Chapter 8 Tactics
 8.1 Invocation of tactics
 8.2 Explicit proof as a term
 8.3 Basics
 8.4 Negation and contradiction
 8.5 Conversion tactics
 8.6 Introductions
 8.7 Induction and Case Analysis
 8.8 Equality
 8.9 Equality and inductive sets
 8.10 Inversion
 8.11 Classical tactics
 8.12 Automatizing
 8.13 Controlling automation
 8.14 Generation of induction principles with Scheme
 8.15 Generation of induction principles with Functional Scheme
 8.16 Simple tactic macros
A deduction rule is a link between some (unique) formula, that we call the conclusion and (several) formulas that we call the premises. Indeed, a deduction rule can be read in two ways. The first one has the shape: “if I know this and this then I can deduce this”. For instance, if I have a proof of A and a proof of B then I have a proof of A ∧ B. This is forward reasoning from premises to conclusion. The other way says: “to prove this I have to prove this and this”. For instance, to prove A ∧ B, I have to prove A and I have to prove B. This is backward reasoning which proceeds from conclusion to premises. We say that the conclusion is the goal to prove and premises are the subgoals. The tactics implement backward reasoning. When applied to a goal, a tactic replaces this goal with the subgoals it generates. We say that a tactic reduces a goal to its subgoal(s).
Each (sub)goal is denoted with a number. The current goal is numbered 1. By default, a tactic is applied to the current goal, but one can address a particular goal in the list by writing n:tactic which means “apply tactic tactic to goal number n”. We can show the list of subgoals by typing Show (see Section 7.3.1).
Since not every rule applies to a given statement, every tactic cannot be used to reduce any goal. In other words, before applying a tactic to a given goal, the system checks that some preconditions are satisfied. If it is not the case, the tactic raises an error message.
Tactics are build from atomic tactics and tactic expressions (which extends the folklore notion of tactical) to combine those atomic tactics. This chapter is devoted to atomic tactics. The tactic language will be described in Chapter 9.
There are, at least, three levels of atomic tactics. The simplest one implements basic rules of the logical framework. The second level is the one of derived rules which are built by combination of other tactics. The third one implements heuristics or decision procedures to build a complete proof of a goal.
8.1 Invocation of tactics
A tactic is applied as an ordinary command. If the tactic does not address the first subgoal, the command may be preceded by the wished subgoal number as shown below:
tactic_invocation  ::=  num : tactic . 
  tactic . 
8.2 Explicit proof as a term
8.2.1 exact term
This tactic applies to any goal. It gives directly the exact proof term of the goal. Let T be our goal, let p be a term of type U then exact p succeeds iff T and U are convertible (see Section 4.3).
Error messages:
Variants:
8.2.2 refine term
This tactic allows to give an exact proof but still with some holes. The holes are noted “_”.
Error messages:
 invalid argument: the tactic refine doesn’t know what to do with the term you gave.
 Refine passed illformed term: the term you gave is not a valid proof (not easy to debug in general). This message may also occur in higherlevel tactics, which call refine internally.
 Cannot infer a term for this placeholder there is a hole in the term you gave which type cannot be inferred. Put a cast around it.
An example of use is given in Section 10.1.
8.3 Basics
Tactics presented in this section implement the basic typing rules of pCic given in Chapter 4.
8.3.1 assumption
This tactic applies to any goal. It implements the “Var” rule given in Section 4.2. It looks in the local context for an hypothesis which type is equal to the goal. If it is the case, the subgoal is proved. Otherwise, it fails.
Error messages:
Variants:
8.3.2 clear ident
This tactic erases the hypothesis named ident in the local context of the current goal. Then ident is no more displayed and no more usable in the proof development.
Variants:
 clear ident_{1} … ident_{n}
This is equivalent to clear ident_{1}. … clear ident_{n}.
 clearbody ident
This tactic expects ident to be a local definition then clears its body. Otherwise said, this tactic turns a definition into an assumption.
 clear  ident_{1} … ident_{n}
This tactic clears all hypotheses except the ones depending in the hypotheses named ident_{1} … ident_{n} and in the goal.
 clear
This tactic clears all hypotheses except the ones depending in goal.
 clear dependent ident
This clears the hypothesis ident and all hypotheses which depend on it.
Error messages:
8.3.3 move ident_{1} after ident_{2}
This moves the hypothesis named ident_{1} in the local context after the hypothesis named ident_{2}.
If ident_{1} comes before ident_{2} in the order of dependences, then all hypotheses between ident_{1} and ident_{2} which (possibly indirectly) depend on ident_{1} are moved also.
If ident_{1} comes after ident_{2} in the order of dependences, then all hypotheses between ident_{1} and ident_{2} which (possibly indirectly) occur in ident_{1} are moved also.
Variants:
 move ident_{1} before ident_{2}
This moves ident_{1} towards and just before the hypothesis named ident_{2}.
 move ident at top
This moves ident at the top of the local context (at the beginning of the context).
 move ident at bottom
This moves ident at the bottom of the local context (at the end of the context).
Error messages:
 ident_{i} not found
 Cannot move ident_{1} after ident_{2}: it occurs in ident_{2}
 Cannot move ident_{1} after ident_{2}: it depends on ident_{2}
8.3.4 rename ident_{1} into ident_{2}
This renames hypothesis ident_{1} into ident_{2} in the current context^{1}
Variants:
 rename ident_{1} into ident_{2}, …,
ident_{2k1} into ident_{2k}
Is equivalent to the sequence of the corresponding atomic rename.
Error messages:
8.3.5 intro
This tactic applies to a goal which is either a product or starts with a let binder. If the goal is a product, the tactic implements the “Lam” rule given in Section 4.2^{2}. If the goal starts with a let binder then the tactic implements a mix of the “Let” and “Conv”.
If the current goal is a dependent product forall x:T, U (resp let x:=t in U) then intro puts x:T (resp x:=t) in the local context. The new subgoal is U.
If the goal is a non dependent product T > U, then it puts in the local context either Hn:T (if T is of type Set or Prop) or Xn:T (if the type of T is Type). The optional index n is such that Hn or Xn is a fresh identifier. In both cases the new subgoal is U.
If the goal is neither a product nor starting with a let definition, the tactic intro applies the tactic red until the tactic intro can be applied or the goal is not reducible.
Error messages:
Variants:
 intros
Repeats intro until it meets the headconstant. It never reduces headconstants and it never fails.
 intro ident
Applies intro but forces ident to be the name of the introduced hypothesis.
Error message: name ident is already used
Remark: If a name used by intro hides the base name of a global constant then the latter can still be referred to by a qualified name (see 2.6.2).  intros ident_{1} … ident_{n}
Is equivalent to the composed tactic intro ident_{1}; … ; intro ident_{n}.
More generally, the intros tactic takes a pattern as argument in order to introduce names for components of an inductive definition or to clear introduced hypotheses; This is explained in 8.7.3.
 intros until ident
Repeats intro until it meets a premise of the goal having form ( ident : term ) and discharges the variable named ident of the current goal.
 intros until num
Repeats intro until the numth nondependent product. For instance, on the subgoal
forall x y:nat, x=y > y=x
the tactic intros until 1 is equivalent to intros x y H, asx=y > y=x
is the first nondependent product. And on the subgoalforall x y z:nat, x=y > y=x
the tactic intros until 1 is equivalent to intros x y z as the product on z can be rewritten as a nondependent product:forall x y:nat, nat > x=y > y=x
Error message: No such hypothesis in current goalHappens when num is 0 or is greater than the number of nondependent products of the goal.
 intro after ident
intro before ident
intro at top
intro at bottomApplies intro and moves the freshly introduced hypothesis respectively after the hypothesis ident, before the hypothesis ident, at the top of the local context, or at the bottom of the local context. All hypotheses on which the new hypothesis depends are moved too so as to respect the order of dependencies between hypotheses. Note that intro at bottom is a synonym for intro with no argument.
Error messages:  intro ident_{1} after ident_{2}
intro ident_{1} before ident_{2}
intro ident_{1} at top
intro ident_{1} at bottomBehaves as previously but naming the introduced hypothesis ident_{1}. It is equivalent to intro ident_{1} followed by the appropriate call to move (see Section 8.3.3).
Error messages:
8.3.6 apply term
This tactic applies to any goal. The argument term is a term wellformed in the local context. The tactic apply tries to match the current goal against the conclusion of the type of term. If it succeeds, then the tactic returns as many subgoals as the number of non dependent premises of the type of term. If the conclusion of the type of term does not match the goal and the conclusion is an inductive type isomorphic to a tuple type, then each component of the tuple is recursively matched to the goal in the lefttoright order.
The tactic apply relies on firstorder unification with dependent types unless the conclusion of the type of term is of the form (P t_{1} … t_{n}) with P to be instantiated. In the latter case, the behavior depends on the form of the goal. If the goal is of the form (fun x => Q) u_{1} … u_{n} and the t_{i} and u_{i} unifies, then P is taken to be (fun x => Q). Otherwise, apply tries to define P by abstracting over t_{1} … t_{n} in the goal. See pattern in Section 8.5.7 to transform the goal so that it gets the form (fun x => Q) u_{1} … u_{n}.
Error messages:

