rewriting, blocking on the fly the reduction in the term t
.
Example
 From Coq Require Import ssreflect ssrfun ssrbool.
 From Coq Require Import List.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable A : Type.
 A is declared
 Fixpoint has (p : A > bool) (l : list A) : bool := if l is cons x l then p x  (has p l) else false.
 has is defined has is recursively defined (guarded on 2nd argument)
 Lemma test p x y l (H : p x = true) : has p ( x :: y :: l) = true.
 1 goal A : Type p : A > bool x, y : A l : list A H : p x = true ============================ has p (x :: y :: l) = true
 rewrite {2}[cons]lock /= lock.
 1 goal A : Type p : A > bool x, y : A l : list A H : p x = true ============================ p x  has p (y :: l) = true
It is sometimes desirable to globally prevent a definition from being
expanded by simplification; this is done by adding locked
in the
definition.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Definition lid := locked (fun x : nat => x).
 lid is defined
 Lemma test : lid 3 = 3.
 1 goal ============================ lid 3 = 3
 rewrite /=.
 1 goal ============================ lid 3 = 3
 unlock lid.
 1 goal ============================ 3 = 3
 Tactic unlock occ_switch? ident¶
This tactic unfolds such definitions while removing “locks”; i.e., it replaces the occurrence(s) of
ident
coded by theocc_switch
with the corresponding body.
We found that it was usually preferable to prevent the expansion of
some functions by the partial evaluation switch /=
, unless this
allowed the evaluation of a condition. This is possible thanks to another
mechanism of term tagging, resting on the following Notation:
The term (nosimpl t)
simplifies to t
except in a definition.
More precisely, given:
the term foo
(or (foo t’)
) will not be expanded by the simpl
tactic unless it is in a forcing context (e.g., in match foo t’ with …
end
, foo t’
will be reduced if this allows match
to be reduced).
Note that nosimpl bar
is simply notation for a term that reduces to
bar
; hence unfold foo
will replace foo
by bar
, and
fold foo
will replace bar
by foo
.
Warning
The nosimpl
trick only works if no reduction is apparent in
t
; in particular, the declaration:
will usually not work. Anyway, the common practice is to tag only the function, and to use the following definition, which blocks the reduction as expected:
A standard example making this technique shine is the case of arithmetic operations. We define for instance:
The operation addn
behaves exactly like plus
, except that
(addn (S n) m)
will not simplify spontaneously to
(S (addn n m))
(the two terms, however, are convertible).
In addition, the unfolding step rewrite /addn
will replace addn
directly with plus
, so the nosimpl
form is
essentially invisible.
Congruence¶
Because of the way matching interferes with parameters of type families, the tactic:
will generally fail to perform congruence simplification, even on rather simple cases. We therefore provide a more robust alternative in which the function is supplied:
 Tactic congr natural? term¶
This tactic:
checks that the goal is a Leibniz equality;
matches both sides of this equality with “term applied to some arguments”, inferring the right number of arguments from the goal and the type of
term
(this may expand some definitions or fixpoints);generates the subgoals corresponding to pairwise equalities of the arguments present in the goal.
The goal can be a nondependent product
P > Q
. In that case, the system asserts the equationP = Q
, uses it to solve the goal, and calls thecongr
tactic on the remaining goalP = Q
. This can be useful for instance to perform a transitivity step, like in the following situation.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test (x y z : nat) (H : x = y) : x = z.
 1 goal x, y, z : nat H : x = y ============================ x = z
 congr (_ = _) : H.
 1 goal x, y, z : nat ============================ y = z
 Abort.
 Lemma test (x y z : nat) : x = y > x = z.
 1 goal x, y, z : nat ============================ x = y > x = z
 congr (_ = _).
 1 goal x, y, z : nat ============================ y = z
The optional
natural
forces the number of arguments for which the tactic should generate equality proof obligations.This tactic supports equalities between applications with dependent arguments. Yet dependent arguments should have exactly the same parameters on both sides, and these parameters should appear as first arguments.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Definition f n := if n is 0 then plus else mult.
 f is defined
 Definition g (n m : nat) := plus.
 g is defined
 Lemma test x y : f 0 x y = g 1 1 x y.
 1 goal x, y : nat ============================ f 0 x y = g 1 1 x y
 congr plus.
 No more goals.
This script shows that the
congr
tactic matchesplus
withf 0
on the left hand side andg 1 1
on the right hand side, and solves the goal.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test n m (Hnm : m <= n) : S m + (S n  S m) = S n.
 1 goal n, m : nat Hnm : m <= n ============================ S m + (S n  S m) = S n
 congr S; rewrite /plus.
 1 goal n, m : nat Hnm : m <= n ============================ m + (S n  S m) = n
The tactic
rewrite /plus
folds back the expansion ofplus
, which was necessary for matching both sides of the equality with an application ofS
.Like most SSReflect arguments,
term
can contain wildcards.Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test x y : x + (y * (y + x  x)) = x * 1 + (y + 0) * y.
 1 goal x, y : nat ============================ x + y * (y + x  x) = x * 1 + (y + 0) * y
 congr ( _ + (_ * _)).
 3 goals x, y : nat ============================ x = x * 1 goal 2 is: y = y + 0 goal 3 is: y + x  x = y
Contextual patterns¶
The simple form of patterns used so far, terms possibly containing
wild cards, often requires an additional occ_switch
to be specified.
