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

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 so-called small scale reflection proof methodology. Despite the original purpose this set of tactic 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 :.

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 ... done] [Loading ML file ssreflect_plugin.cmxs ... done] Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Compatibility issues¶

Requiring the above modules creates an environment which 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 quasi-keywords 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) or rewrite .. in * or any other variant of rewrite will not work, and the SSReflect syntax and semantics for occurrence selection and rule chaining is different. Use an explicit rewrite direction (rewrite <- … or rewrite -> …) 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 imports SsrSyntax, 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", parenthesis free, syntax of tactics such as have, set and pose.
The generalization of if statements to non-Boolean conditions is turned off by SSReflect, because it is mostly subsumed by Coercion to bool of the sumXXX types (declared in ssrfun.v) and the if term is pattern then term else term construct (see Pattern conditional). To use the generalized form, turn off the SSReflect Boolean if 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 SSReflect-generated 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¶

Small-scale 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 well-established 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 in that

The pattern can be nested (deep pattern matching), in particular, this allows expression of the form:

let: exist (x, y) p_xy := Hp in … .

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 for f, whereas with a usual let 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 ρ-calculus), 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 if-is-then-else construct supports the dependent match annotations:

term+=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:

term+=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 ML-style polymorphism, but unfortunately this emulation breaks down for higher-order 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 higher-order expressions are quite frequent in representation functions, especially those which 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 of ident_i are implicit; then it assigns Maximal 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 incantation

Set 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:

binder+=& termof term

Caveat: & T and of T abbreviations have to appear at the end of a binder list. For instance, the usual two-constructor 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 tactic supports open syntax: the body of the definition does not need surrounding parentheses. For instance:

pose t := x + y.

is a valid tactic expression.

The pose tactic is also improved for the local definition of higher order terms. Local definitions of functions can use the same syntax as global ones. For example, the tactic pose supports parameters:

Example

From Coq Require Import ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Lemma test : True.: 1 subgoal ============================ True
pose f x y := x + y.: 1 subgoal f := fun x y : nat => x + y : nat -> nat -> nat ============================ True

The SSReflect pose tactic also supports (co)fixpoints, by providing the local counterpart of the Fixpoint f := … and CoFixpoint f := … constructs. For instance, the following tactic:

pose fix f (x y : nat) {struct x} : nat := if x is S p then S (f p y) else 0.

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:

pose cofix f (arg : T) := … .

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:

pose f := _ + 1.

is shorthand for:

pose f n := n + 1.

When the local definition of a function involves both arguments and holes, hole abstractions appear first. For instance, the tactic:

pose f x := x + _.

is shorthand for:

pose f n x := x + n.

The interaction of the pose tactic with the interpretation of implicit arguments results in a powerful and concise syntax for local definitions involving dependent types. For instance, the tactic:

pose f x y := (x, y).

adds to the context the local definition:

pose f (Tx Ty : Type) (x : Tx) (y : Ty) := (x, y).

The generalization of wildcards makes the use of the pose tactic resemble ML-like 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.

occ_switch::={ +-? natural* }

where:

ident is a fresh identifier chosen by the user.
term 1 is an optional type annotation. The type annotation term 1 can be given in open syntax (no surrounding parentheses). If no occ_switch (described hereafter) is present, it is also the case for the second term. On the other hand, in presence of occ_switch, parentheses surrounding the second term 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 subgoal x : nat ============================ f x + f x = f x
set t := f _.: 1 subgoal x : nat t := f x : nat ============================ t + t = t

set t := {2}(f _).: 1 subgoal x : nat t := f x : nat ============================ f x + t = f x

The type annotation may contain wildcards, which will be filled with the appropriate value 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 …).
If the head of term is a hole, the subterm should have at least as many arguments as term.

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 subgoal x, y, z : nat ============================ x + y = z
set t := _ x.: 1 subgoal 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 pattern term 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 head t0 of term.

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 subgoal ============================ (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 subgoal t := unkeyed (fun y z : nat => S y + z) 2 : nat -> nat ============================ t 3 = 6

The notation unkeyed defined in ssreflect.v is a shorthand for the degenerate term let 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 subgoal x, y, z : nat ============================ x + y = z

set t := _ + _.
1 subgoal 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 subgoal ============================ 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 which require control on subterm selection like rewriting, generalization, …

An occurrence switch can be:

A list 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 subgoal ============================ f 2 + f 8 = f 2 + f 2

set x := {+1 3}(f 2).
1 subgoal 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 subgoal x, y : nat ============================ x < y -> S x < S y

set t := S x.
1 subgoal x, y : nat t := S x : nat ============================ t <= y -> t < S y
A list of natural numbers between {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 subgoal ============================ f 2 + f 8 = f 2 + f 2

set x := {-2}(f 2).
1 subgoal 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 which 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 definition x := 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 subgoal x, y, z : nat ============================ x + y = x + y + z

set a := {2}(_ + _).
1 subgoal 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 subgoal 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+

This variant of set (ssreflect) introduces a defined constant called ident in the context, and folds it in the context entries mentioned on the right hand side of in. The body of ident is the first subterm matching these context entries (taken in the given order).

Example

From Coq Require Import ssreflect.

Lemma test x t (Hx : x = 3) : x + t = 4.: 1 subgoal x, t : nat Hx : x = 3 ============================ x + t = 4
set z := 3 in Hx.: 1 subgoal x, t : nat z := 3 : nat Hx : x = z ============================ x + t = 4

Variant set ident := term in ident+ *

This variant matches term and then folds ident similarly in all the given context entries but also folds ident in the goal.

Example

From Coq Require Import ssreflect.

Lemma test x t (Hx : x = 3) : x + t = 4.: 1 subgoal x, t : nat Hx : x = 3 ============================ x + t = 4
set z := 3 in Hx * .: 1 subgoal 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 data-flow of a proof script. This is especially true for reflection-based 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 pseudo-tacticals), shared across most SSReflect tactics, and thus form the foundation of the SSReflect proof language.

Bookkeeping¶

During the course of a proof Coq always present 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 P n ->, 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 others. 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 α-conversion.

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 subgoal ============================ forall m n : nat, n <= m -> m - n + n = m

might start with

move=> m n le_n_m.: 1 subgoal 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 subgoal 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

move: m le_n_m => p le_n_p.