Impossible to unify … with …
The apply tactic failed to match the conclusion of term and the current goal. You can help the apply tactic by transforming your goal with the change or pattern tactics (see sections 8.5.7, 8.3.12).
 Unable to find an instance for the variables
ident …ident
This occurs when some instantiations of the premises of term are not deducible from the unification. This is the case, for instance, when you want to apply a transitivity property. In this case, you have to use one of the variants below:
Variants:
 apply term with term_{1} … term_{n}
Provides apply with explicit instantiations for all dependent premises of the type of term which do not occur in the conclusion and consequently cannot be found by unification. Notice that term_{1} … term_{n} must be given according to the order of these dependent premises of the type of term.
 apply term with (ref_{1} := term_{1}) … (ref_{n}
:= term_{n})
This also provides apply with values for instantiating premises. Here, variables are referred by names and nondependent products by increasing numbers (see syntax in Section 8.3.22).
 apply term_{1} , …, term_{n}
This is a shortcut for apply term_{1} ; [ ..  … ; [ ..  apply term_{n} ] … ], i.e. for the successive applications of term_{i+1} on the last subgoal generated by apply term_{i}, starting from the application of term_{1}.
 eapply term
The tactic eapply behaves as apply but does not fail when no instantiation are deducible for some variables in the premises. Rather, it turns these variables into socalled existential variables which are variables still to instantiate. An existential variable is identified by a name of the form ?n where n is a number. The instantiation is intended to be found later in the proof.
An example of use of eapply is given in Section 10.2.
 simple apply term
This behaves like apply but it reasons modulo conversion only on subterms that contain no variables to instantiate. For instance, if id := fun x:nat => x and H : forall y, id y = y then simple apply H on goal O = O does not succeed because it would require the conversion of f ?y and O where ?y is a variable to instantiate. Tactic simple apply does not either traverse tuples as apply does.
Because it reasons modulo a limited amount of conversion, simple apply fails quicker than apply and it is then wellsuited for uses in useddefined tactics that backtrack often.
 [simple] apply term_{1} [with
bindings_list_{1}] , …, term_{n} [with
bindings_list_{n}]
[simple] eapply term_{1} [with bindings_list_{1}] , …, term_{n} [with bindings_list_{n}]This summarizes the different syntaxes for apply and eapply.
 lapply term
This tactic applies to any goal, say G. The argument term has to be wellformed in the current context, its type being reducible to a nondependent product A > B with B possibly containing products. Then it generates two subgoals B>G and A. Applying lapply H (where H has type A>B and B does not start with a product) does the same as giving the sequence cut B. 2:apply H. where cut is described below.
Warning: When term contains more than one non dependent product the tactic lapply only takes into account the first product.
8.3.7 set ( ident := term )
This replaces term by ident in the conclusion or in the hypotheses of the current goal and adds the new definition ident:= term to the local context. The default is to make this replacement only in the conclusion.
Variants:
 set ( ident := term ) in goal_occurrences
This notation allows to specify which occurrences of term have to be substituted in the context. The in goal_occurrences clause is an occurrence clause whose syntax and behavior is described in Section 8.3.23.
 set ( ident binder … binder := term )
This is equivalent to set ( ident := fun binder … binder => term ).
 set term
This behaves as set ( ident := term ) but ident is generated by Coq. This variant also supports an occurrence clause.
 set ( ident_{0} binder … binder := term
) in goal_occurrences
set term in goal_occurrencesThese are the general forms which combine the previous possibilities.
 remember term as ident
This behaves as set ( ident := term ) in * and using a logical (Leibniz’s) equality instead of a local definition.
 remember term as ident in goal_occurrences
This is a more general form of remember that remembers the occurrences of term specified by an occurrences set.
 pose ( ident := term )
This adds the local definition ident := term to the current context without performing any replacement in the goal or in the hypotheses. It is equivalent to set ( ident := term ) in .
 pose ( ident binder … binder := term )
This is equivalent to pose ( ident := fun binder … binder => term ).
 pose term
This behaves as pose ( ident := term ) but ident is generated by Coq.
8.3.8 assert ( ident : form )
This tactic applies to any goal. assert (H : U) adds a new hypothesis of name H asserting U to the current goal and opens a new subgoal U^{3}. The subgoal U comes first in the list of subgoals remaining to prove.
Error messages:
Variants:
 assert form
This behaves as assert ( ident : form ) but ident is generated by Coq.
 assert ( ident := term )
This behaves as assert (ident : type);[exact termidtac] where type is the type of term.
 cut form
This tactic applies to any goal. It implements the non dependent case of the “App” rule given in Section 4.2. (This is Modus Ponens inference rule.) cut U transforms the current goal T into the two following subgoals: U > T and U. The subgoal U > T comes first in the list of remaining subgoal to prove.
 assert form by tactic
This tactic behaves like assert but applies tactic to solve the subgoals generated by assert.
 assert form as intro_pattern
If intro_pattern is a naming introduction pattern (see Section 8.7.3), the hypothesis is named after this introduction pattern (in particular, if intro_pattern is ident, the tactic behaves like assert (ident : form)).
If intro_pattern is a disjunctive/conjunctive introduction pattern, the tactic behaves like assert form then destructing the resulting hypothesis using the given introduction pattern.
 assert form as intro_pattern by tactic
This combines the two previous variants of assert.
 pose proof term as intro_pattern
This tactic behaves like assert T as intro_pattern by exact term where T is the type of term.
In particular, pose proof term as ident behaves as assert (ident:T) by exact term (where T is the type of term) and pose proof term as disj_conj_intro_pattern behaves like destruct term as disj_conj_intro_pattern.
 specialize (ident term_{1} … term_{n})
specialize ident with bindings_listThe tactic specialize works on local hypothesis ident. The premises of this hypothesis (either universal quantifications or nondependent implications) are instantiated by concrete terms coming either from arguments term_{1} … term_{n} or from a bindings list (see Section 8.3.22 for more about bindings lists). In the second form, all instantiation elements must be given, whereas in the first form the application to term_{1} … term_{n} can be partial. The first form is equivalent to assert (ident’:=identterm_{1} … term_{n}); clear ident; rename ident’ into ident.
The name ident can also refer to a global lemma or hypothesis. In this case, for compatibility reasons, the behavior of specialize is close to that of generalize: the instantiated statement becomes an additional premise of the goal.
8.3.9 apply term in ident
This tactic applies to any goal. The argument term is a term
wellformed in the local context and the argument ident is an
hypothesis of the context. The tactic apply term in ident
tries to match the conclusion of the type of ident against a non
dependent premise of the type of term, trying them from right to
left. If it succeeds, the statement of hypothesis ident is
replaced by the conclusion of the type of term. The tactic also
returns as many subgoals as the number of other non dependent premises
in the type of term and of the non dependent premises of the type
of ident. If the conclusion of the type of term does not match
the goal and the conclusion is an inductive type isomorphic to a
tuple type, then the tuple is (recursively) decomposed and the first
component of the tuple of which a non dependent premise matches the
conclusion of the type of ident. Tuples are decomposed in a
widthfirst lefttoright order (for instance if the type of H1
is a A <> B
statement, and the type of H2 is A
then apply H1 in H2 transforms the type of H2 into B). The tactic apply relies on firstorder patternmatching
with dependent types.
Error messages:

Statement without assumptions
This happens if the type of term has no non dependent premise.
 Unable to apply
This happens if the conclusion of ident does not match any of the non dependent premises of the type of term.
Variants:

apply term , … , term in ident
This applies each of term in sequence in ident.
 apply term with bindings_list , … , term with bindings_list in ident
This does the same but uses the bindings in each bindings_list to instantiate the parameters of the corresponding type of term (see syntax of bindings in Section 8.3.22).
 eapply term with bindings_list , … , term with bindings_list in ident
This works as apply term with bindings_list , … , term with bindings_list in ident but turns unresolved bindings into existential variables, if any, instead of failing.
 apply term, with bindings_list , … , term, with bindings_list in ident as disj_conj_intro_pattern
This works as apply term, with bindings_list , … , term, with bindings_list in ident then destructs the hypothesis ident along disj_conj_intro_pattern as destruct ident as disj_conj_intro_pattern would.
 eapply term, with bindings_list , … , term, with bindings_list in ident as disj_conj_intro_pattern
This works as apply term, with bindings_list , … , term, with bindings_list in ident as disj_conj_intro_pattern but using eapply.
 simple apply term in ident
This behaves like apply term in ident but it reasons modulo conversion only on subterms that contain no variables to instantiate. For instance, if id := fun x:nat => x and H : forall y, id y = y > True and H0 : O = O then simple apply H in H0 does not succeed because it would require the conversion of f ?y and O where ?y is a variable to instantiate. Tactic simple apply term in ident does not either traverse tuples as apply term in ident does.
 [simple] apply term [with bindings_list] , … , term [with bindings_list] in ident [as disj_conj_intro_pattern]
[simple] eapply term [with bindings_list] , … , term [with bindings_list] in ident [as disj_conj_intro_pattern]This summarizes the different syntactic variants of apply term in ident and eapply term in ident.
8.3.10 generalize term
This tactic applies to any goal. It generalizes the conclusion w.r.t. one subterm of it. For example:
1 subgoal
x : nat
y : nat
============================
0 <= x + y + y
Coq < generalize (x + y + y).
1 subgoal
x : nat
y : nat
============================
forall n : nat, 0 <= n
If the goal is G and t is a subterm of type T in the goal, then generalize t replaces the goal by forall (x:T), G′ where G′ is obtained from G by replacing all occurrences of t by x. The name of the variable (here n) is chosen based on T.
Variants:

generalize term_{1} , … , term_{n}
Is equivalent to generalize term_{n}; … ; generalize term_{1}. Note that the sequence of term_{i}’s are processed from n to 1.
 generalize term at num_{1} … num_{i}
Is equivalent to generalize term but generalizing only over the specified occurrences of term (counting from left to right on the expression printed using option Set Printing All).
 generalize term as ident
Is equivalent to generalize term but use ident to name the generalized hypothesis.
 generalize term_{1} at num_{11} … num_{1i1}
as ident_{1}
, … ,
term_{n} at num_{n1} … num_{nin}
as ident_{2}
This is the most general form of generalize that combines the previous behaviors.
 generalize dependent term
This generalizes term but also all hypotheses which depend on term. It clears the generalized hypotheses.
8.3.11 revert ident_{1} … ident_{n}
This applies to any goal with variables ident_{1} … ident_{n}. It moves the hypotheses (possibly defined) to the goal, if this respects dependencies. This tactic is the inverse of intro.
Error messages:
Variants:

revert dependent ident
This moves to the goal the hypothesis ident and all hypotheses which depend on it.
8.3.12 change term
This tactic applies to any goal. It implements the rule “Conv” given in Section 4.3. change U replaces the current goal T with U providing that U is wellformed and that T and U are convertible.
Error messages:

change term_{1} with term_{2}
This replaces the occurrences of term_{1} by term_{2} in the current goal. The terms term_{1} and term_{2} must be convertible.
 change term_{1} at num_{1} … num_{i} with term_{2}
This replaces the occurrences numbered num_{1} … num_{i} of term_{1} by term_{2} in the current goal. The terms term_{1} and term_{2} must be convertible.
Error message: Too few occurrences  change term in ident
 change term_{1} with term_{2} in ident
 change term_{1} at num_{1} … num_{i} with term_{2} in
ident
This applies the change tactic not to the goal but to the hypothesis ident.
See also: 8.5
8.3.13 fix ident num
This tactic is a primitive tactic to start a proof by induction. In general, it is easier to rely on higherlevel induction tactics such as the ones described in Section 8.7.
In the syntax of the tactic, the identifier ident is the name given to the induction hypothesis. The natural number num tells on which premise of the current goal the induction acts, starting from 1 and counting both dependent and non dependent products. Especially, the current lemma must be composed of at least num products.
Like in a fix expression, the induction hypotheses have to be used on structurally smaller arguments. The verification that inductive proof arguments are correct is done only at the time of registering the lemma in the environment. To know if the use of induction hypotheses is correct at some time of the interactive development of a proof, use the command Guarded (see Section 7.3.2).
Variants:

fix ident_{1} num with ( ident_{2}
binder_{2} … binder_{2} [{ struct ident′_{2}
}] : type_{2} ) … ( ident_{1}
binder_{n} … binder_{n} [{ struct ident′_{n} }]
: type_{n} )
This starts a proof by mutual induction. The statements to be simultaneously proved are respectively forall binder_{2} … binder_{2}, type_{2}, …, forall binder_{n} … binder_{n}, type_{n}. The identifiers ident_{1} … ident_{n} are the names of the induction hypotheses. The identifiers ident′_{2} … ident′_{n} are the respective names of the premises on which the induction is performed in the statements to be simultaneously proved (if not given, the system tries to guess itself what they are).
8.3.14 cofix ident
This tactic starts a proof by coinduction. The identifier ident is the name given to the coinduction hypothesis. Like in a cofix expression, the use of induction hypotheses have to guarded by a constructor. The verification that the use of coinductive hypotheses is correct is done only at the time of registering the lemma in the environment. To know if the use of coinduction hypotheses is correct at some time of the interactive development of a proof, use the command Guarded (see Section 7.3.2).
Variants:

cofix ident_{1} with ( ident_{2}
binder_{2} … binder_{2} : type_{2} ) … ( ident_{1} binder_{1} … binder_{1} : type_{n}
)
This starts a proof by mutual coinduction. The statements to be simultaneously proved are respectively forall binder_{2} … binder_{2}, type_{2}, …, forall binder_{n} … binder_{n}, type_{n}. The identifiers ident_{1} … ident_{n} are the names of the coinduction hypotheses.
8.3.15 evar (ident:term)
The evar tactic creates a new local definition named ident with type term in the context. The body of this binding is a fresh existential variable.
8.3.16 instantiate (num:= term)
The instantiate tactic allows to solve an existential variable with the term term. The num argument is the position of the existential variable from right to left in the conclusion. This cannot be the number of the existential variable since this number is different in every session.
Variants:
 instantiate (num:=term) in ident
 instantiate (num:=term) in (Value of ident)
 instantiate (num:=term) in (Type of ident)
These allow to refer respectively to existential variables occurring in a hypothesis or in the body or the type of a local definition.
 instantiate
Without argument, the instantiate tactic tries to solve as many existential variables as possible, using information gathered from other tactics in the same tactical. This is automatically done after each complete tactic (i.e. after a dot in proof mode), but not, for example, between each tactic when they are sequenced by semicolons.
8.3.17 admit
The admit tactic “solves” the current subgoal by an axiom. This typically allows to temporarily skip a subgoal so as to progress further in the rest of the proof. To know if some proof still relies on unproved subgoals, one can use the command Print Assumptions (see Section 6.3.5). Admitted subgoals have names of the form ident_admitted possibly followed by a number.
8.3.18 constr_eq term_{1} term_{2}
This tactic applies to any goal. It checks whether its arguments are equal modulo alpha conversion and casts.
8.3.19 is_evar term
This tactic applies to any goal. It checks whether its argument is an existential variable. Existential variables are uninstantiated variables generated by e.g. eapply (see Section 8.3.6).
8.3.20 has_evar term
This tactic applies to any goal. It checks whether its argument has an existential variable as a subterm. Unlike context patterns combined with is_evar, this tactic scans all subterms, including those under binders.
8.3.21 is_var term
This tactic applies to any goal. It checks whether its argument is a variable or hypothesis in the current goal context or in the opened sections.
Error message: Not a variable or hypothesis
8.3.22 Bindings list
Tactics that take a term as argument may also support a bindings list, so as to instantiate some parameters of the term by name or position. The general form of a term equipped with a bindings list is term with bindings_list where bindings_list may be of two different forms:
 In a bindings list of the form (ref_{1} := term_{1}) … (ref_{n} := term_{n}), ref is either an ident or a num. The references are determined according to the type of term. If ref_{i} is an identifier, this identifier has to be bound in the type of term and the binding provides the tactic with an instance for the parameter of this name. If ref_{i} is some number n, this number denotes the nth non dependent premise of the term, as determined by the type of term.
 A bindings list can also be a simple list of terms term_{1} …term_{n}. In that case the references to which these terms correspond are determined by the tactic. In case of induction, destruct, elim and case (see Section 11) the terms have to provide instances for all the dependent products in the type of term while in the case of apply, or of constructor and its variants, only instances for the dependent products which are not bound in the conclusion of the type are required.
8.3.23 Occurrences sets and occurrences clauses
An occurrences clause is a modifier to some tactics that obeys the following syntax:
occurrence_clause  ::=  in goal_occurrences 
goal_occurrences  ::=  [ident_{1} [at_occurrences] , 
… ,  
ident_{m} [at_occurrences]]  
[ [* [at_occurrences]]]  
  *  [* [at_occurrences]]  
  *  
at_occurrences  ::=  at occurrences 
occurrences  ::=  [] num_{1} … num_{n} 
The role of an occurrence clause is to select a set of occurrences of a term in a goal. In the first case, the ident_{i} [at num_{1}^{i} … num_{ni}^{i}] parts indicate that occurrences have to be selected in the hypotheses named ident_{i}. If no numbers are given for hypothesis ident_{i}, then all occurrences of term in the hypothesis are selected. If numbers are given, they refer to occurrences of term when the term is printed using option Set Printing All (see Section 2.9), counting from left to right. In particular, occurrences of term in implicit arguments (see Section 2.7) or coercions (see Section 2.8) are counted.
If a minus sign is given between at and the list of occurrences, it negates the condition so that the clause denotes all the occurrences except the ones explicitly mentioned after the minus sign.
As an exception to the lefttoright order, the occurrences in the return subexpression of a match are considered before the occurrences in the matched term.
In the second case, the * on the left of  means that all occurrences of term are selected in every hypothesis.
In the first and second case, if * is mentioned on the right of , the occurrences of the conclusion of the goal have to be selected. If some numbers are given, then only the occurrences denoted by these numbers are selected. In no numbers are given, all occurrences of term in the goal are selected.
Finally, the last notation is an abbreviation for *  *. Note also that  is optional in the first case when no * is given.
Here are some tactics that understand occurrences clauses: set, remember, induction, destruct.
See also: Sections 8.3.7, 8.7, 2.9.
8.4 Negation and contradiction
8.4.1 absurd term
This tactic applies to any goal. The argument term is any proposition P of type Prop. This tactic applies False elimination, that is it deduces the current goal from False, and generates as subgoals ∼P and P. It is very useful in proofs by cases, where some cases are impossible. In most cases, P or ∼P is one of the hypotheses of the local context.
8.4.2 contradiction
This tactic applies to any goal. The contradiction tactic attempts to find in the current context (after all intros) one hypothesis which is equivalent to False. It permits to prune irrelevant cases. This tactic is a macro for the tactics sequence intros; elimtype False; assumption.
Error messages:
Variants:

contradiction ident
The proof of False is searched in the hypothesis named ident.
8.4.3 contradict ident
This tactic allows to manipulate negated hypothesis and goals. The name ident should correspond to a hypothesis. With contradict H, the current goal and context is transformed in the following way:
 H:¬A ⊢ B becomes ⊢ A
 H:¬A ⊢ ¬B becomes H: B ⊢ A
 H: A ⊢ B becomes ⊢ ¬A
 H: A ⊢ ¬B becomes H: B ⊢ ¬A
8.4.4 exfalso
This tactic implements the “ex falso quodlibet” logical principle: an elimination of False is performed on the current goal, and the user is then required to prove that False is indeed provable in the current context. This tactic is a macro for elimtype False.
8.5 Conversion tactics
This set of tactics implements different specialized usages of the tactic change.
All conversion tactics (including change) can be parameterized by the parts of the goal where the conversion can occur. This is done using goal clauses which consists in a list of hypotheses and, optionally, of a reference to the conclusion of the goal. For defined hypothesis it is possible to specify if the conversion should occur on the type part, the body part or both (default).
Goal clauses are written after a conversion tactic (tactics set 8.3.7, rewrite 8.8.1, replace 8.8.3 and autorewrite 8.12.13 also use goal clauses) and are introduced by the keyword in. If no goal clause is provided, the default is to perform the conversion only in the conclusion.
The syntax and description of the various goal clauses is the following:
 in ident_{1} … ident_{n}  only in hypotheses ident_{1} …ident_{n}
 in ident_{1} … ident_{n}  * in hypotheses ident_{1} …ident_{n} and in the conclusion
 in *  in every hypothesis
 in * (equivalent to in *  *) everywhere
 in (type of ident_{1}) (value of ident_{2}) …  in type part of ident_{1}, in the value part of ident_{2}, etc.
For backward compatibility, the notation in ident_{1}…ident_{n} performs the conversion in hypotheses ident_{1}…ident_{n}.
8.5.1 cbv flag_{1} … flag_{n}, lazy flag_{1} … flag_{n} and compute
These parameterized reduction tactics apply to any goal and perform the normalization of the goal according to the specified flags. In correspondence with the kinds of reduction considered in Coq namely β (reduction of functional application), δ (unfolding of transparent constants, see 6.10.2), ι (reduction of patternmatching over a constructed term, and unfolding of fix and cofix expressions) and ζ (contraction of local definitions), the flag are either beta, delta, iota or zeta. The delta flag itself can be refined into delta [qualid_{1}…qualid_{k}] or delta [qualid_{1}…qualid_{k}], restricting in the first case the constants to unfold to the constants listed, and restricting in the second case the constant to unfold to all but the ones explicitly mentioned. Notice that the delta flag does not apply to variables bound by a letin construction inside the term itself (use here the zeta flag). In any cases, opaque constants are not unfolded (see Section 6.10.1).
The goal may be normalized with two strategies: lazy (lazy tactic), or callbyvalue (cbv tactic). The lazy strategy is a callbyneed strategy, with sharing of reductions: the arguments of a function call are partially evaluated only when necessary, and if an argument is used several times then it is computed only once. This reduction is efficient for reducing expressions with dead code. For instance, the proofs of a proposition exists x. P(x) reduce to a pair of a witness t, and a proof that t satisfies the predicate P. Most of the time, t may be computed without computing the proof of P(t), thanks to the lazy strategy.
The callbyvalue strategy is the one used in ML languages: the arguments of a function call are evaluated first, using a weak reduction (no reduction under the λabstractions). Despite the lazy strategy always performs fewer reductions than the callbyvalue strategy, the latter is generally more efficient for evaluating purely computational expressions (i.e. with few dead code).
Variants:

compute
cbvThese are synonyms for cbv beta delta iota zeta.
 lazy
This is a synonym for lazy beta delta iota zeta.
 compute [qualid_{1}…qualid_{k}]
cbv [qualid_{1}…qualid_{k}]These are synonyms of cbv beta delta [qualid_{1}…qualid_{k}] iota zeta.
 compute [qualid_{1}…qualid_{k}]
cbv [qualid_{1}…qualid_{k}]These are synonyms of cbv beta delta [qualid_{1}…qualid_{k}] iota zeta.
 lazy [qualid_{1}…qualid_{k}]
lazy [qualid_{1}…qualid_{k}]These are respectively synonyms of cbv beta delta [qualid_{1}…qualid_{k}] iota zeta and cbv beta delta [qualid_{1}…qualid_{k}] iota zeta.
 vm_compute
This tactic evaluates the goal using the optimized callbyvalue evaluation bytecodebased virtual machine. This algorithm is dramatically more efficient than the algorithm used for the cbv tactic, but it cannot be finetuned. It is specially interesting for full evaluation of algebraic objects. This includes the case of reflexionbased tactics.
8.5.2 red
This tactic applies to a goal which has the form forall (x:T1)…(xk:Tk), c t1 … tn where c is a constant. If c is transparent then it replaces c with its definition (say t) and then reduces (t t1 … tn) according to βιζreduction rules.
Error messages:
8.5.3 hnf
This tactic applies to any goal. It replaces the current goal with its head normal form according to the βδιζreduction rules, i.e. it reduces the head of the goal until it becomes a product or an irreducible term.
Example: The term forall n:nat, (plus (S n) (S n))
is not reduced by hnf.
Remark: The δ rule only applies to transparent constants
(see Section 6.10.1 on transparency and opacity).
8.5.4 simpl
This tactic applies to any goal. The tactic simpl first applies βιreduction rule. Then it expands transparent constants and tries to reduce T’ according, once more, to βι rules. But when the ι rule is not applicable then possible δreductions are not applied. For instance trying to use simpl on (plus n O)=n changes nothing. Notice that only transparent constants whose name can be reused as such in the recursive calls are possibly unfolded. For instance a constant defined by plus’ := plus is possibly unfolded and reused in the recursive calls, but a constant such as succ := plus (S O) is never unfolded.

simpl term
This applies simpl only to the occurrences of term in the current goal.
 simpl term at num_{1} … num_{i}
This applies simpl only to the num_{1}, …, num_{i} occurrences of term in the current goal.
Error message: Too few occurrences  simpl ident
This applies simpl only to the applicative subterms whose head occurrence is ident.
 simpl ident at num_{1} … num_{i}
This applies simpl only to the num_{1}, …, num_{i} applicative subterms whose head occurrence is ident.
8.5.5 unfold qualid
This tactic applies to any goal. The argument qualid must denote a defined transparent constant or local definition (see Sections 1.3.2 and 6.10.2). The tactic unfold applies the δ rule to each occurrence of the constant to which qualid refers in the current goal and then replaces it with its βιnormal form.
Error messages:
Variants:

unfold qualid_{1}, …, qualid_{n}
Replaces simultaneously qualid_{1}, …, qualid_{n} with their definitions and replaces the current goal with its βι normal form.
 unfold qualid_{1} at num_{1}^{1}, …, num_{i}^{1},
…, qualid_{n} at num_{1}^{n} … num_{j}^{n}
The lists num_{1}^{1}, …, num_{i}^{1} and num_{1}^{n}, …, num_{j}^{n} specify the occurrences of qualid_{1}, …, qualid_{n} to be unfolded. Occurrences are located from left to right.
Error message: bad occurrence number of qualid_{i}
Error message: qualid_{i} does not occur  unfold string
If string denotes the discriminating symbol of a notation (e.g. "+") or an expression defining a notation (e.g.
"_ + _"
), and this notation refers to an unfoldable constant, then the tactic unfolds it.  unfold string%key
This is variant of unfold string where string gets its interpretation from the scope bound to the delimiting key key instead of its default interpretation (see Section 12.2.2).
 unfold qualid_or_string_{1} at num_{1}^{1}, …, num_{i}^{1},
…, qualid_or_string_{n} at num_{1}^{n} … num_{j}^{n}
This is the most general form, where qualid_or_string is either a qualid or a string referring to a notation.
8.5.6 fold term
This tactic applies to any goal. The term term is reduced using the red tactic. Every occurrence of the resulting term in the goal is then replaced by term.
Variants:

fold term_{1} … term_{n}
Equivalent to fold term_{1};…; fold term_{n}.
8.5.7 pattern term
This command applies to any goal. The argument term must be a free subterm of the current goal. The command pattern performs βexpansion (the inverse of βreduction) of the current goal (say T) by
 replacing all occurrences of term in T with a fresh variable
 abstracting this variable
 applying the abstracted goal to term
For instance, if the current goal T is expressible has φ(t) where the notation captures all the instances of t in φ(t), then pattern t transforms it into (fun x:A => φ(x)) t. This command can be used, for instance, when the tactic apply fails on matching.
Variants:

pattern term at num_{1} … num_{n}
Only the occurrences num_{1} … num_{n} of term are considered for βexpansion. Occurrences are located from left to right.
 pattern term at  num_{1} … num_{n}
All occurrences except the occurrences of indexes num_{1} … num_{n} of term are considered for βexpansion. Occurrences are located from left to right.
 pattern term_{1}, …, term_{m}
Starting from a goal φ(t_{1} … t_{m}), the tactic pattern t_{1}, …, t_{m} generates the equivalent goal (fun (x_{1}:A_{1}) … (x_{m}:A_{m}) => φ(x_{1}… x_{m})) t_{1} … t_{m}.
If t_{i} occurs in one of the generated types A_{j} these occurrences will also be considered and possibly abstracted.  pattern term_{1} at num_{1}^{1} … num_{n1}^{1}, …,
term_{m} at num_{1}^{m} … num_{nm}^{m}
This behaves as above but processing only the occurrences num_{1}^{1}, …, num_{i}^{1} of term_{1}, …, num_{1}^{m}, …, num_{j}^{m} of term_{m} starting from term_{m}.
 pattern term_{1} [at [] num_{1}^{1} … num_{n1}^{1}] , …,
term_{m} [at [] num_{1}^{m} … num_{nm}^{m}]
This is the most general syntax that combines the different variants.
8.5.8 Conversion tactics applied to hypotheses
conv_tactic in ident_{1} … ident_{n}
Applies the conversion tactic conv_tactic to the hypotheses ident_{1}, …, ident_{n}. The tactic conv_tactic is any of the conversion tactics listed in this section.
If ident_{i} is a local definition, then ident_{i} can be replaced by (Type of ident_{i}) to address not the body but the type of the local definition. Example: unfold not in (Type of H1) (Type of H3).
Error messages:
8.6 Introductions
Introduction tactics address goals which are inductive constants. They are used when one guesses that the goal can be obtained with one of its constructors’ type.
8.6.1 constructor num
This tactic applies to a goal such that the head of its conclusion is an inductive constant (say I). The argument num must be less or equal to the numbers of constructor(s) of I. Let ci be the ith constructor of I, then constructor i is equivalent to intros; apply ci.
Error messages:
Variants:

constructor
This tries constructor 1 then constructor 2, … , then constructor n where n if the number of constructors of the head of the goal.
 constructor num with bindings_list
Let ci be the ith constructor of I, then constructor i with bindings_list is equivalent to intros; apply ci with bindings_list.
Warning: the terms in the bindings_list are checked in the context where constructor is executed and not in the context where apply is executed (the introductions are not taken into account).  split
Applies if I has only one constructor, typically in the case of conjunction A∧ B. Then, it is equivalent to constructor 1.
 exists bindings_list
Applies if I has only one constructor, for instance in the case of existential quantification ∃ x· P(x). Then, it is equivalent to intros; constructor 1 with bindings_list.
 exists bindings_list , … , bindings_list
This iteratively applies exists bindings_list.
 left
rightApply if I has two constructors, for instance in the case of disjunction A∨ B. Then, they are respectively equivalent to constructor 1 and constructor 2.
 left with bindings_list
right with bindings_list
split with bindings_listAs soon as the inductive type has the right number of constructors, these expressions are equivalent to calling constructor i with bindings_list for the appropriate i.
 econstructor
eexists
esplit
eleft
eright
These tactics and their variants behave like constructor, exists, split, left, right and their variants but they introduce existential variables instead of failing when the instantiation of a variable cannot be found (cf eapply and Section 10.2).
8.7 Induction and Case Analysis
The tactics presented in this section implement induction or case analysis on inductive or coinductive objects (see Section 4.5).
8.7.1 induction term
This tactic applies to any goal. The type of the argument term must be an inductive constant. Then, the tactic induction generates subgoals, one for each possible form of term, i.e. one for each constructor of the inductive type.
The tactic induction automatically replaces every occurrences of term in the conclusion and the hypotheses of the goal. It automatically adds induction hypotheses (using names of the form IHn1) to the local context. If some hypothesis must not be taken into account in the induction hypothesis, then it needs to be removed first (you can also use the tactics elim or simple induction, see below).
There are particular cases:
 If term is an identifier ident denoting a quantified variable of the conclusion of the goal, then induction ident behaves as intros until ident; induction ident.
 If term is a num, then induction num behaves as
intros until num followed by induction applied to the
last introduced hypothesis.
Remark: For simple induction on a numeral, use syntax induction (num) (not very interesting anyway).
Example:
1 subgoal
============================
forall n : nat, n = n > n <= n
Coq < intros n H.
1 subgoal
n : nat
H : n = n
============================
n <= n
Coq < induction n.
2 subgoals
H : 0 = 0
============================
0 <= 0
subgoal 2 is:
S n <= S n
Error messages:
 Not an inductive product
 Unable to find an instance for the variables
ident …ident
Use in this case the variant elim … with … below.
Variants:

induction term as disj_conj_intro_pattern
This behaves as induction term but uses the names in disj_conj_intro_pattern to name the variables introduced in the context. The disj_conj_intro_pattern must typically be of the form [ p_{11} …p_{1n1}  …  p_{m1} …p_{mnm} ] with m being the number of constructors of the type of term. Each variable introduced by induction in the context of the i^{th} goal gets its name from the list p_{i1} …p_{ini} in order. If there are not enough names, induction invents names for the remaining variables to introduce. More generally, the p_{ij} can be any disjunctive/conjunctive introduction pattern (see Section 8.7.3). For instance, for an inductive type with one constructor, the pattern notation (p_{1},…,p_{n}) can be used instead of [ p_{1} …p_{n} ].
 induction term as naming_intro_pattern
This behaves as induction term but adds an equation between term and the value that term takes in each of the induction case. The name of the equation is built according to naming_intro_pattern which can be an identifier, a “?”, etc, as indicated in Section 8.7.3.
 induction term as naming_intro_pattern disj_conj_intro_pattern
This combines the two previous forms.
 induction term with bindings_list
This behaves like induction term providing explicit instances for the premises of the type of term (see the syntax of bindings in Section 8.3.22).
 einduction term
This tactic behaves like induction term excepts that it does not fail if some dependent premise of the type of term is not inferable. Instead, the unresolved premises are posed as existential variables to be inferred later, in the same way as eapply does (see Section 10.2).
 induction term_{1} using term_{2}
This behaves as induction term_{1} but using term_{2} as induction scheme. It does not expect the conclusion of the type of term_{1} to be inductive.
 induction term_{1} using term_{2} with bindings_list
This behaves as induction term_{1} using term_{2} but also providing instances for the premises of the type of term_{2}.
 induction term_{1} … term_{n} using qualid
This syntax is used for the case qualid denotes an induction principle with complex predicates as the induction principles generated by Function or Functional Scheme may be.
 induction term in goal_occurrences
This syntax is used for selecting which occurrences of term the induction has to be carried on. The in at_occurrences clause is an occurrence clause whose syntax and behavior is described in Section 8.3.23.
When an occurrence clause is given, an equation between term and the value it gets in each case of the induction is added to the context of the subgoals corresponding to the induction cases (even if no clause as naming_intro_pattern is given).
 induction term_{1} with bindings_list_{1} as naming_intro_pattern disj_conj_intro_pattern using term_{2} with bindings_list_{2} in goal_occurrences
einduction term_{1} with bindings_list_{1} as naming_intro_pattern disj_conj_intro_pattern using term_{2} with bindings_list_{2} in goal_occurrencesThese are the most general forms of induction and einduction. It combines the effects of the with, as, using, and in clauses.
 elim term
This is a more basic induction tactic. Again, the type of the argument term must be an inductive type. Then, according to the type of the goal, the tactic elim chooses the appropriate destructor and applies it as the tactic apply would do. For instance, if the proof context contains n:nat and the current goal is T of type Prop, then elim n is equivalent to apply nat_ind with (n:=n). The tactic elim does not modify the context of the goal, neither introduces the induction loading into the context of hypotheses.
More generally, elim term also works when the type of term is a statement with premises and whose conclusion is inductive. In that case the tactic performs induction on the conclusion of the type of term and leaves the nondependent premises of the type as subgoals. In the case of dependent products, the tactic tries to find an instance for which the elimination lemma applies and fails otherwise.
 elim term with bindings_list
Allows to give explicit instances to the premises of the type of term (see Section 8.3.22).
 eelim term
In case the type of term has dependent premises, this turns them into existential variables to be resolved later on.
 elim term_{1} using term_{2}
elim term_{1} using term_{2} with bindings_listAllows the user to give explicitly an elimination predicate term_{2} which is not the standard one for the underlying inductive type of term_{1}. The bindings_list clause allows to instantiate premises of the type of term_{2}.
 elim term_{1} with bindings_list_{1} using term_{2} with bindings_list_{2}
eelim term_{1} with bindings_list_{1} using term_{2} with bindings_list_{2}These are the most general forms of elim and eelim. It combines the effects of the using clause and of the two uses of the with clause.
 elimtype form
The argument form must be inductively defined. elimtype I is equivalent to cut I. intro Hn; elim Hn; clear Hn. Therefore the hypothesis Hn will not appear in the context(s) of the subgoal(s). Conversely, if t is a term of (inductive) type I and which does not occur in the goal then elim t is equivalent to elimtype I; 2: exact t.
 simple induction ident
This tactic behaves as intros until ident; elim ident when ident is a quantified variable of the goal.
 simple induction num
This tactic behaves as intros until num; elim ident where ident is the name given by intros until num to the numth nondependent premise of the goal.
8.7.2 destruct term
The tactic destruct is used to perform case analysis without recursion. Its behavior is similar to induction except that no induction hypothesis is generated. It applies to any goal and the type of term must be inductively defined. There are particular cases:
 If term is an identifier ident denoting a quantified variable of the conclusion of the goal, then destruct ident behaves as intros until ident; destruct ident.
 If term is a num, then destruct num behaves as
intros until num followed by destruct applied to the
last introduced hypothesis.
Remark: For destruction of a numeral, use syntax destruct (num) (not very interesting anyway).
Variants:

destruct term as disj_conj_intro_pattern
This behaves as destruct term but uses the names in intro_pattern to name the variables introduced in the context. The intro_pattern must have the form [ p_{11} …p_{1n1}  …  p_{m1} …p_{mnm} ] with m being the number of constructors of the type of term. Each variable introduced by destruct in the context of the i^{th} goal gets its name from the list p_{i1} …p_{ini} in order. If there are not enough names, destruct invents names for the remaining variables to introduce. More generally, the p_{ij} can be any disjunctive/conjunctive introduction pattern (see Section 8.7.3). This provides a concise notation for nested destruction.
 destruct term as disj_conj_intro_pattern _eqn
This behaves as destruct term but adds an equation between term and the value that term takes in each of the possible cases. The name of the equation is chosen by Coq. If disj_conj_intro_pattern is simply [], it is automatically considered as a disjunctive pattern of the appropriate size.
 destruct term as disj_conj_intro_pattern _eqn: naming_intro_pattern
This behaves as destruct term as disj_conj_intro_pattern _eqn but use naming_intro_pattern to name the equation (see Section 8.7.3). Note that spaces can generally be removed around _eqn.
 destruct term with bindings_list
This behaves like destruct term providing explicit instances for the dependent premises of the type of term (see syntax of bindings in Section 8.3.22).
 edestruct term
This tactic behaves like destruct term excepts that it does not fail if the instance of a dependent premises of the type of term is not inferable. Instead, the unresolved instances are left as existential variables to be inferred later, in the same way as eapply does (see Section 10.2).
 destruct term_{1} using term_{2}
destruct term_{1} using term_{2} with bindings_listThese are synonyms of induction term_{1} using term_{2} and induction term_{1} using term_{2} with bindings_list.
 destruct term in goal_occurrences
This syntax is used for selecting which occurrences of term the case analysis has to be done on. The in goal_occurrences clause is an occurrence clause whose syntax and behavior is described in Section 8.3.23.
When an occurrence clause is given, an equation between term and the value it gets in each case of the analysis is added to the context of the subgoals corresponding to the cases (even if no clause as naming_intro_pattern is given).
 destruct term_{1} with bindings_list_{1} as disj_conj_intro_pattern _eqn: naming_intro_pattern using term_{2} with bindings_list_{2} in goal_occurrences
edestruct term_{1} with bindings_list_{1} as disj_conj_intro_pattern _eqn: naming_intro_pattern using term_{2} with bindings_list_{2} in goal_occurrencesThese are the general forms of destruct and edestruct. They combine the effects of the with, as, using, and in clauses.
 case term
The tactic case is a more basic tactic to perform case analysis without recursion. It behaves as elim term but using a caseanalysis elimination principle and not a recursive one.
 case_eq term
The tactic case_eq is a variant of the case tactic that allow to perform case analysis on a term without completely forgetting its original form. This is done by generating equalities between the original form of the term and the outcomes of the case analysis. The effect of this tactic is similar to the effect of destruct term in  * with the exception that no new hypotheses are introduced in the context.
 case term with bindings_list
Analogous to elim term with bindings_list above.
 ecase term
ecase term with bindings_listIn case the type of term has dependent premises, or dependent premises whose values are not inferable from the with bindings_list clause, ecase turns them into existential variables to be resolved later on.
 simple destruct ident
This tactic behaves as intros until ident; case ident when ident is a quantified variable of the goal.
 simple destruct num
This tactic behaves as intros until num; case ident where ident is the name given by intros until num to the numth nondependent premise of the goal.
8.7.3 intros intro_pattern … intro_pattern
This extension of the tactic intros combines introduction of variables or hypotheses and case analysis. An introduction pattern is either:

A naming introduction pattern, i.e. either one of:
 the pattern ?
 the pattern ?ident
 an identifier
 A disjunctive/conjunctive introduction pattern, i.e. either one of:
 a disjunction of lists of patterns: [p_{11} … p_{1m1}  …  p_{11} … p_{nmn}]
 a conjunction of patterns: ( p_{1} , … , p_{n} )
 a list of patterns ( p_{1} & … & p_{n} ) for sequence of rightassociative binary constructs
 the wildcard: _
 the rewriting orientations: > or <
Assuming a goal of type Q > P (non dependent product), or of type forall x:T, P (dependent product), the behavior of intros p is defined inductively over the structure of the introduction pattern p:
 introduction on ? performs the introduction, and lets Coq choose a fresh name for the variable;
 introduction on ?ident performs the introduction, and lets Coq choose a fresh name for the variable based on ident;
 introduction on ident behaves as described in Section 8.3.5;
 introduction over a disjunction of list of patterns [p_{11} … p_{1m1}  …  p_{11} … p_{nmn}] expects the product to be over an inductive type whose number of constructors is n (or more generally over a type of conclusion an inductive type built from n constructors, e.g. C > A\/B if n=2): it destructs the introduced hypothesis as destruct (see Section 8.7.2) would and applies on each generated subgoal the corresponding tactic; intros p_{i1} … p_{imi}; if the disjunctive pattern is part of a sequence of patterns and is not the last pattern of the sequence, then Coq completes the pattern so as all the argument of the constructors of the inductive type are introduced (for instance, the list of patterns [  ] H applied on goal forall x:nat, x=0 > 0=x behaves the same as the list of patterns [  ? ] H);
 introduction over a conjunction of patterns (p_{1}, …, p_{n}) expects the goal to be a product over an inductive type I with a single constructor that itself has at least n arguments: it performs a case analysis over the hypothesis, as destruct would, and applies the patterns p_{1} … p_{n} to the arguments of the constructor of I (observe that (p_{1}, …, p_{n}) is an alternative notation for [p_{1} … p_{n}]);
 introduction via ( p_{1} & …& p_{n} )
is a shortcut for introduction via
(p_{1},(…,(…,p_{n})…)); it expects the
hypothesis to be a sequence of rightassociative binary inductive
constructors such as conj or ex_intro; for instance, an
hypothesis with type A
/\
exists x, B/\
C/\
D can be introduced via pattern (a & x & b & c & d);  introduction on the wildcard depends on whether the product is dependent or not: in the non dependent case, it erases the corresponding hypothesis (i.e. it behaves as an intro followed by a clear, cf Section 8.3.2) while in the dependent case, it succeeds and erases the variable only if the wildcard is part of a more complex list of introduction patterns that also erases the hypotheses depending on this variable;
 introduction over > (respectively <) expects the hypothesis to be an equality and the righthandside (respectively the lefthandside) is replaced by the lefthandside (respectively the righthandside) in both the conclusion and the context of the goal; if moreover the term to substitute is a variable, the hypothesis is removed.
Remark: intros p_{1} … p_{n} is not equivalent to intros
p_{1};…; intros p_{n} for the following reasons:
 A wildcard pattern never succeeds when applied isolated on a dependent product, while it succeeds as part of a list of introduction patterns if the hypotheses that depends on it are erased too.
 A disjunctive or conjunctive pattern followed by an introduction
pattern forces the introduction in the context of all arguments of
the constructors before applying the next pattern while a terminal
disjunctive or conjunctive pattern does not. Here is an exampleCoq < Goal forall n:nat, n = 0 > n = 0.
1 subgoal
============================
forall n : nat, n = 0 > n = 0
Coq < intros [  ] H.
2 subgoals
H : 0 = 0
============================
0 = 0
subgoal 2 is:
S n = 0
Coq < Show 2.
subgoal 2 is:
n : nat
H : S n = 0
============================
S n = 0
Coq < Undo.
1 subgoal
============================
forall n : nat, n = 0 > n = 0
Coq < intros [  ]; intros H.
2 subgoals
H : 0 = 0
============================
0 = 0
subgoal 2 is:
S H = 0 > S H = 0
Coq < Show 2.
subgoal 2 is:
H : nat
============================
S H = 0 > S H = 0
1 subgoal
============================
forall A B C : Prop, A \/ B /\ C > (A > C) > C
Coq < intros A B C [a [_ c]] f.
2 subgoals
A : Prop
B : Prop
C : Prop
a : A
f : A > C
============================
C
subgoal 2 is:
C
Coq < apply (f a).
1 subgoal
A : Prop
B : Prop
C : Prop
c : C
f : A > C
============================
C
Coq < exact c.
Proof completed.
Coq < Qed.
intros A B C [a (_, c)] f.
apply (f a).
exact c.
intros_test is defined
8.7.4 double induction ident_{1} ident_{2}
This tactic is deprecated and should be replaced by induction ident_{1}; induction ident_{2} (or induction ident_{1}; destruct ident_{2} depending on the exact needs).
Variant:
 double induction num_{1} num_{2}
This tactic is deprecated and should be replaced by induction num_{1}; induction num_{3} where num_{3} is the result of num_{2}num_{1}.
8.7.5 dependent induction ident
The experimental tactic dependent induction performs inductioninversion on an instantiated inductive predicate. One needs to first require the Coq.Program.Equality module to use this tactic. The tactic is based on the BasicElim tactic by Conor McBride [98] and the work of Cristina Cornes around inversion [35]. From an instantiated inductive predicate and a goal it generates an equivalent goal where the hypothesis has been generalized over its indexes which are then constrained by equalities to be the right instances. This permits to state lemmas without resorting to manually adding these equalities and still get enough information in the proofs. A simple example is the following:
1 subgoal
============================
forall n : nat, n < 1 > n = 0
Coq < intros n H ; induction H.
2 subgoals
n : nat
============================
n = 0
subgoal 2 is:
n = 0
Here we didn’t get any information on the indexes to help fulfill this proof. The problem is that when we use the induction tactic we lose information on the hypothesis instance, notably that the second argument is 1 here. Dependent induction solves this problem by adding the corresponding equality to the context.
Coq < Lemma le_minus : forall n:nat, n < 1 > n = 0.
1 subgoal
============================
forall n : nat, n < 1 > n = 0
Coq < intros n H ; dependent induction H.
2 subgoals
============================
0 = 0
subgoal 2 is:
n = 0
The subgoal is cleaned up as the tactic tries to automatically simplify the subgoals with respect to the generated equalities. In this enriched context it becomes possible to solve this subgoal.
1 subgoal
n : nat
H : S n <= 0
IHle : 0 = 1 > n = 0
============================
n = 0
Now we are in a contradictory context and the proof can be solved.
Proof completed.
This technique works with any inductive predicate. In fact, the dependent induction tactic is just a wrapper around the induction tactic. One can make its own variant by just writing a new tactic based on the definition found in Coq.Program.Equality. Common useful variants are the following, defined in the same file:
Variants:

dependent induction ident generalizing ident_{1} …ident_{n}
Does dependent induction on the hypothesis ident but first generalizes the goal by the given variables so that they are universally quantified in the goal. This is generally what one wants to do with the variables that are inside some constructors in the induction hypothesis. The other ones need not be further generalized.
 dependent destruction ident
Does the generalization of the instance ident but uses destruct instead of induction on the generalized hypothesis. This gives results equivalent to inversion or dependent inversion if the hypothesis is dependent.
A larger example of dependent induction and an explanation of the underlying technique are developed in section 10.6.
8.7.6 decompose [ qualid_{1} … qualid_{n} ] term
This tactic allows to recursively decompose a complex proposition in order to obtain atomic ones. Example:
1 subgoal
============================
forall A B C : Prop, A /\ B /\ C \/ B /\ C \/ C /\ A > C
Coq < intros A B C H; decompose [and or] H; assumption.
Proof completed.
Coq < Qed.
decompose does not work on righthand sides of implications or products.
Variants:
 decompose sum term This decomposes sum types (like or).
 decompose record term This decomposes record types (inductive types with one constructor, like and and exists and those defined with the Record macro, see Section 2.1).
8.7.7 functional induction (qualid term_{1} … term_{n}).
The experimental tactic functional induction performs case analysis and induction following the definition of a function. It makes use of a principle generated by Function (see Section 2.3) or Functional Scheme (see Section 8.15).
minus_equation is defined
minus_ind is defined
Coq <
Coq < Lemma le_minus : forall n m:nat, (n  m <= n).
1 subgoal
============================
forall n m : nat, n  m <= n
Coq < intros n m.
1 subgoal
n : nat
m : nat
============================
n  m <= n
Coq < functional induction (minus n m); simpl; auto.
Proof completed.
Coq < Qed.
Remark: (qualid term_{1} … term_{n}) must be a correct
full application of qualid. In particular, the rules for implicit
arguments are the same as usual. For example use @qualid if
you want to write implicit arguments explicitly.
Remark: Parenthesis over qualid…term_{n} are mandatory.
Remark: functional induction (f x1 x2 x3) is actually a wrapper
for induction x1 x2 x3 (f x1 x2 x3) using qualid followed by
a cleaning phase, where qualid is the induction principle
registered for f (by the Function (see Section 2.3)
or Functional Scheme (see Section 8.15) command)
corresponding to the sort of the goal. Therefore functional
induction may fail if the induction scheme (qualid) is
not defined. See also Section 2.3 for the function terms
accepted by Function.
Remark: There is a difference between obtaining an induction scheme for a
function by using Function (see Section 2.3) and by
using Functional Scheme after a normal definition using
Fixpoint or Definition. See 2.3 for
details.
See also: 2.3,8.15,10.4,
8.10.3
Error messages:
Variants:

functional induction (qualid term_{1} … term_{n})
using term_{m+1} with term_{n+1} …term_{m}
Similar to Induction and elim (see Section 8.7), allows to give explicitly the induction principle and the values of dependent premises of the elimination scheme, including predicates for mutual induction when qualid is part of a mutually recursive definition.
 functional induction (qualid term_{1} … term_{n})
using term_{m+1} with ref_{1} := term_{n+1} …
ref_{m} := term_{n}
Similar to induction and elim (see Section 8.7).
 All previous variants can be extended by the usual as intro_pattern construction, similar for example to induction and elim (see Section 8.7).
8.8 Equality
These tactics use the equality eq:forall A:Type, A>A>Prop defined in file Logic.v (see Section 3.1.2). The notation for eq T t u is simply t=u dropping the implicit type of t and u.
8.8.1 rewrite term
This tactic applies to any goal. The type of term must have the form
forall (x_{1}:A_{1}) … (x_{n}:A_{n})eq term_{1} term_{2}.
where eq is the Leibniz equality or a registered setoid equality.
Then rewrite term finds the first subterm matching term_{1} in the goal, resulting in instances term_{1}′ and term_{2}′ and then replaces every occurrence of term_{1}′ by term_{2}′. Hence, some of the variables x_{i} are solved by unification, and some of the types A_{1}, …, A_{n} become new subgoals.
Error messages:
 The term provided does not end with an equation
 Tactic generated a subgoal identical to the original goal
This happens if term_{1} does not occur in the goal.
Variants:

rewrite > term
Is equivalent to rewrite term  rewrite < term
Uses the equality term_{1}=term_{2} from right to left  rewrite term in clause
Analogous to rewrite term but rewriting is done following clause (similarly to 8.5). For instance: rewrite H in H1 will rewrite H in the hypothesis H1 instead of the current goal.
 rewrite H in H1 at 1, H2 at  2  * means rewrite H; rewrite H in H1 at 1; rewrite H in H2 at  2. In particular a failure will happen if any of these three simpler tactics fails.
 rewrite H in *  will do rewrite H in H_{i} for all hypothesis H_{i} <> H. A success will happen as soon as at least one of these simpler tactics succeeds.
 rewrite H in * is a combination of rewrite H and rewrite H in *  that succeeds if at least one of these two tactics succeeds.
 rewrite term at occurrences
Rewrite only the given occurrences of term_{1}′. Occurrences are specified from left to right as for pattern (§8.5.7). The rewrite is always performed using setoid rewriting, even for Leibniz’s equality, so one has to Import Setoid to use this variant.
 rewrite term by tactic
Use tactic to completely solve the sideconditions arising from the rewrite.
 rewrite term_{1}, …, term_{n}
Is equivalent to the n successive tactics rewrite term_{1} up to rewrite term_{n}, each one working on the first subgoal generated by the previous one. Orientation > or < can be inserted before each term to rewrite. One unique clause can be added at the end after the keyword in; it will then affect all rewrite operations.  In all forms of rewrite described above, a term to rewrite
can be immediately prefixed by one of the following modifiers:
 ? : the tactic rewrite ?term performs the rewrite of term as many times as possible (perhaps zero time). This form never fails.
 n? : works similarly, except that it will do at most n rewrites.
 ! : works as ?, except that at least one rewrite should succeed, otherwise the tactic fails.
 n! (or simply n) : precisely n rewrites of term will be done, leading to failure if these n rewrites are not possible.
 erewrite term
This tactic works as rewrite term but turning unresolved bindings into existential variables, if any, instead of failing. It has the same variants as rewrite has.
8.8.2 cutrewrite > term_{1} = term_{2}
This tactic acts like replace term_{1} with term_{2} (see below).
8.8.3 replace term_{1} with term_{2}
This tactic applies to any goal. It replaces all free occurrences of term_{1} in the current goal with term_{2} and generates the equality term_{2}=term_{1} as a subgoal. This equality is automatically solved if it occurs amongst the assumption, or if its symmetric form occurs. It is equivalent to cut term_{2}=term_{1}; [intro Hn; rewrite < Hn; clear Hn assumption  symmetry; try assumption].
Error messages:
Variants:

replace term_{1} with term_{2} by tactic
This acts as replace term_{1} with term_{2} but applies tactic to solve the generated subgoal term_{2}=term_{1}.  replace term
Replace term with term’ using the first assumption whose type has the form term=term’ or term’=term  replace > term
Replace term with term’ using the first assumption whose type has the form term=term’  replace < term
Replace term with term’ using the first assumption whose type has the form term’=term  replace term_{1} with term_{2} clause
replace term_{1} with term_{2} clause by tactic
replace term clause
replace > term clause
replace < term clause
Act as before but the replacements take place in clause (see Section 8.5) and not only in the conclusion of the goal.
The clause argument must not contain any type of nor value of.
8.8.4 reflexivity
This tactic applies to a goal which has the form t=u. It checks that t and u are convertible and then solves the goal. It is equivalent to apply refl_equal.
Error messages:
8.8.5 symmetry
This tactic applies to a goal which has the form t=u and changes it into u=t.
Variant: symmetry in ident
If the statement of the hypothesis ident has the form t=u,
the tactic changes it to u=t.
8.8.6 transitivity term
This tactic applies to a goal which has the form t=u and transforms it into the two subgoals t=term and term=u.
8.8.7 subst ident
This tactic applies to a goal which has ident in its context and (at least) one hypothesis, say H, of type ident=t or t=ident. Then it replaces ident by t everywhere in the goal (in the hypotheses and in the conclusion) and clears ident and H from the context.
Remark: When several hypotheses have the form ident=t or t=ident, the first one is used.
Variants:

subst ident_{1} …ident_{n}
Is equivalent to subst ident_{1}; …; subst ident_{n}.  subst
Applies subst repeatedly to all identifiers from the context for which an equality exists.
8.8.8 stepl term
This tactic is for chaining rewriting steps. It assumes a goal of the form “R term_{1} term_{2}” where R is a binary relation and relies on a database of lemmas of the form forall x y z, R x y > eq x z > R z y where eq is typically a setoid equality. The application of stepl term then replaces the goal by “R term term_{2}” and adds a new goal stating “eq term term_{1}”.
Lemmas are added to the database using the command
Declare Left Step term.
The tactic is especially useful for parametric setoids which are not accepted as regular setoids for rewrite and setoid_replace (see Chapter 25).