While this may work pretty fine for small goals, the use of
polymorphic functions and dependent types may lead to an invisible
duplication of function arguments. These copies usually end up in
types hidden by the implicitarguments machinery or by userdefined
notations. In these situations, computing the right occurrence numbers
is very tedious, because they must be counted on the goal as printed
after setting the Printing All
flag. Moreover, the resulting script is
not really informative for the reader, since it refers to occurrence
numbers he cannot easily see.
Contextual patterns mitigate these issues by allowing to specify occurrences according to the context they occur in.
Syntax¶
The following table summarizes the full syntax of c_pattern
and the
corresponding subterm(s) identified by the pattern. In the third
column, we use s.m.r. for “the subterms matching the redex” specified
in the second column.
redex 
subterms affected 




all occurrences of 

subterm of 
all the subterms identified by 


in all the subterms identified by



in all the subterms identified by 
The rewrite tactic supports two more patterns obtained prefixing the
first two with in
. The intended meaning is that the pattern identifies
all subterms of the specified context. The rewrite
tactic will infer a
pattern for the redex looking at the rule used for rewriting.
redex 
subterms affected 



inferred from rule 
in all s.m.r. in all occurrences of 

inferred from rule 
in all s.m.r. in all the subterms identified by 
The first c_pattern
is the simplest form matching any context but
selecting a specific redex and has been described in the previous
sections. We have seen so far that the possibility of selecting a
redex using a term with holes is already a powerful means of redex
selection. Similarly, any terms provided by the user in the more
complex forms of c_pattern
s
presented in the tables above can contain
holes.
For a quick glance at what can be expressed with the last
r_pattern
,
consider the goal a = b
and the tactic
It rewrites all occurrences of the left hand side of rule
inside b
only (a
, and the hidden type of the equality, are ignored). Note that the
variant rewrite [X in _ = X]rule
would have rewritten b
exactly (i.e., it would only work if b
and the lefthand side
of rule can be unified).
Matching contextual patterns¶
The c_pattern
and r_pattern
involving terms
with holes are matched
against the goal in order to find a closed instantiation. This
matching proceeds as follows:
instantiation order and place for 










In the following patterns, the redex is intended to be inferred from the rewrite rule.
instantiation order and place for 






Examples¶
Contextual pattern in set and the : tactical¶
As already mentioned in Section Abbreviations, the set
tactic takes as an
argument a term in open syntax. This term is interpreted as the
simplest form of c_pattern
. To avoid confusion in the grammar, open
syntax is supported only for the simplest form of patterns, while
parentheses are required around more complex patterns.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test a b : a + b + 1 = b + (a + 1).
 1 goal a, b : nat ============================ a + b + 1 = b + (a + 1)
 set t := (X in _ = X).
 1 goal a, b : nat t := b + (a + 1) : nat ============================ a + b + 1 = t
 rewrite {}/t.
 1 goal a, b : nat ============================ a + b + 1 = b + (a + 1)
 set t := (a + _ in X in _ = X).
 1 goal a, b : nat t := a + 1 : nat ============================ a + b + 1 = b + t
Since the user may define an infix notation for in
, the result of the former
tactic may be ambiguous. The disambiguation rule implemented is to prefer
patterns over simple terms, but to interpret a pattern with double
parentheses as a simple term. For example, the following tactic would
capture any occurrence of the term a in A
.
Contextual patterns can also be used as arguments of the :
tactical.
For example:
Contextual patterns in rewrite¶
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Notation "n .+1" := (Datatypes.S n) (at level 2, left associativity, format "n .+1") : nat_scope.
 Axiom addSn : forall m n, m.+1 + n = (m + n).+1.
 addSn is declared
 Axiom addn0 : forall m, m + 0 = m.
 addn0 is declared
 Axiom addnC : forall m n, m + n = n + m.
 addnC is declared
 Lemma test x y z f : (x.+1 + y) + f (x.+1 + y) (z + (x + y).+1) = 0.
 1 goal x, y, z : nat f : nat > nat > nat ============================ x.+1 + y + f (x.+1 + y) (z + (x + y).+1) = 0
 rewrite [in f _ _]addSn.
 1 goal x, y, z : nat f : nat > nat > nat ============================ x.+1 + y + f (x + y).+1 (z + (x + y).+1) = 0
Note: the simplification rule addSn
is applied only under the f
symbol.
Then, we simplify also the first addition and expand 0
into 0 + 0
.
 rewrite addSn [X in _ = X]addn0.
 1 goal x, y, z : nat f : nat > nat > nat ============================ (x + y).+1 + f (x + y).+1 (z + (x + y).+1) = 0 + 0
Note that the righthand side of addn0
is undetermined, but the
rewrite pattern specifies the redex explicitly. The righthand side
of addn0
is unified with the term identified by X
, here 0
.
The following pattern does not specify a redex, since it identifies an entire region; hence the rewrite rule has to be instantiated explicitly. Thus the tactic:
 rewrite {2}[in X in _ = X](addn0 0).
 1 goal x, y, z : nat f : nat > nat > nat ============================ (x + y).+1 + f (x + y).+1 (z + (x + y).+1) = 0 + (0 + 0)
The following tactic is quite tricky:
 rewrite [_.+1 in X in f _ X](addnC x.+1).
 1 goal x, y, z : nat f : nat > nat > nat ============================ (x + y).+1 + f (x + y).+1 (z + (y + x.+1)) = 0 + (0 + 0)
The explicit redex _.+1
is important, since its head constant S
differs from the head constant inferred from
(addnC x.+1)
(that is +
).