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 there SSReflect variants, these tactic 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 with elim: n. Instead of elim n (note, no colon), this has the advantage of removing n from the context. Better yet, this elim can be combined with previous move 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 subgoal ============================ forall m n : nat, n <= m -> m - n + n = m
move=> m n le_n_m.: 1 subgoal 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 subgoals m : nat ============================ m - 0 + 0 = m subgoal 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 constants a, b, c as goal variables before performing tactic.
tactic => a b c pops the top three goal variables as context constants a, b, c, after 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

elim=> [|m IHm] n le_n.

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

move: a H1 H2; tactic => a H1 H2.

with two differences: the in tactical will preserve the body of an 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

elim: n => [|n IHn] in m le_n_m *.

instead of

elim: n m le_n_m => [|n IHn] m le_n_m.

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 subgoal ============================ ~ False
move.: 1 subgoal ============================ 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 a let … in, and performs hnf otherwise.

Of course this tactic is most often used in combination with the bookkeeping tacticals (see section Introduction in the context and Discharge). These combinations mostly subsume the intros, generalize, revert, rename, clear and pattern 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, the case 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 the spurious-ssr-injection key.

Finally the case tactic of SSReflect performs False elimination, even if no branch is generated by this case operation. Hence the tactic case on a goal of the form False -> 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 subgoal m : nat ============================ forall n : nat, m <= n
elim.: 2 subgoals m : nat ============================ m <= 0 subgoal 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 of term 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 of refine tactics tries to catch the appropriate number of wildcards to be inserted. Note that this use of the refine tactic implies that the tactic tries to match the goal up to expansion of constants and evaluation of subterms.

apply (ssreflect) 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 subgoal ============================ 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 focused subgoals (shelved: 2) y : nat y_gt1 : 1 < y y_lt2 : y < 2 ============================ y < 3 subgoal 2 is: forall Hyp0 : y < 3, 0 < proj1_sig (exist (fun n : nat => n < 3) y Hyp0)
by apply: lt_trans y_lt2 _.: 1 focused subgoal (shelved: 1) 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 subgoals.

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?¶

d_item::=occ_switchclear_switch? termclear_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 or apply, the exact tactic (section Terminators), the congr tactic (section Congruence), or the application of the view tactical ‘/’ (section Interpreting assumptions) to one of move, case, or elim.
The optional ident specifies equation generation (section Generation of equations), and is only allowed if tactic is move, case or elim, or the application of the view tactical ‘/’ (section Interpreting assumptions) to case or elim.
An occ_switch selects occurrences of term, as in Abbreviations; occ_switch is not allowed if tactic is apply or exact.
A clear item clear_switch specifies facts and constants to be deleted from the proof context (as per the clear tactic).

The : tactical first discharges all the d_item, right to left, and then performs 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 term; however if tactic is apply or exact a different matching algorithm, described below, is used 14.
These occurrences are replaced by a new variable; in particular, if term is a fact, this adds an assumption to the goal.
If 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, 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 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 term.

For example, the tactic:

move: n {2}n (refl_equal n).

first generalizes (refl_equal n : n = n);
then generalizes the second occurrence of n.
finally generalizes all the other occurrences of n, and clears n from the proof context (assuming n is a proof constant).

Therefore this tactic changes any goal G into

forall n n0 : nat, n = n0 -> G.

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 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 tactic is move or case and an equation ident is given, then clear (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 clear 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 subgoal Hfg : forall x : nat, f x = g x a, b : nat ============================ f a = g b
apply: trans_equal (Hfg _) _.: 1 focused subgoal (shelved: 1) 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

Tactic tactic => i_item+¶

i_item::=i_patterns_itemclear_switchi_viewi_blocks_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_items, left to right. An s_item specifies a simplification operation; a clear_switch specifies context pruning as in Discharge. The i_patterns 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 tactic done described in Terminators, i.e., it executes try done.
/= simplifies the goal by performing partial evaluation, as per the tactic simpl 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.: Identifier 'myop' now a keyword Toplevel input, characters 0-59: > Notation "'myop'" := (ltac:(my ltac code)) : ssripat_scope. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: This notation contains Ltac expressions: it will not be used for printing. [non-reversible-notation,parsing] Setting notation at level 0.

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_patterns:

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 δ-expansion is required to expose the top variable or assumption. A clear_switch (even an empty one) immediately preceding an ident is complemented with that ident if and only if the identifier is a simple proof context entry 20. As a consequence by prefixing the ident with {} one can replace a context entry. This behavior can be avoided by separating the clear_switch from the ident with the - intro pattern.

>

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 to move=> ? ? on a goal such as:

forall x y, x < y -> G

A typical use if move=>> H to name H the first assumption, in the example above x < 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 last i_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 ? and move the * i_item does not expand definitions in the goal to expose quantifiers, so it may be useful to repeat a move=> * 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 second move=> * 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 SSReflect rewrite 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 tactic is not a move, is a branchingi_pattern. It executes the sequence i_item_i on the i-th subgoal produced by tactic. The execution of 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 tactic, in which case exactly m subgoals should remain after the s_item, or we have the trivial branching i_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 tactic is a move, is a destructing i_pattern. It starts by destructing the top variable, using the SSReflect case tactic described in The defective tactics. It then behaves as the corresponding branching i_pattern, executing the sequence i_item_i in the i-th subgoal generated by the case analysis; unless we have the trivial destructing i_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 the i_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 in move=> -[H1 H2]. It can also be used to indicate explicitly the link between a view and a name like in move=> /eqP-H1. Last, it can serve as a separator between views. Section Views and reflection 19 explains in which respect the tactic move=> /v1/v2 differs from the tactic move=> /v1-/v2.

[: ident …]

introduces in the context an abstract constant for each ident. Its type has to be fixed later on by using the abstract 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.: Overwriting previous delimiting key bool in scope bool_scope
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 subgoal x, y : nat ============================ Nat.leb 0 x = true -> Nat.leb 0 x && Nat.leb y 2 = true
move=> {x} ->.: 1 subgoal 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 tactic is a move, and in most of the remaining cases tactic is case or elim, which implies destruction. 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_blocks:

[^ 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 with ident. If the intro pattern immediately follows a call to elim 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 subgoal ============================ r -> True
Proof. move => [^ x ].: 1 subgoal 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:

move En: (size l) => n.

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:

pose n := (size l).

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 subgoal a, b : nat ============================ a <> b
case E : a => [|n].: 2 subgoals a, b : nat E : a = 0 ============================ 0 <> b subgoal 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 subgoal a, b : nat ============================ a <> b
case E : a => H.: 2 subgoals a, b : nat E : a = 0 H : 0 = b ============================ False subgoal 2 is: False
Show 2.: subgoal 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 a d_item immediately following this / switch. The d_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 arguments 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 looking at the type of the top assumption. This allows for the compact syntax:

case: {2}_ / eqP.

where _ is interpreted as (_ == _) since eqP T a b : reflect (a = b) (a == b) and reflect 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 which 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 subgoal A : Type ============================ forall l, last_spec l
Admitted.: lastP is declared

We are now ready to use lastP in conjunction with case.