Moreover, the pattern f _ X
is important to rule out
the first occurrence of (x + y).+1
.
Last, only the subterms of f _ X
identified by X
are rewritten; thus the first argument of
f
is skipped too.
Also note that the pattern _.+1
is interpreted in the context
identified by X
; thus it gets instantiated to
(y + x).+1
and not (x + y).+1
.
The last rewrite pattern allows to specify exactly the shape of the term identified by X, which is thus unified with the lefthand side of the rewrite rule.
 rewrite [x.+1 + y as X in f X _]addnC.
 1 goal x, y, z : nat f : nat > nat > nat ============================ (x + y).+1 + f (y + x.+1) (z + (y + x.+1)) = 0 + (0 + 0)
Patterns for recurrent contexts¶
The user can define shortcuts for recurrent contexts corresponding to
the ident in term
part. The notation scope identified with
%pattern
provides a special notation (X in t)
the user must adopt
in order to define
context shortcuts.
The following example is taken from ssreflect.v
, where the
LHS
and RHS
shortcuts are defined.
Shortcuts defined this way can be freely used in place of the trailing
ident in term
part of any contextual pattern. Some examples follow:
Views and reflection¶
The bookkeeping facilities presented in Section Basic tactics are crafted to ease simultaneous introductions and generalizations of facts and operations of casing, naming, etc. It is also a common practice to make a stack operation immediately followed by an interpretation of the fact being pushed, that is, to apply a lemma to this fact before passing it to a tactic for decomposition, application and so on.
SSReflect provides a convenient, unified syntax to combine these
interpretation operations with the proof stack operations. This view
mechanism relies on the combination of the /
view switch with
bookkeeping tactics and tacticals.
Interpreting eliminations¶
The view syntax combined with the elim
tactic specifies an elimination
scheme to be used instead of the default, generated, one. Hence, the
SSReflect tactic:
is a synonym for:
where top is a fresh name and V any secondorder lemma.
Since an elimination view supports the two bookkeeping tacticals of discharge and introduction (see Section Basic tactics), the SSReflect tactic:
is a synonym for:
where x
is a variable in the context, y
a fresh name and V
any second order lemma; SSReflect relaxes the syntactic restrictions of the Coq
elim
. The first pattern following :
can be a _
wildcard if the
conclusion of the view V
specifies a pattern for its last argument
(e.g., if V
is a functional induction lemma generated by the
Function
command).
The elimination view mechanism is compatible with the equationname generation (see Section Generation of equations).
Example
The following script illustrates a toy example of this feature. Let us define a function adding an element at the end of a list:
 From Coq Require Import ssreflect List.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variable d : Type.
 d is declared
 Fixpoint add_last (s : list d) (z : d) {struct s} : list d := if s is cons x s' then cons x (add_last s' z) else z :: nil.
 add_last is defined add_last is recursively defined (guarded on 1st argument)
One can define an alternative, reversed, induction principle on inductively defined lists, by proving the following lemma:
 Axiom last_ind_list : forall P : list d > Prop, P nil > (forall s (x : d), P s > P (add_last s x)) > forall s : list d, P s.
 last_ind_list is declared
Then, the combination of elimination views with equation names results in a concise syntax for reasoning inductively using the userdefined elimination scheme.
 Lemma test (x : d) (l : list d): l = l.
 1 goal d : Type x : d l : list d ============================ l = l
 elim/last_ind_list E : l=> [ u v]; last first.
 2 goals d : Type x : d u : list d v : d l : list d E : l = add_last u v ============================ u = u > add_last u v = add_last u v goal 2 is: nil = nil
Userprovided eliminators (potentially generated with Coq’s Function
command) can be combined with the type family switches described
in Section Type families.
Consider an eliminator foo_ind
of type:
and consider the tactic:
The elim/
tactic distinguishes two cases.
 truncated eliminator
when
x
does not occur inP p1 … pm
and the type ofen
unifies withT
anden
is not_
. In that case,en
is passed to the eliminator as the last argument (x
infoo_ind
) anden−1 … e1
are used as patterns to select in the goal the occurrences that will be bound by the predicateP
; thus it must be possible to unify the subterm of the goal matched byen−1
withpm
, the one matched byen−2
withpm−1
and so on. regular eliminator
in all the other cases. Here it must be possible to unify the term matched by
en
withpm
, the one matched byen−1
withpm−1
and so on. Note that standard eliminators have the shape…forall x, P … x
; thusen
is the pattern identifying the eliminated term, as expected.
As explained in Section Type families, the initial prefix of
ei
can be omitted.
Here is an example of a regular, but nontrivial, eliminator.
Example
Here is a toy example illustrating this feature.
 From Coq Require Import ssreflect FunInd.
 [Loading ML file extraction_plugin.cmxs (using legacy method) ... done] [Loading ML file funind_plugin.cmxs (using legacy method) ... done]
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Function plus (m n : nat) {struct n} : nat := if n is S p then S (plus m p) else m.
 plus is defined plus is recursively defined (guarded on 2nd argument) plus_equation is defined plus_rect is defined plus_ind is defined plus_rec is defined R_plus_correct is defined R_plus_complete is defined
 About plus_ind.
 plus_ind : forall [m : nat] [P : nat > nat > Prop], (forall n p : nat, n = S p > P p (plus m p) > P (S p) (S (plus m p))) > (forall n _x : nat, n = _x > match _x with  0 => True  S _ => False end > P _x m) > forall n : nat, P n (plus m n) plus_ind is not universe polymorphic Arguments plus_ind [m]%nat_scope [P]%function_scope (f f0)%function_scope n%nat_scope plus_ind is transparent Expands to: Constant Top.Test.plus_ind
 Lemma test x y z : plus (plus x y) z = plus x (plus y z).