Lemma test l : (length l) * 2 = length (l ++ l).: 1 subgoal A : Type l : list A ============================ length l * 2 = length (l ++ l)
case: (lastP l).: 2 subgoals A : Type l : list A ============================ length nil * 2 = length (nil ++ nil) subgoal 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 tactc case: l / (lastP l) generates the same subgoals but l has been cleared from both contexts:

case: l / (lastP l).: 2 subgoals A : Type ============================ length nil * 2 = length (nil ++ nil) subgoal 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 subgoals A : Type l : list A ============================ length nil * 2 = length (l ++ nil) subgoal 2 is: forall (s : list A) (x : A), length (add_last x s) * 2 = length (l ++ add_last x s)

Note that selected occurrences on the left of the / switch have been substituted with l 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 dependent d_item specified by the user. Again starting with the above goal, the command:

Example

Lemma test l : (length l) * 2 = length (l ++ l).: 1 subgoal A : Type l : list A ============================ length l * 2 = length (l ++ l)
case E: {1 3}l / (lastP l) => [|s x].: 2 subgoals A : Type l : list A E : l = nil ============================ length nil * 2 = length (l ++ nil) subgoal 2 is: length (add_last x s) * 2 = length (l ++ add_last x s)
Show 2.: subgoal 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, the d_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 above 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 which fails if it does not solve the current goal, like discriminate, contradiction or assumption.

In fact, SSReflect provides a generic tactical which turns any tactic into a closing one (similar to now). Its general syntax is:

Tactic by tactic¶

In the script provided as example in section Indentation and bullets, the paragraph corresponding to each sub-case ends with a tactic line prefixed with a by, like in:

by apply/eqP; rewrite -dvdn1.

Tactic done¶

The by tactical is implemented using the user-defined, and extensible done tactic. This done tactic tries to solve the current goal by some trivial means and fails if it doesn’t succeed. Indeed, the tactic expression by tactic is equivalent to tactic; done.

Conversely, the tactic by [ ] is equivalent to done.

The default implementation of the done tactic, in the ssreflect.v file, is:

The lemma not_locked_false_eq_true is needed to discriminate locked boolean predicates (see section Locking, unlocking). The iterator tactical do is presented in section Iteration. This tactic can be customized by the user, for instance to include an auto tactic.

A natural and common way of closing a goal is to apply a lemma which is the exact one needed for the goal to be solved. The defective form of the tactic:

exact.

is equivalent to:

do [done | by move=> top; apply top].

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:

exact: MyLemma.

is equivalent to:

by apply: MyLemma.

(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:

by rewrite my_lemma1.

succeeds, then the tactic:

by rewrite my_lemma1; apply my_lemma2.

usually fails since it is equivalent to:

by (rewrite my_lemma1; apply my_lemma2).

Selectors¶

Tactic last¶

Tactic first¶

When composing tactics, the two tacticals first and last 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 analogue tactic ; 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.

Variant last natural first: If natural's value is $k$, this tactic rotates the $n$ subgoals $G_1$ , …, $G_n$ in focus. Subgoal $G_{n + 1 − k}$ becomes the first, and the circular order of subgoals remains unchanged.

Tactic first natural last¶: If natural's value is $k$, this tactic rotates the $n$ subgoals $G_1$ , …, $G_n$ in focus. Subgoal $G_{k + 1 \bmod n}$ becomes the first, and the circular order of subgoals remains unchanged.

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

tactic ; last k [ tactic1 |…| tacticm ] || tacticn.

where natural denotes the integer $k$ as above, applies tactic1 to the $n−k+1$-th goal, … tacticm to the $n−k+2$-th goal and tacticn to the others.

Example

Here is a small example on lists. We define first a function which 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 subgoal n : nat t : test n ============================ True
case: t; last 2 [move=> k| move=> l]; idtac.: 4 subgoals n : nat ============================ forall n0 : nat, n0 = 1 -> True subgoal 2 is: k = 2 -> True subgoal 3 is: l = 3 -> True subgoal 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:

do [ tactic 1 | … | tactic n ].

is equivalent to the standard Ltac expression:

first [ tactic 1 | … | tactic n ].

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:

n! the step tactic is repeated exactly n times (where n is a positive integer argument).
! the step tactic is repeated as many times as possible, and done at least once.
? the step tactic is repeated as many times as possible, optionally.
n? the step tactic is repeated up to n times, optionally.

For instance, the tactic:

tactic; do 1? rewrite mult_comm.

rewrites at most one time the lemma mult_comm in all the subgoals generated by tactic, whereas the tactic:

tactic; do 2! rewrite mult_comm.

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:

Tactic tactic in ident+ *?¶

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 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 subgoal x, y : nat H1 : x = y H2 : y = 3 ============================ x + y = 6
do [mytac H2] in H1 *.: 1 subgoal 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 over x and reintroduces the generalized variable under the name y (and does not clear x). 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 make 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 goal G, it generates a first subgoal requiring a proof of term in the context of G. The second generated subgoal is of the form term -> G, where term becomes the new top assumption, instead of being introduced with a fresh name. At the proof-term level, the have 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 subgoal ============================ True
have: _ * 0 = 0.: 2 subgoals ============================ forall n : nat, n * 0 = 0 subgoal 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 subgoal ============================ True

have: forall n : nat, n * 0 = 0.: 2 subgoals ============================ forall n : nat, n * 0 = 0 subgoal 2 is: (forall n : nat, n * 0 = 0) -> True

The have tactic also enjoys the same abstraction mechanism as the pose tactic for the non-inferred 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 subgoal ============================ True

have: forall x y, (x, y) = (x, y + 0).: 2 subgoals ============================ forall (T : Type) (x : T) (y : nat), (x, y) = (x, y + 0) subgoal 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:

Tactic have i_item* i_pattern? s_itemssr_binder+? : term? := termby tactic?¶

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 sub-proof for an intermediate result, uses tactics of the form:

Variant have clear_switch i_item : term by tactic

which behave like:

have: term ; first by tactic. move=> clear_switch i_item.

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:

have H23 : 3 + 2 = 2 + 3 by rewrite addnC.

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 subgoal a : nat ============================ 3 * a - 1 = a
have -> : forall x, x * a = a.: 2 subgoals a : nat ============================ forall x : nat, x * a = a subgoal 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):

have {x} -> : x = y.

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 subgoal ============================ True
have [x Px]: exists x : nat, x > 0; last first.: 2 subgoals x : nat Px : x > 0 ============================ True subgoal 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 optional ident is present, this assumption is introduced under the name ident. Note that the body of the constant is lost for the user.

Again, non inferred 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 subgoal ============================ True
have H := forall x, (x, x) = (x, x).: 1 subgoal 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 ... done] [Loading ML file zify_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done]
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Lemma test : True.: 1 subgoal ============================ True
have H x (y : nat) : 2 * x + y = x + x + y by lia.: 1 subgoal 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 subgoal ============================ (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 which 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 subgoals H : 2 + 2 = 3 -> True ============================ True subgoal 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 subgoal n, m : nat H : m + 1 < n ============================ True
have @i : 'I_n by apply: (Sub m); lia.: 1 subgoal n, m : nat H : m + 1 < n i := Sub m ((fun ... => Morphisms.iff_flip_impl_subrelation ... ...%Z lemma) (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 subgoal n, m : nat H : m + 1 < n ============================ True
have [:pm] @i : 'I_n by apply: (Sub m); abstract: pm; lia.: 1 subgoal 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 subgoal n, m : nat H : m + 1 < n ============================ True
have [:pm] @i : 'I_n := Sub m pm.: 2 subgoals n, m : nat H : m + 1 < n ============================ S m <= n subgoal 2 is: True
by lia.: 1 subgoal 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 subgoal n, m : nat H : m + 1 < n ============================ True
have [:pm] @i k : 'I_(n+k) by apply: (Sub m); abstract: pm k; lia.: 1 subgoal 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 subgoal ============================ True

have foo : ty.
2 subgoals ============================ ty subgoal 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 in ty. 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 subgoal foo : ty ============================ True

No inference for ty and t.

have foo := t.
1 subgoal foo : ty ============================ True

No inference for t. Unresolved instances are quantified in the (inferred) type of t and abstracted in t.

Flag SsrHave NoTCResolution¶: This flag restores the behavior of SSReflect 1.4 and below (never resolve typeclasses).

Variants: the suff and wlog tactics¶

As it 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. This 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 subgoal ============================ G
suff have H : P.: 2 subgoals H : P ============================ G subgoal 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 which 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:

Variant wlog: / term
Variant without loss: / term

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:

move=> clear_switch i_item.

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 subgoal 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 subgoals 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) subgoal 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 subgoal 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.: Overwriting previous delimiting key bool in scope bool_scope
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 subgoal n : nat ngt0 : 0 < n ============================ P n
gen have ltnV, /andP[nge0 neq0] : n ngt0 / (0 <= n) && (n != 0); last first.: 2 subgoals n : nat ngt0 : 0 < n ltnV : forall n : nat, 0 < n -> (0 <= n) && (n != 0) nge0 : 0 <= n neq0 : n != 0 ============================ P n subgoal 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 let-in redex is created, which keeps track if 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 let-in 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 subgoal x : nat ============================ x <= addx x
wlog H : (y := x) (@twoy := addx x) / twoy = 2 * y.: 2 subgoals x : nat ============================ (forall y : nat, let twoy := y + y in twoy = 2 * y -> y <= twoy) -> x <= addx x subgoal 2 is: y <= twoy

To avoid unfolding the term captured by the pattern add x one can use the pattern id (addx x), that 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 subgoal x : nat ============================ x <= addx x

wlog H : (y := x) (@twoy := id (addx x)) / twoy = 2 * y.: 2 subgoals x : nat ============================ (forall y : nat, let twoy := addx y in twoy = 2 * y -> y <= addx y) -> x <= addx x subgoal 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 mean 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:

Tactic rewrite rstep+¶

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:

rstep::=r_prefix? r_itemr_prefix::=-? mult? occ_switchclear_switch? [ r_pattern ]?r_pattern::=termin ident in? termterm interm as ident in termr_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 of r_item: if present the direction is right-to-left and it is left-to-right 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 the r_item.
This optional term, or the r_item, may be preceded by an occ_switch (see section Selectors) or a clear_switch (see section Discharge), these two possibilities being exclusive.

An occurrence switch selects the occurrences of the rewrite pattern which 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 an only if it is a simple proof context entry 20. As a consequence one can write rewrite {}H to rewrite with H and dispose H immediately afterwards. This behavior can be avoided by putting parentheses around the rewrite rule.

An r_item can be:

A simplification r_item, represented by a s_item (see section Introduction in the context). Simplification operations are intertwined with the possible other rewrite operations specified by the list of r_item.
A folding/unfolding r_item. The tactic: rewrite /term unfolds the head constant of term in every occurrence of the first matching of term in the goal. In particular, if my_def is a (local or global) defined constant, the tactic: rewrite /my_def. is analogous to: unfold my_def. Conversely: rewrite -/my_def. is equivalent to: fold my_def. When an unfold r_item is combined with a redex pattern, a conversion operation is performed. A tactic of the form: rewrite -[term1]/term2. is equivalent to: change term1 with term2. If term2 is a single constant and term1 head symbol is not term2, then the head symbol of term1 is repeatedly unfolded until term2 appears.
A term, which can be:
- A term whose type has the form: forall (x1 : A1 )…(xn : An ), eq term1 term2 where eq is the Leibniz equality or a registered setoid equality.
- A list of terms (t1 ,…,tn), each ti having a type above. The tactic: rewrite r_prefix (t1 ,…,tn ). is equivalent to: do [rewrite r_prefix t1 | … | rewrite r_prefix tn ].
- An anonymous rewrite lemma (_ : term), where term 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 subgoal x : nat ============================ ddouble x = 4 * x

rewrite [ddouble _]/double.
1 subgoal 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 subgoal x, y : nat ============================ x + y = f y x

rewrite -[f y]/(y + _).
Toplevel input, characters 0-22: > 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 subgoal x, y : nat ============================ x + y = y + x

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:

rewrite occ_switch [term1]term2.

term1 is the explicit rewrite redex and term2 is the rewrite rule. This execution of this tactic unfolds as follows:

First term1 and term2 are βι normalized. Then term2 is put in head normal form if the Leibniz equality constructor eq 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 of term2 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 left hand 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 right hand side of term2.
Finally the goal is βι normalized.

In the case term2 is a list of terms, the first top-down (in the goal) left-to-right (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 which 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:

rewrite /my_def {2}[f _]/= my_eq //=.