 1 goal x, y, z : nat ============================ plus (plus x y) z = plus x (plus y z)
The following tactics are all valid and perform the same elimination on this goal.
 From Coq Require Import ssreflect FunInd.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Function plus (m n : nat) {struct n} : nat := if n is S p then S (plus m p) else m.
 plus is defined plus is recursively defined (guarded on 2nd argument) plus_equation is defined plus_rect is defined plus_ind is defined plus_rec is defined R_plus_correct is defined R_plus_complete is defined
 About plus_ind.
 plus_ind : forall [m : nat] [P : nat > nat > Prop], (forall n p : nat, n = S p > P p (plus m p) > P (S p) (S (plus m p))) > (forall n _x : nat, n = _x > match _x with  0 => True  S _ => False end > P _x m) > forall n : nat, P n (plus m n) plus_ind is not universe polymorphic Arguments plus_ind [m]%nat_scope [P]%function_scope (f f0)%function_scope n%nat_scope plus_ind is transparent Expands to: Constant Top.Test.plus_ind
 Lemma test x y z : plus (plus x y) z = plus x (plus y z).
 1 goal x, y, z : nat ============================ plus (plus x y) z = plus x (plus y z)
 elim/plus_ind: z / _.
 2 goals x, y : nat ============================ forall n p : nat, n = S p > plus (plus x y) p = plus x (plus y p) > S (plus (plus x y) p) = plus x (plus y (S p)) goal 2 is: forall n _x : nat, n = _x > match _x with  0 => True  S _ => False end > plus x y = plus x (plus y _x)
The two latter examples feature a wildcard pattern: in this case,
the resulting pattern is inferred from the type of the eliminator.
In both of these examples, it is (plus _ _)
that matches the subterm
plus (plus x y) z
, thus instantiating the last _
with z
.
Note that the tactic:
 From Coq Require Import ssreflect FunInd.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Function plus (m n : nat) {struct n} : nat := if n is S p then S (plus m p) else m.
 plus is defined plus is recursively defined (guarded on 2nd argument) plus_equation is defined plus_rect is defined plus_ind is defined plus_rec is defined R_plus_correct is defined R_plus_complete is defined
 About plus_ind.
 plus_ind : forall [m : nat] [P : nat > nat > Prop], (forall n p : nat, n = S p > P p (plus m p) > P (S p) (S (plus m p))) > (forall n _x : nat, n = _x > match _x with  0 => True  S _ => False end > P _x m) > forall n : nat, P n (plus m n) plus_ind is not universe polymorphic Arguments plus_ind [m]%nat_scope [P]%function_scope (f f0)%function_scope n%nat_scope plus_ind is transparent Expands to: Constant Top.Test.plus_ind
 Lemma test x y z : plus (plus x y) z = plus x (plus y z).
 1 goal x, y, z : nat ============================ plus (plus x y) z = plus x (plus y z)
 Fail elim/plus_ind: y / _.
 The command has indeed failed with message: The given pattern matches the term y while the inferred pattern z doesn't
triggers an error: in the conclusion
of the plus_ind
eliminator, the first argument of the predicate
P
should be the same as the second argument of plus
, in the
second argument of P
, but y
and z
do no unify.
Here is an example of a truncated eliminator:
Example
Consider the goal:
 From Coq Require Import ssreflect FunInd.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
where the type of the big_prop
eliminator is
Since the pattern for the argument of Pb is not specified, the
inferred one, big[_/_]_(i < _  _ i) _ i
, is used instead,
and after the introductions, the following goals are generated:
Note that the pattern matching algorithm instantiated all the variables occurring in the pattern.
Interpreting assumptions¶
Interpreting an assumption in the context of a proof consists in
applying to it a lemma before generalizing and/or decomposing this
assumption. For instance, with the extensive use of boolean reflection
(see Section Views and reflection), it is quite frequent
to need to decompose the logical interpretation of (the boolean
expression of) a fact, rather than the fact itself. This can be
achieved by a combination of move : _ => _
switches, like in the
following example, where 
is a notation for the boolean
disjunction.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables P Q : bool > Prop.
 P is declared Q is declared
 Hypothesis P2Q : forall a b, P (a  b) > Q a.
 P2Q is declared
 Lemma test a : P (a  a) > True.
 1 goal P, Q : bool > Prop P2Q : forall a b : bool, P (a  b) > Q a a : bool ============================ P (a  a) > True
 move=> HPa; move: {HPa}(P2Q HPa) => HQa.
 1 goal P, Q : bool > Prop P2Q : forall a b : bool, P (a  b) > Q a a : bool HQa : Q a ============================ True
which transforms the hypothesis HPa : P a
, which has been introduced
from the initial statement, into HQa : Q a
.
This operation is so common that the tactic shell has specific
syntax for it. The following scripts:
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables P Q : bool > Prop.
 P is declared Q is declared
 Hypothesis P2Q : forall a b, P (a  b) > Q a.
 P2Q is declared
 Lemma test a : P (a  a) > True.