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 subgoal m, n, p : nat ============================ m + (n + p) = n + (m + p)
by elim: m p => [ | m Hrec] p; rewrite ?addSnnS -?addnS.: No more subgoals.
Qed.
Lemma addnC n m : m + n = n + m.: 1 subgoal n, m : nat ============================ m + n = n + m
by rewrite -{1}[n]addn0 addnCA addn0.: No more subgoals.
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 left hand 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 subgoal H : forall t u : nat, t + u = u + t x, y : nat ============================ x + y = y + x
rewrite [y + _]H.: 1 subgoal 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 left hand 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 subgoal 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 left hand 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 subgoal x, y : nat ============================ x + y + 0 = x + y + y + 0 + 0 + (x + y + 0)
rewrite {2}[_ + y + 0](_: forall z, z + 0 = z).: 2 subgoals x, y : nat ============================ forall z : nat, z + 0 = z subgoal 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 subgoal x, y, z : nat ============================ x + 1 = x + y + 1
rewrite 2!(_ : _ + 1 = z).: 4 subgoals x, y, z : nat ============================ x + 1 = z subgoal 2 is: z = z subgoal 3 is: x + y + 1 = z subgoal 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.

Multi-rule 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 subgoal a, b, c : nat eqab : a = b eqac : a = c ============================ a = a
rewrite (eqab, eqac).: 1 subgoal 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 left-to-right traversal of the goal, with the first rule from the multirule tree in left-to-right 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 subgoal a, b, c : nat eqab : a = b eqac : a = c eqd0 : d = 0 ============================ d = b
rewrite multi2.: 1 subgoal 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 subgoal 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 subgoal 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 left-hand 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, 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:

Lemma addE : add =2 addn. Lemma doubleE : double =1 doublen. Lemma add_mulE n m s : add_mul n m s = addn (muln n m) s. Lemma mulE : mul =2 muln. Lemma mul_expE m n p : mul_exp m n p = muln (expn m n) p. Lemma expE : exp =2 expn. Lemma oddE : odd =1 oddn.

The operation on the left hand side of each lemma is the efficient version, and the corresponding naive implementation is on the right hand side. In order to reason conveniently on expressions involving the efficient operations, we gather all these rules in the definition trecE:

Definition trecE := (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))).

The tactic: rewrite !trecE. restores the naive versions 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_items 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 subgoal y, z : nat ============================ y * 0 + y * (z * 0) = 0
rewrite (_ : _ * 0 = 0).: 2 subgoals y, z : nat ============================ y * 0 = 0 subgoal 2 is: 0 + y * (z * 0) = 0

while the other tactic results in

rewrite (_ : forall x, x * 0 = 0).: 2 subgoals y, z : nat ============================ forall x : nat, x * 0 = 0 subgoal 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 which 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 subgoal g : nat -> nat ============================ f 3 + f 3 = f 6

rewrite -> H.
1 subgoal 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 0-9: > rewrite H. > ^^^^^^^^^ Error: The LHS of H (g _) does not match any subterm of the goal

Rewriting with H first requires unfolding the occurrences of f where the substitution is to be performed (here there is a single such occurrence), using tactic rewrite /f (for a global replacement of f by g) or rewrite pattern/f, for a finer selection.

rewrite /f H.
1 subgoal g : nat -> nat ============================ 0 + 0 = g 6

alternatively one can override the pattern inferred from H

rewrite [f _]H.
1 subgoal 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.: Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Set Warnings "-notation-overridden".

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 subgoal x : 'I_2 y : nat ============================ Some x = insub 2 y
rewrite insubT.: 2 subgoals x : 'I_2 y : nat ============================ forall Hyp0 : y < 2, Some x = Some (Sub y Hyp0) subgoal 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 subgoal x : 'I_2 y : nat H : y < 2 ============================ Some x = insub 2 y
rewrite insubT.: 1 subgoal 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 higher-order 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 subgoal 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 0-14: > 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 non-inferable 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 subgoals l : list nat ============================ forall n : nat, n - n = 0 subgoal 2 is: sumlist (map (fun _ : nat => 0) l) = 0
by move=> m; rewrite subnn.: 1 subgoal 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 subgoal l : list nat ============================ sumlist (map (fun m : nat => m - m) l) = 0
under eq_map => m do rewrite subnn.: 1 subgoal 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*| ]?¶

Operate 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 one-liner 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 subgoal l : list nat ============================ sumlist (map (fun m : nat => m - m) l) = 0
under eq_map => m.: 2 focused subgoals (shelved: 2) l : list nat m : nat ============================ 'Under[ m - m ] subgoal 2 is: sumlist (map ?Goal l) = 0
rewrite subnn.: 2 focused subgoals (shelved: 2) l : list nat m : nat ============================ 'Under[ 0 ] subgoal 2 is: sumlist (map ?Goal l) = 0
over.: 1 focused subgoal (shelved: 1) l : list nat ============================ sumlist (map (fun _ : nat => 0) l) = 0

The execution of the Ltac expression:

under term => [ i_item₁ | … | i_item_n ].

involves the following steps:

It performs a rewrite term without failing like in the first example with rewrite eq_map., but creating evars (see evar). If term is prefixed by a pattern or an occurrence selector, then the modifiers are honoured.
As a n-branches 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 as forall n, F1 n = F2 n for eq_map).
If so under puts these n goals in head normal form (using the defective form of the tactic move), then executes the corresponding intro pattern i_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. Class RewriteRelation) between a term and an evar, e.g. m - m = ?F2 m in the running example. (This support for setoid-like relations is enabled as soon as we do both Require Import ssreflect. and Require 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 regular rewrite 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, but PreOrder relations are also supported, for example.)
Finally, a post-processing 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 the over 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).

One-liner mode¶

The Ltac expression:

under term => [ i_item₁ | … | i_item_n ] do [ tactic₁ | … | tactic_n ].

can be seen as a shorter form for the following expression:

Notes:

The beta-iota reduction here is useful to get rid of the beta redexes that could be introduced after the substitution of the evars by the under tactic.
Note that the provided tactics can as well involve other under tactics. See below for a typical example involving the bigop 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 default i_item * is applied to each branch. E.g., the Ltac expression: under term do [ tactic₁ | … | tactic_n ] is equivalent to: under term => [ * | … | * ] do [ tactic₁ | … | tactic_n ] (and it can be noted here that the under tactic performs a move. 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 subgoal 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 subgoal 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:

Lemma master_key : unit. Proof. exact tt. Qed. Definition locked A := let: tt := master_key in fun x : A => x. Lemma lock : forall A x, x = locked x :> A.