 1 goal P, Q : bool > Prop P2Q : forall a b : bool, P (a  b) > Q a a : bool ============================ P (a  a) > True
 move=> HPa; move/P2Q: HPa => HQa.
 1 goal P, Q : bool > Prop P2Q : forall a b : bool, P (a  b) > Q a a : bool HQa : Q a ============================ True
or more directly:
 move/P2Q=> HQa.
 1 goal P, Q : bool > Prop P2Q : forall a b : bool, P (a  b) > Q a a : bool HQa : Q a ============================ True
are equivalent to the former one. The former script shows how to interpret a fact (already in the context), thanks to the discharge tactical (see Section Discharge), and the latter, how to interpret the top assumption of a goal. Note that the number of wildcards to be inserted to find the correct application of the view lemma to the hypothesis has been automatically inferred.
The view mechanism is compatible with the case
tactic and with the
equationname generation mechanism (see Section Generation of equations):
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables P Q: bool > Prop.
 P is declared Q is declared
 Hypothesis Q2P : forall a b, Q (a  b) > P a \/ P b.
 Q2P is declared
 Lemma test a b : Q (a  b) > True.
 1 goal P, Q : bool > Prop Q2P : forall a b : bool, Q (a  b) > P a \/ P b a, b : bool ============================ Q (a  b) > True
 case/Q2P=> [HPa  HPb].
 2 goals P, Q : bool > Prop Q2P : forall a b : bool, Q (a  b) > P a \/ P b a, b : bool HPa : P a ============================ True goal 2 is: True
This view tactic performs:
The term on the right of the /
view switch is called a view lemma.
Any SSReflect term coercing to a product type can be used as a view
lemma.
The examples we have given so far explicitly provide the direction of the translation to be performed. In fact, view lemmas need not to be oriented. The view mechanism is able to detect which application is relevant for the current goal.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables P Q: bool > Prop.
 P is declared Q is declared
 Hypothesis PQequiv : forall a b, P (a  b) <> Q a.
 PQequiv is declared
 Lemma test a b : P (a  b) > True.
 1 goal P, Q : bool > Prop PQequiv : forall a b : bool, P (a  b) <> Q a a, b : bool ============================ P (a  b) > True
 move/PQequiv=> HQab.
 1 goal P, Q : bool > Prop PQequiv : forall a b : bool, P (a  b) <> Q a a, b : bool HQab : Q a ============================ True
has the same behavior as the first example above.
The view mechanism can insert automatically a view hint to transform the double implication into the expected simple implication. The last script is in fact equivalent to:
where:
Specializing assumptions¶
The special case when the head symbol of the view lemma is a wildcard is used to interpret an assumption by specializing it. The view mechanism hence offers the possibility to apply a higherorder assumption to some given arguments.
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test z : (forall x y, x + y = z > z = x) > z = 0.
 1 goal z : nat ============================ (forall x y : nat, x + y = z > z = x) > z = 0
 move/(_ 0 z).
 1 goal z : nat ============================ (0 + z = z > z = 0) > z = 0
Interpreting goals¶
In a similar way, it is also often convenient to
change a goal by turning it into an equivalent proposition. The view
mechanism of SSReflect has a special syntax apply/
for combining in a
single tactic simultaneous goal interpretation operations and
bookkeeping steps.
Example
The following example use the
~~
prenex notation for boolean negation:
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables P Q: bool > Prop.
 P is declared Q is declared
 Hypothesis PQequiv : forall a b, P (a  b) <> Q a.
 PQequiv is declared
 Lemma test a : P ((~~ a)  a).
 1 goal P, Q : bool > Prop PQequiv : forall a b : bool, P (a  b) <> Q a a : bool ============================ P (~~ a  a)
 apply/PQequiv.
 1 goal P, Q : bool > Prop PQequiv : forall a b : bool, P (a  b) <> Q a a : bool ============================ Q (~~ a)
thus in this case, the tactic apply/PQequiv
is equivalent to
apply: (iffRL (PQequiv _ _))
, where iffRL
is the analogue of
iffLR
for the converse implication.
Any SSReflect term whose type coerces to a double implication can be used as a view for goal interpretation.
Note that the goal interpretation view mechanism supports both apply
and exact
tactics. As expected, a goal interpretation view command
exact
/term should solve the current goal or it will fail.
Warning
Goalinterpretation view tactics are not compatible with
the bookkeeping tactical =>
, since this would be redundant with the
apply: term => _
construction.
Boolean reflection¶
In the Calculus of Inductive Constructions, there is an obvious
distinction between logical propositions and boolean values. On the
one hand, logical propositions are objects of sort Prop
, which is
the carrier of intuitionistic reasoning. Logical connectives in
Prop
are types, which give precise information on the structure
of their proofs; this information is automatically exploited by Coq
tactics. For example, Coq knows that a proof of A \/ B
is
either a proof of A
or a proof of B
. The tactics left
and
right
change the goal A \/ B
to A
and B
, respectively;
dually, the tactic case
reduces the goal A \/ B => G
to two
subgoals A => G
and B => G
.
On the other hand, bool is an inductive datatype with two
constructors: true
and false
. Logical connectives on bool are
computable functions, defined by their truth tables, using case
analysis:
Example
 From Coq Require Import ssreflect.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Definition orb (b1 b2 : bool) := if b1 then true else b2.
 orb is defined
Properties of such connectives are also established using case analysis
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test b : b  ~~ b = true.
 1 goal b : bool ============================ b  ~~ b = true
 by case: b.
 No more goals.
Once b
is replaced by true
in the first goal and by false
in the
second one, the goals reduce by computation to the trivial true = true
.
Thus, Prop
and bool
are truly complementary: the former supports
robust natural deduction; the latter allows bruteforce
evaluation. SSReflect supplies a generic mechanism to have the best of
the two worlds and move freely from a propositional version of a
decidable predicate to its boolean version.
First, booleans are injected into propositions using the coercion mechanism:
This allows any boolean formula b
to be used in a context where Coq
would expect a proposition, e.g., after Lemma … :
. It is then
interpreted as (is_true b)
, i.e., the proposition b = true
. Coercions
are elided by the prettyprinter; so they are essentially transparent
to the user.
The reflect predicate¶
To get all the benefits of the boolean reflection, it is in fact
convenient to introduce the following inductive predicate reflect
to
relate propositions and booleans:
The statement (reflect P b)
asserts that (is_true b)
and P
are
logically equivalent propositions.
For instance, the following lemma:
relates the boolean conjunction to the logical one /\
. Note that in
andP
, b1
and b2
are two boolean variables and the
proposition b1 /\ b2
hides two coercions. The conjunction of
b1
and b2
can then be viewed as b1 /\ b2
or as b1 && b2
.
Expressing logical equivalences through this family of inductive types makes possible to take benefit from rewritable equations associated to the case analysis of Coq’s inductive types.
Since the equivalence predicate is defined in Coq as:
where /\
is a notation for and
:
This makes case analysis very different according to the way an equivalence property has been defined.
Let us compare the respective behaviors of andE
and andP
.
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Axiom andE : forall (b1 b2 : bool), (b1 /\ b2) <> (b1 && b2).
 andE is declared
 Lemma test (b1 b2 : bool) : if (b1 && b2) then b1 else ~~(b1b2).
 1 goal b1, b2 : bool ============================ if b1 && b2 then b1 else ~~ (b1  b2)
 case: (@andE b1 b2).
 1 goal b1, b2 : bool ============================ (b1 /\ b2 > b1 && b2) > (b1 && b2 > b1 /\ b2) > if b1 && b2 then b1 else ~~ (b1  b2)
 Restart.
 1 goal b1, b2 : bool ============================ if b1 && b2 then b1 else ~~ (b1  b2)
 case: (@andP b1 b2).
 2 goals b1, b2 : bool ============================ b1 /\ b2 > b1 goal 2 is: ~ (b1 /\ b2) > ~~ (b1  b2)
Expressing reflection relations through the reflect
predicate is hence
a very convenient way to deal with classical reasoning, by case
analysis. Using the reflect
predicate allows, moreover, to program rich
specifications inside its two constructors, which will be
automatically taken into account during destruction. This
formalisation style gives far more efficient specifications than
quantified (double) implications.
A naming convention in SSReflect is to postfix the name of view lemmas
with P
. For example, orP
relates 
and \/
;
negP
relates ~~
and ~
.
The view mechanism is compatible with reflect predicates.
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test (a b : bool) (Ha : a) (Hb : b) : a /\ b.
 1 goal a, b : bool Ha : a Hb : b ============================ a /\ b
 apply/andP.
 1 goal a, b : bool Ha : a Hb : b ============================ a && b
Conversely
 Lemma test (a b : bool) : a /\ b > a.
 1 goal a, b : bool ============================ a /\ b > a
 move/andP.
 1 goal a, b : bool ============================ a && b > a
The same tactics can also be used to perform the converse operation, changing a boolean conjunction into a logical one. The view mechanism guesses the direction of the transformation to be used, i.e., the constructor of the reflect predicate that should be chosen.
General mechanism for interpreting goals and assumptions¶
Specializing assumptions¶
The SSReflect tactic:
is equivalent to the tactic:
where top
is a fresh name for introducing the top assumption of the
current goal.
Interpreting assumptions¶
The general form of an assumption view tactic is:
 Variant movecase / term
The term, called the view lemma, can be:
a (term coercible to a) function;
a (possibly quantified) implication;
a (possibly quantified) double implication;
a (possibly quantified) instance of the reflect predicate (see Section Views and reflection).
Let top
be the top assumption in the goal.
There are three steps in the behavior of an assumption view tactic.
It first introduces
top
.If the type of
term
is neither a double implication nor an instance of the reflect predicate, then the tactic automatically generalises a term of the formterm term1 … termn
, where the termsterm1 … termn
instantiate the possible quantified variables ofterm
, in order for(term term1 … termn top)
to be well typed.If the type of
term
is an equivalence, or an instance of the reflect predicate, it generalises a term of the form(termvh (term term1 … termn ))
, where the termtermvh
inserted is called an assumption interpretation view hint.It finally clears top.
For a case/term
tactic, the generalisation step is replaced by a
case analysis step.
View hints are declared by the user (see Section Views and reflection) and
stored in the Hint View database. The proof engine automatically
detects from the shape of the top assumption top
and of the view lemma
term
provided to the tactic the appropriate view hint in the
database to be inserted.
If term
is a double implication, then the view hint will be one of
the defined view hints for implication. These hints are by default the
ones present in the file ssreflect.v
:
which transforms a double implication into the lefttoright one, or:
which produces the converse implication. In both cases, the two
first Prop
arguments are implicit.
If term
is an instance of the reflect
predicate, then A
will be one
of the defined view hints for the reflect
predicate, which are by
default the ones present in the file ssrbool.v
. These hints are not
only used for choosing the appropriate direction of the translation,
but they also allow complex transformation, involving negations.