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.: Overwriting previous delimiting key bool in scope bool_scope
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 subgoal 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 subgoal 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 subgoal ============================ lid 3 = 3
rewrite /=.: 1 subgoal ============================ lid 3 = 3
unlock lid.: 1 subgoal ============================ 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 the occ_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:

Notation "'nosimpl' t" := (let: tt := tt in t).

The term (nosimpl t) simplifies to t except in a definition. More precisely, given:

Definition foo := (nosimpl bar).

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:

Definition foo x := nosimpl (bar x).

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:

Definition foo x := nosimpl bar x.

A standard example making this technique shine is the case of arithmetic operations. We define for instance:

Definition addn := nosimpl plus.

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:

apply: my_congr_property.

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 non dependent product P -> Q. In that case, the system asserts the equation P = Q, uses it to solve the goal, and calls the congr tactic on the remaining goal P = 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 subgoal x, y, z : nat H : x = y ============================ x = z
congr (_ = _) : H.: 1 focused subgoal (shelved: 1) x, y, z : nat ============================ y = z
Abort.
Lemma test (x y z : nat) : x = y -> x = z.: 1 subgoal x, y, z : nat ============================ x = y -> x = z
congr (_ = _).: 1 focused subgoal (shelved: 1) 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 subgoal x, y : nat ============================ f 0 x y = g 1 1 x y
congr plus.: No more subgoals.

This script shows that the congr tactic matches plus with f 0 on the left hand side and g 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 subgoal n, m : nat Hnm : m <= n ============================ S m + (S n - S m) = S n
congr S; rewrite -/plus.: 1 subgoal n, m : nat Hnm : m <= n ============================ m + (S n - S m) = n

The tactic rewrite -/plus folds back the expansion of plus which was necessary for matching both sides of the equality with an application of S.

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 subgoal x, y : nat ============================ x + y * (y + x - x) = x * 1 + (y + 0) * y
congr ( _ + (_ * _)).: 3 focused subgoals (shelved: 3) x, y : nat ============================ x = x * 1 subgoal 2 is: y = y + 0 subgoal 3 is: y + x - x = y

Contextual patterns¶

The simple form of patterns used so far, terms possibly containing wild cards, often require 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 implicit arguments machinery or by user-defined 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 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.

`c_pattern`	redex	subterms affected
`term`	`term`	all occurrences of `term`
`ident in term`	subterm of `term` selected by `ident`	all the subterms identified by `ident` in all the occurrences of `term`
`term1 in ident in term2`	`term1` in all s.m.r.	in all the subterms identified by `ident` in all the occurrences of `term2`
`term1 as ident in term2`	`term 1`	in all the subterms identified by `ident` in all the occurrences of `term2[term 1 /ident]`

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.

`r_pattern`	redex	subterms affected
`in term`	inferred from rule	in all s.m.r. in all occurrences of `term`
`in ident in term`	inferred from rule	in all s.m.r. in all the subterms identified by `ident` in all the occurrences of `term`

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_patterns 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

rewrite [in X in _ = X]rule.

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 left hand 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:

`c_pattern`	instantiation order and place for `term_i` and redex
`term`	`term` is matched against the goal, redex is unified with the instantiation of `term`
`ident in term`	`term` is matched against the goal, redex is unified with the subterm of the instantiation of `term` identified by `ident`
`term1 in ident in term2`	`term2` is matched against the goal, `term1` is matched against the subterm of the instantiation of `term1` identified by `ident`, redex is unified with the instantiation of `term1`
`term1 as ident in term2`	`term2[term1/ident]` is matched against the goal, redex is unified with the instantiation of `term1`

In the following patterns, the redex is intended to be inferred from the rewrite rule.

`r_pattern`	instantiation order and place for `term_i` and redex
`in ident in term`	`term` is matched against the goal, the redex is matched against the subterm of the instantiation of `term` identified by `ident`
`in term`	`term` is matched against the goal, redex is matched against the instantiation of `term`

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 subgoal a, b : nat ============================ a + b + 1 = b + (a + 1)
set t := (X in _ = X).: 1 subgoal a, b : nat t := b + (a + 1) : nat ============================ a + b + 1 = t
rewrite {}/t.: 1 subgoal a, b : nat ============================ a + b + 1 = b + (a + 1)
set t := (a + _ in X in _ = X).: 1 subgoal 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.

set t := ((a in A)).

Contextual patterns can also be used as arguments of the : tactical. For example:

elim: n (n in _ = n) (refl_equal n).

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 subgoal 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 subgoal 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 subgoal x, y, z : nat f : nat -> nat -> nat ============================ (x + y).+1 + f (x + y).+1 (z + (x + y).+1) = 0 + 0

Note that the right hand side of addn0 is undetermined, but the rewrite pattern specifies the redex explicitly. The right hand 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 subgoal 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 subgoal 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 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, that is thus unified with the left hand side of the rewrite rule.

rewrite [x.+1 + y as X in f X _]addnC.: 1 subgoal 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.

Notation RHS := (X in _ = X)%pattern. Notation LHS := (X in X = _)%pattern.

Shortcuts defined this way can be freely used in place of the trailing ident in term part of any contextual pattern. Some examples follow:

set rhs := RHS. rewrite [in RHS]rule. case: (a + _ in RHS).

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 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:

elim/V.

is a synonym for:

intro top; elim top using V; clear top.

where top is a fresh name and V any second-order lemma.

Since an elimination view supports the two bookkeeping tacticals of discharge and introduction (see section Basic tactics), the SSReflect tactic:

elim/V: x => y.

is a synonym for:

elim x using V; clear x; intro y.

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 equation name 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 result in a concise syntax for reasoning inductively using the user-defined elimination scheme.

Lemma test (x : d) (l : list d): l = l.: 1 subgoal d : Type x : d l : list d ============================ l = l
elim/last_ind_list E : l=> [| u v]; last first.: 2 subgoals 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 subgoal 2 is: nil = nil

User-provided 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:

foo_ind : forall …, forall x : T, P p1 … pm.

and consider the tactic:

elim/foo_ind: e1 … / en.