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Check introN.
 introN : forall (P : Prop) (b : bool), reflect P b > ~ P > ~~ b
 Lemma test (a b : bool) (Ha : a) (Hb : b) : ~~ (a && b).
 1 goal a, b : bool Ha : a Hb : b ============================ ~~ (a && b)
 apply/andP.
 1 goal a, b : bool Ha : a Hb : b ============================ ~ (a /\ b)
In fact, this last script does not
exactly use the hint introN
, but the more general hint:
 Check introNTF.
 introNTF : forall (P : Prop) (b c : bool), reflect P b > (if c then ~ P else P) > ~~ b = c
The lemma introN
is an instantiation of introNF
using c := true
.
Note that views, being part of i_pattern
, can be used to interpret
assertions too. For example, the following script asserts a && b
, but
actually uses its propositional interpretation.
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test (a b : bool) (pab : b && a) : b.
 1 goal a, b : bool pab : b && a ============================ b
 have /andP [pa >] : (a && b) by rewrite andbC.
 1 goal a, b : bool pab : b && a pa : a ============================ true
Interpreting goals¶
A goal interpretation view tactic of the form:
 Variant apply/term
applied to a goal top
is interpreted in the following way.
If the type of
term
is not an instance of the reflect predicate, nor an equivalence, then the termterm
is applied to the current goaltop
, possibly inserting implicit arguments.If the type of
term
is an instance of the reflect predicate or an equivalence, then a goal interpretation view hint can possibly be inserted, which corresponds to the application of a term(termvh (term _ … _))
to the current goal, possibly inserting implicit arguments.
Like assumption interpretation view hints, goal interpretation ones
are userdefined lemmas stored (see Section Views and reflection) in the Hint View
database, bridging the possible gap between the type of term
and the
type of the goal.
Interpreting equivalences¶
Equivalent boolean propositions are simply equal boolean terms. A special construction helps the user to prove boolean equalities by considering them as logical double implications (between their coerced versions), while performing at the same time logical operations on both sides.
The syntax of double views is:
The first term is the view lemma applied to the lefthand side of the equality, while the second term is the one applied to the righthand side.
In this context, the identity view can be used when no view has to be applied:
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test (b1 b2 b3 : bool) : ~~ (b1  b2) = b3.
 1 goal b1, b2, b3 : bool ============================ ~~ (b1  b2) = b3
 apply/idP/idP.
 2 goals b1, b2, b3 : bool ============================ ~~ (b1  b2) > b3 goal 2 is: b3 > ~~ (b1  b2)
The same goal can be decomposed in several ways, and the user may choose the most convenient interpretation.
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Lemma test (b1 b2 b3 : bool) : ~~ (b1  b2) = b3.
 1 goal b1, b2, b3 : bool ============================ ~~ (b1  b2) = b3
 apply/norP/idP.
 2 goals b1, b2, b3 : bool ============================ ~~ b1 /\ ~~ b2 > b3 goal 2 is: b3 > ~~ b1 /\ ~~ b2
Declaring new Hint Views¶
 Command Hint View for move / ident  natural?¶
 Command Hint View for apply / ident  natural?¶
This command can be used to extend the database of hints for the view mechanism.
As library
ssrbool.v
already declares a corpus of hints, this feature is probably useful only for users who define their own logical connectives.The
ident
is the name of the lemma to be declared as a hint. Ifmove
is used as tactic, the hint is declared for assumption interpretation tactics;apply
declares hints for goal interpretations. Goal interpretation view hints are declared for both simple views and lefthand side views. The optional natural number is the number of implicit arguments to be considered for the declared hint view lemma.
Multiple views¶
The hypotheses and the goal can be interpreted by applying multiple views
in sequence. Both move
and apply
can be followed by an arbitrary
number of /term
. The main difference between the following two
tactics
is that the former applies all the views to the principal goal.
Applying a view with hypotheses generates new goals, and the second
line would apply the view v2
to all the goals generated by apply/v1
.
Note that the NOOP intro pattern 
can be used to separate two views,
making the two following examples equivalent:
The tactic move
can be used together with the in
tactical to
pass a given hypothesis to a lemma.
Example
 From Coq Require Import ssreflect ssrbool.
 Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 Section Test.
 Variables P Q R : Prop.
 P is declared Q is declared R is declared
 Variable P2Q : P > Q.
 P2Q is declared
 Variable Q2R : Q > R.
 Q2R is declared
 Lemma test (p : P) : True.
 1 goal P, Q, R : Prop P2Q : P > Q Q2R : Q > R p : P ============================ True
 move/P2Q/Q2R in p.
 1 goal P, Q, R : Prop P2Q : P > Q Q2R : Q > R p : R ============================ True
If the list of views is of length two, Hint Views
for interpreting
equivalences are indeed taken into account; otherwise only single
Hint Views
are used.