The elim/ tactic distinguishes two cases:

truncated eliminator: when x does not occur in P p1 … pm and the type of en unifies with T and en is not _. In that case, en is passed to the eliminator as the last argument (x in foo_ind) and en−1 … e1 are used as patterns to select in the goal the occurrences that will be bound by the predicate P, thus it must be possible to unify the subterm of the goal matched by en−1 with pm , the one matched by en−2 with pm−1 and so on.
regular eliminator: in all the other cases. Here it must be possible to unify the term matched by en with pm , the one matched by en−1 with pm−1 and so on. Note that standard eliminators have the shape …forall x, P … x, thus en 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 ... done] [Loading ML file recdef_plugin.cmxs ... 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 (_ _)%function_scope _%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 subgoal 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.

elim/plus_ind: z / (plus _ z). elim/plus_ind: {z}(plus _ z). elim/plus_ind: {z}_. elim/plus_ind: z / _.

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 (_ _)%function_scope _%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 subgoal x, y, z : nat ============================ plus (plus x y) z = plus x (plus y z)

elim/plus_ind: z / _.: 2 subgoals 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)) subgoal 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 these examples, it is (plus _ _), which 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 (_ _)%function_scope _%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 subgoal 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.

Lemma test p n (n_gt0 : 0 < n) (pr_p : prime p) : p %| \prod_(i <- prime_decomp n | i \in prime_decomp n) i.1 ^ i.2 -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1. Proof. elim/big_prop: _ => [| u v IHu IHv | [q e] /=].

where the type of the big_prop eliminator is

big_prop: forall (R : Type) (Pb : R -> Type) (idx : R) (op1 : R -> R -> R), Pb idx -> (forall x y : R, Pb x -> Pb y -> Pb (op1 x y)) -> forall (I : Type) (r : seq I) (P : pred I) (F : I -> R), (forall i : I, P i -> Pb (F i)) -> Pb (\big[op1/idx]_(i <- r | P i) F i).

Since the pattern for the argument of Pb is not specified, the inferred one is used instead: big[_/_]_(i <- _ | _ i) _ i, and after the introductions, the following goals are generated:

subgoal 1 is: p %| 1 -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1 subgoal 2 is: p %| u * v -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1 subgoal 3 is: (q, e) \in prime_decomp n -> p %| q ^ e -> exists2 x : nat * nat, x \in prime_decomp n & p = x.1.

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 subgoal 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 subgoal 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 subgoal 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 subgoal 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 subgoal 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 equation name 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 subgoal 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 subgoals P, Q : bool -> Prop Q2P : forall a b : bool, Q (a || b) -> P a \/ P b a, b : bool HPa : P a ============================ True subgoal 2 is: True

This view tactic performs:

move=> HQ; case: {HQ}(Q2P HQ) => [HPa | HPb].

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 subgoal 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 subgoal 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:

Lemma test a b : P (a || b) -> True. move/(iffLR (PQequiv _ _)).

where:

Lemma iffLR P Q : (P <-> Q) -> P -> Q.

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 higher-order 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 subgoal z : nat ============================ (forall x y : nat, x + y = z -> z = x) -> z = 0
move/(_ 0 z).: 1 subgoal z : nat ============================ (0 + z = z -> z = 0) -> z = 0

Interpreting goals¶

In a similar way, it is also often convenient to changing 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.: Overwriting previous delimiting key bool in scope bool_scope
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 subgoal P, Q : bool -> Prop PQequiv : forall a b : bool, P (a || b) <-> Q a a : bool ============================ P (~~ a || a)
apply/PQequiv.: 1 focused subgoal (shelved: 1) 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 iffRL 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

Goal interpretation 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.: Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Test.

Lemma test b : b || ~~ b = true.: 1 subgoal b : bool ============================ b || ~~ b = true
by case: b.: No more subgoals.

Once b is replaced by true in the first goal and by false in the second one, the goals reduce by computations to the trivial true = true.

Thus, Prop and bool are truly complementary: the former supports robust natural deduction, the latter allows brute-force 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:

Coercion is_true (b : bool) := b = true.

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 pretty-printer, 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:

Inductive reflect (P: Prop): bool -> Type := | Reflect_true : P -> reflect P true | Reflect_false : ~P -> reflect P false.

The statement (reflect P b) asserts that (is_true b) and P are logically equivalent propositions.

For instance, the following lemma:

Lemma andP: forall b1 b2, reflect (b1 /\ b2) (b1 && b2).

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:

Definition iff (A B:Prop) := (A -> B) /\ (B -> A).

where /\ is a notation for and:

Inductive and (A B:Prop) : Prop := conj : A -> B -> and A B.

This make case analysis very different according to the way an equivalence property has been defined.

Lemma andE (b1 b2 : bool) : (b1 /\ b2) <-> (b1 && b2).

Let us compare the respective behaviors of andE and andP.

Example

From Coq Require Import ssreflect ssrbool.: Overwriting previous delimiting key bool in scope bool_scope
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 ~~(b1||b2).: 1 subgoal b1, b2 : bool ============================ if b1 && b2 then b1 else ~~ (b1 || b2)

case: (@andE b1 b2).: 1 subgoal b1, b2 : bool ============================ (b1 /\ b2 -> b1 && b2) -> (b1 && b2 -> b1 /\ b2) -> if b1 && b2 then b1 else ~~ (b1 || b2)

Restart.: 1 subgoal b1, b2 : bool ============================ if b1 && b2 then b1 else ~~ (b1 || b2)

case: (@andP b1 b2).: 2 subgoals b1, b2 : bool ============================ b1 /\ b2 -> b1 subgoal 2 is: ~ (b1 /\ b2) -> ~~ (b1 || b2)

Expressing reflection relation 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.: Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Test.

Lemma test (a b : bool) (Ha : a) (Hb : b) : a /\ b.: 1 subgoal a, b : bool Ha : a Hb : b ============================ a /\ b
apply/andP.: 1 focused subgoal (shelved: 1) a, b : bool Ha : a Hb : b ============================ a && b

Conversely

Lemma test (a b : bool) : a /\ b -> a.: 1 subgoal a, b : bool ============================ a /\ b -> a
move/andP.: 1 subgoal 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 which should be chosen.

General mechanism for interpreting goals and assumptions¶

Specializing assumptions¶

The SSReflect tactic:

move/(_ term1 … termn).

is equivalent to the tactic:

intro top; generalize (top term1 … termn); clear top.

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 form: term term1 … termn where the terms term1 … termn instantiate the possible quantified variables of term , 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 term termvh 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 are 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:

Lemma iffLR : forall P Q, (P <-> Q) -> P -> Q.

which transforms a double implication into the left-to-right one, or:

Lemma iffRL : forall P Q, (P <-> Q) -> Q -> P.