Synopsis and Index¶
Parameters¶
SSReflect tactics
d_tactic::=
elimcasecongrapplyexactmove
Notation scope
key::=
ident
Module name
modname::=
qualid
Natural number
nat_or_ident::=
naturalident
where ident
is an Ltac variable denoting a standard Coq number
(should not be the name of a tactic that can be followed by a
bracket [
, such as do
, have
,…)
Items and switches¶
ssr_binder::=
ident( ident : term? )
binder (see Abbreviations)
clear_switch::=
{ ident+ }
clear switch (see Discharge)
c_pattern::=
term interm as? ident in term
context pattern (see Contextual patterns)
d_item::=
occ_switchclear_switch? term( c_pattern )?
discharge item (see Discharge)
gen_item::=
@? ident( ident )( @? ident := c_pattern )
generalization item (see Structure)
i_pattern::=
ident>_?*+occ_switch? ><[ i_item? ][: ident+ ]
intro pattern (see Introduction in the context)
i_item::=
clear_switchs_itemi_patterni_viewi_block
view (see Introduction in the context)
i_view::=
{}? /term/ltac:( tactic )
intro block (see Introduction in the context)
i_block::=
[^ ident ][^~ identnatural ]
intro item (see Introduction in the context)
int_mult::=
natural? mult_mark
multiplier (see Iteration)
occ_switch::=
{ +? natural* }
occur. switch (see Occurrence selection)
mult::=
natural? mult_mark
multiplier (see Iteration)
mult_mark::=
?!
multiplier mark (see Iteration)
r_item::=
/? terms_item
rewrite item (see Rewriting)
r_prefix::=
? int_mult? occ_switchclear_switch? [ r_pattern ]?
rewrite prefix (see Rewriting)
r_pattern::=
termc_patternin ident in? term
rewrite pattern (see Rewriting)
r_step::=
r_prefix? r_item
rewrite step (see Rewriting)
s_item::=
/=////=
simplify switch (see Introduction in the context)
Tactics¶
Note: without loss
and suffices
are synonyms for wlog
and suff
,
respectively.
 Tactic move¶
idtac
orhnf
(see Bookkeeping)
 Tactic apply¶
 Tactic exact¶
application (see The defective tactics)
 Variant abstract: d_item+
(see The abstract tactic and Generating let in context entries with have)
 Variant elim
induction (see The defective tactics)
 Variant case
case analysis (see The defective tactics)
 Tactic under r_prefix? term => i_item+? do tactic[ tactic* ]?¶
under (see Rewriting under binders)
 Tactic over¶
over (see The over tactic)
 Tactic have i_item* i_pattern? s_itemssr_binder+? : term? := term¶
 Tactic have i_item* i_pattern? s_itemssr_binder+? : term by tactic?
 Tactic have suff clear_switch? i_pattern? : term? := term
 Tactic have suff clear_switch? i_pattern? : term by tactic?
 Tactic gen have ident ,? i_pattern? : gen_item+ / term by tactic?
 Tactic generally have ident ,? i_pattern? : gen_item+ / term by tactic?¶
forward chaining (see Structure)
 Tactic suff i_item* i_pattern? ssr_binder+ : term by tactic?¶
 Tactic suffices i_item* i_pattern? ssr_binder+ : term by tactic?¶
 Tactic suff have? clear_switch? i_pattern? : term by tactic?¶
 Tactic suffices have? clear_switch? i_pattern? : term by tactic?
backchaining (see Structure)
 Variant pose ident := term
local definition (see Definitions)
 Variant pose ident ssr_binder+ := term
local function definition
 Variant pose fix fix_decl
local fix definition
 Variant pose cofix fix_decl
local cofix definition
 Tactic set ident : term? := occ_switch? term( c_pattern)¶
abbreviation (see Abbreviations)
 Tactic unlock r_prefix? ident*¶
unlock (see Locking, unlocking)
 Tactic congr natural? term¶
congruence (see Congruence)
Tacticals¶
tactic+=
d_tactic ident? : d_item+ clear_switch?
discharge (see Discharge)
tactic+=
tactic => i_item+
introduction (see Introduction in the context)
tactic+=
tactic in gen_itemclear_switch+ *?
localization (see Localization)
tactic+=
do mult? tactic[ tactic+ ]
iteration (see Iteration)
tactic+=
tactic ; firstlast natural? tactic[ tactic+ ]
selector (see Selectors)
tactic+=
tactic ; firstlast natural?
rotation (see Selectors)
tactic+=
by tactic[ tactic* ]
closing (see Terminators)
Commands¶
 Command Hint View for moveapply / ident  natural?¶
view hint declaration (see Declaring new Hint Views)
 Command Hint View for apply // ident natural?¶
right hand side double , view hint declaration (see Declaring new Hint Views)
 Command Prenex Implicits ident+¶
prenex implicits declaration (see Parametric polymorphism)
Settings¶
 Flag Debug Ssreflect¶
Developer only. Print debug information on reflect.
 Flag Debug SsrMatching¶
Developer only. Print debug information on SSR matching.
Footnotes
 11
Unfortunately, even after a call to the
Set Printing All
command, some occurrences are still not displayed to the user, essentially the ones possibly hidden in the predicate of a dependent match structure. 12
Thus scripts that depend on bound variable names, e.g., via intros or with, are inherently fragile.
 13
The name
subnK
reads as “right cancellation rule fornat
subtraction”. 14
Also, a slightly different variant may be used for the first
d_item
ofcase
andelim
; see Section Type families. 15
Except that
/=
does not expand the local definitions created by the SSReflectin
tactical. 16
SSReflect reserves all identifiers of the form “_x_”, which is used for such generated names.
 17
More precisely, it should have a quantified inductive type with a assumptions and m − a constructors.
 18
This is an implementation feature: there is no such obstruction in the metatheory.
 19
The current state of the proof shall be displayed by the
Show Proof
command of Coq proof mode. 20(1,2,3)
A simple proof context entry is a naked identifier (i.e., not between parentheses) designating a context entry that is not a section variable.