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.: Overwriting previous delimiting key bool in scope bool_scope
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 subgoal a, b : bool Ha : a Hb : b ============================ ~~ (a && b)
apply/andP.: 1 focused subgoal (shelved: 1) 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.: Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Test.

Lemma test (a b : bool) (pab : b && a) : b.: 1 subgoal a, b : bool pab : b && a ============================ b
have /andP [pa ->] : (a && b) by rewrite andbC.: 1 subgoal 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 term term is applied to the current goal top, 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 user-defined 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:

Variant apply/term/term

The first term is the view lemma applied to the left hand side of the equality, while the second term is the one applied to the right hand side.

In this context, the identity view can be used when no view has to be applied:

Lemma idP : reflect b1 b1.

Example

From Coq Require Import ssreflect ssrbool.: Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Test.

Lemma test (b1 b2 b3 : bool) : ~~ (b1 || b2) = b3.: 1 subgoal b1, b2, b3 : bool ============================ ~~ (b1 || b2) = b3
apply/idP/idP.: 2 focused subgoals (shelved: 2) b1, b2, b3 : bool ============================ ~~ (b1 || b2) -> b3 subgoal 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.: Overwriting previous delimiting key bool in scope bool_scope
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Test.

Lemma test (b1 b2 b3 : bool) : ~~ (b1 || b2) = b3.: 1 subgoal b1, b2, b3 : bool ============================ ~~ (b1 || b2) = b3
apply/norP/idP.: 2 focused subgoals (shelved: 2) b1, b2, b3 : bool ============================ ~~ b1 /\ ~~ b2 -> b3 subgoal 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. If move 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 left hand side views. The optional natural number is the number of implicit arguments to be considered for the declared hint view lemma.

Variant Hint View for apply//ident | natural?

This variant with a double slash //, declares hint views for right hand sides of double views.

See the files ssreflect.v and ssrbool.v for examples.

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

apply/v1/v2/v3. apply/v1; apply/v2; apply/v3.

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 NO-OP intro pattern - can be used to separate two views, making the two following examples equivalent:

move=> /v1; move=> /v2. move=> /v1 - /v2.

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.: Overwriting previous delimiting key bool in scope bool_scope
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 subgoal P, Q, R : Prop P2Q : P -> Q Q2R : Q -> R p : P ============================ True
move/P2Q/Q2R in p.: 1 subgoal 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.

SSReflect searching tool¶

Command Search pattern? -? stringpattern % ident?* in -? qualid+?¶

Changed in version 8.12: This command is only available when loading a separate plugin (ssrsearch).

Deprecated since version 8.12: This command is deprecated since all the additional features it provides have been integrated in the standard Search command.

This is the SSReflect extension of the Search command. qualid is the name of an open module. This command returns the list of lemmas:

whose conclusion contains a subterm matching the optional first pattern. A - reverses the test, producing the list of lemmas whose conclusion does not contain any subterm matching the pattern;
whose name contains the given string. A - prefix reverses the test, producing the list of lemmas whose name does not contain the string. A string that contains symbols or is followed by a scope key, is interpreted as the constant whose notation involves that string (e.g., + for addn), if this is unambiguous; otherwise the diagnostic includes the output of the Locate command.
whose statement, including assumptions and types, contains a subterm matching the next patterns. If a pattern is prefixed by -, the test is reversed;
contained in the given list of modules, except the ones in the modules prefixed by a -.

Note

As for regular terms, patterns can feature scope indications. For instance, the command: Search _ (_ + _)%N. lists all the lemmas whose statement (conclusion or hypotheses) involves an application of the binary operation denoted by the infix + symbol in the N scope (which is SSReflect scope for natural numbers).
Patterns with holes should be surrounded by parentheses.
Search always volunteers the expansion of the notation, avoiding the need to execute Locate independently. Moreover, a string fragment looks for any notation that contains fragment as a substring. If the ssrbool.v library is imported, the command: Search "~~". answers :

From Coq Require Import ssreflect ssrbool.
Overwriting previous delimiting key bool in scope bool_scope

Set Implicit Arguments.

Unset Strict Implicit.

Unset Printing Implicit Defensive.

Require Import ssrsearch.
[Loading ML file ssrsearch_plugin.cmxs ... done]

Search "~~".
Toplevel input, characters 0-12: > Search "~~". > ^^^^^^^^^^^^ Warning: SSReflect's Search command is deprecated. [deprecated-ssr-search,deprecated] "~~" is part of notation ("~~ _") In bool_scope, ("~~ b") denotes negb b Toplevel input, characters 0-12: > Search "~~". > ^^^^^^^^^^^^ Warning: Listing only lemmas with conclusion matching (~~ ?b) negbT: forall b : bool, b = false -> ~~ b contra: forall c b : bool, (c -> b) -> ~~ b -> ~~ c contraNN: forall c b : bool, (c -> b) -> ~~ b -> ~~ c contraL: forall c b : bool, (c -> ~~ b) -> b -> ~~ c contraTN: forall c b : bool, (c -> ~~ b) -> b -> ~~ c contraFN: forall c b : bool, (c -> b) -> b = false -> ~~ c contra_notN: forall (P : Prop) (b : bool), (b -> P) -> ~ P -> ~~ b contraPN: forall (P : Prop) (b : bool), (b -> ~ P) -> P -> ~~ b introN: forall (P : Prop) (b : bool), reflect P b -> ~ P -> ~~ b
A diagnostic is issued if there are different matching notations; it is an error if all matches are partial.
Similarly, a diagnostic warns about multiple interpretations, and signals an error if there is no default one.
The command Search in M. is a way of obtaining the complete signature of the module M.
Strings and pattern indications can be interleaved, but the first indication has a special status if it is a pattern, and only filters the conclusion of lemmas:
- The command : Search (_ =1 _) "bij". lists all the lemmas whose conclusion features a =1 and whose name contains the string bij.
- The command : Search "bij" (_ =1 _). lists all the lemmas whose statement, including hypotheses, features a =1 and whose name contains the string bij.

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 which can be followed by a bracket [, like 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 Introduction in the context

i_item::=clear_switchs_itemi_patterni_viewi_block

view Introduction in the context

i_view::={}? /term/ltac:( tactic )

intro block 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 or hnf (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)

Variant rewrite r_step+: rewrite (see Rewriting)

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 wlog suff? i_item? : gen_itemclear_switch* / term¶: specializing (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 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 for nat subtraction”.
14: Also, a slightly different variant may be used for the first d_item of case and elim; see section Type families.
15: Except /= does not expand the local definitions created by the SSReflect in 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.