stepl term by tactic
This applies stepl term then applies tactic to the second goal.  stepr term
stepr term by tactic
This behaves as stepl but on the righthandside of the binary relation. Lemmas are expected to be of the form “forall x y z, R x y > eq y z > R x z” and are registered using the commandDeclare Right Step term.
8.8.9 f_equal
This tactic applies to a goal of the form f a_{1} … a_{n} = f′ a′_{1} … a′_{n}. Using f_equal on such a goal leads to subgoals f=f′ and a_{1}=a′_{1} and so on up to a_{n}=a′_{n}. Amongst these subgoals, the simple ones (e.g. provable by reflexivity or congruence) are automatically solved by f_equal.
8.9 Equality and inductive sets
We describe in this section some special purpose tactics dealing with equality and inductive sets or types. These tactics use the equality eq:forall (A:Type), A>A>Prop, simply written with the infix symbol =.
8.9.1 decide equality
This tactic solves a goal of the form
forall x y:R, {x=y}+{~
x=y}, where R
is an inductive type such that its constructors do not take proofs or
functions as arguments, nor objects in dependent types.
Variants:

decide equality term_{1} term_{2} .
Solves a goal of the form {term_{1}=term_{2}}+{~
term_{1}=term_{2}}.
8.9.2 compare term_{1} term_{2}
This tactic compares two given objects term_{1} and term_{2}
of an inductive datatype. If G is the current goal, it leaves the subgoals
term_{1}=term_{2} > G and ~
term_{1}=term_{2}
> G. The type
of term_{1} and term_{2} must satisfy the same restrictions as in the tactic
decide equality.
8.9.3 discriminate term
This tactic proves any goal from an assumption stating that two structurally different terms of an inductive set are equal. For example, from (S (S O))=(S O) we can derive by absurdity any proposition.
The argument term is assumed to be a proof of a statement of conclusion term_{1} = term_{2} with term_{1} and term_{2} being elements of an inductive set. To build the proof, the tactic traverses the normal forms^{4} of term_{1} and term_{2} looking for a couple of subterms u and w (u subterm of the normal form of term_{1} and w subterm of the normal form of term_{2}), placed at the same positions and whose head symbols are two different constructors. If such a couple of subterms exists, then the proof of the current goal is completed, otherwise the tactic fails.
Remark: The syntax discriminate ident can be used to refer to a
hypothesis quantified in the goal. In this case, the quantified
hypothesis whose name is ident is first introduced in the local
context using intros until ident.
Error messages:
Variants:

discriminate num
This does the same thing as intros until num followed by discriminate ident where ident is the identifier for the last introduced hypothesis.
 discriminate term with bindings_list
This does the same thing as discriminate term but using the given bindings to instantiate parameters or hypotheses of term.
 ediscriminate num
ediscriminate term [with bindings_list]This works the same as discriminate but if the type of term, or the type of the hypothesis referred to by num, has uninstantiated parameters, these parameters are left as existential variables.
 discriminate
This behaves like discriminate ident if ident is the name of an hypothesis to which discriminate is applicable; if the current goal is of the form term_{1} <> term_{2}, this behaves as intro ident; injection ident.
Error messages:
8.9.4 injection term
The injection tactic is based on the fact that constructors of inductive sets are injections. That means that if c is a constructor of an inductive set, and if (c t_{1}) and (c t_{2}) are two terms that are equal then t_{1} and t_{2} are equal too.
If term is a proof of a statement of conclusion term_{1} = term_{2}, then injection applies injectivity as deep as possible to derive the equality of all the subterms of term_{1} and term_{2} placed in the same positions. For example, from (S (S n))=(S (S (S m)) we may derive n=(S m). To use this tactic term_{1} and term_{2} should be elements of an inductive set and they should be neither explicitly equal, nor structurally different. We mean by this that, if n_{1} and n_{2} are their respective normal forms, then:
 n_{1} and n_{2} should not be syntactically equal,
 there must not exist any pair of subterms u and w, u subterm of n_{1} and w subterm of n_{2} , placed in the same positions and having different constructors as head symbols.
If these conditions are satisfied, then, the tactic derives the equality of all the subterms of term_{1} and term_{2} placed in the same positions and puts them as antecedents of the current goal.
Example: Consider the following goal:
Coq <  nil : list
Coq <  cons : nat > list > list.
Coq < Variable P : list > Prop.
1 subgoal
l : list
n : nat
H : P nil
H0 : cons n l = cons 0 nil
============================
P l
Coq < injection H0.
1 subgoal
l : list
n : nat
H : P nil
H0 : cons n l = cons 0 nil
============================
l = nil > n = 0 > P l
Beware that injection yields always an equality in a sigma type whenever the injected object has a dependent type.
Remark: There is a special case for dependent pairs. If we have a decidable
equality over the type of the first argument, then it is safe to do
the projection on the second one, and so injection will work fine.
To define such an equality, you have to use the Scheme command
(see 8.14).
Remark: If some quantified hypothesis of the goal is named ident, then
injection ident first introduces the hypothesis in the local
context using intros until ident.
Error messages:
 Not a projectable equality but a discriminable one
 Nothing to do, it is an equality between convertible terms
 Not a primitive equality
Variants:

injection num
This does the same thing as intros until num followed by injection ident where ident is the identifier for the last introduced hypothesis.
 injection term with bindings_list
This does the same as injection term but using the given bindings to instantiate parameters or hypotheses of term.
 einjection num
einjection term [with bindings_list]This works the same as injection but if the type of term, or the type of the hypothesis referred to by num, has uninstantiated parameters, these parameters are left as existential variables.
 injection
If the current goal is of the form term_{1} <> term_{2}, this behaves as intro ident; injection ident.
Error message: goal does not satisfy the expected preconditions  injection term [with bindings_list] as intro_pattern … intro_pattern
injection num as intro_pattern … intro_pattern
injection as intro_pattern … intro_pattern
einjection term [with bindings_list] as intro_pattern … intro_pattern
einjection num as intro_pattern … intro_pattern
einjection as intro_pattern … intro_pattern
These variants apply intros intro_pattern … intro_pattern after the call to injection or einjection.
8.9.5 simplify_eq term
Let term be the proof of a statement of conclusion term_{1}=term_{2}. If term_{1} and term_{2} are structurally different (in the sense described for the tactic discriminate), then the tactic simplify_eq behaves as discriminate term, otherwise it behaves as injection term.
Remark: If some quantified hypothesis of the goal is named ident, then
simplify_eq ident first introduces the hypothesis in the local
context using intros until ident.
Variants:

simplify_eq num
This does the same thing as intros until num then simplify_eq ident where ident is the identifier for the last introduced hypothesis.
 simplify_eq term with bindings_list
This does the same as simplify_eq term but using the given bindings to instantiate parameters or hypotheses of term.
 esimplify_eq num
esimplify_eq term [with bindings_list]This works the same as simplify_eq but if the type of term, or the type of the hypothesis referred to by num, has uninstantiated parameters, these parameters are left as existential variables.
 simplify_eq
If the current goal has form t_{1}
<>
t_{2}, it behaves as intro ident; simplify_eq ident.
8.9.6 dependent rewrite > ident
This tactic applies to any goal. If ident has type
(existT B a b)=(existT B a' b')
in the local context (i.e. each term of the
equality has a sigma type { a:A & (B a)}) this tactic rewrites
a
into a'
and b
into b'
in the current
goal. This tactic works even if B is also a sigma type. This kind
of equalities between dependent pairs may be derived by the injection
and inversion tactics.
Variants:

dependent rewrite < ident
Analogous to dependent rewrite > but uses the equality from right to left.
8.10 Inversion
8.10.1 inversion ident
Let the type of ident in the local context be (I t), where I is a (co)inductive predicate. Then, inversion applied to ident derives for each possible constructor c_{i} of (I t), all the necessary conditions that should hold for the instance (I t) to be proved by c_{i}.
Remark: If ident does not denote a hypothesis in the local context
but refers to a hypothesis quantified in the goal, then the
latter is first introduced in the local context using
intros until ident.
Variants:

inversion num
This does the same thing as intros until num then inversion ident where ident is the identifier for the last introduced hypothesis.
 inversion_clear ident
This behaves as inversion and then erases ident from the context.
 inversion ident as intro_pattern
This behaves as inversion but using names in intro_pattern for naming hypotheses. The intro_pattern must have the form [ p_{11} …p_{1n1}  …  p_{m1} …p_{mnm} ] with m being the number of constructors of the type of ident. Be careful that the list must be of length m even if inversion discards some cases (which is precisely one of its roles): for the discarded cases, just use an empty list (i.e. n_{i}=0).
The arguments of the i^{th} constructor and the equalities that inversion introduces in the context of the goal corresponding to the i^{th} constructor, if it exists, get their names from the list p_{i1} …p_{ini} in order. If there are not enough names, induction invents names for the remaining variables to introduce. In case an equation splits into several equations (because inversion applies injection on the equalities it generates), the corresponding name p_{ij} in the list must be replaced by a sublist of the form [p_{ij1} …p_{ijq}] (or, equivalently, (p_{ij1}, …, p_{ijq})) where q is the number of subequalities obtained from splitting the original equation. Here is an example.
Coq < Inductive contains0 : list nat > Prop :=
Coq <  in_hd : forall l, contains0 (0 :: l)
Coq <  in_tl : forall l b, contains0 l > contains0 (b :: l).
contains0 is defined
contains0_ind is defined
Coq < Goal forall l:list nat, contains0 (1 :: l) > contains0 l.
1 subgoal
============================
forall l : Datatypes.list nat, contains0 (1 :: l) > contains0 l
Coq < intros l H; inversion H as [  l’ p Hl’ [Heqp Heql’] ].
1 subgoal
l : Datatypes.list nat
H : contains0 (1 :: l)
l’ : Datatypes.list nat
p : nat
Hl’ : contains0 l
Heqp : p = 1
Heql’ : l’ = l
============================
contains0 l
 inversion num as intro_pattern
This allows to name the hypotheses introduced by inversion num in the context.
 inversion_clear
ident as intro_pattern
This allows to name the hypotheses introduced by inversion_clear in the context.
 inversion ident
in ident_{1} … ident_{n}
Let ident_{1} … ident_{n}, be identifiers in the local context. This tactic behaves as generalizing ident_{1} … ident_{n}, and then performing inversion.
 inversion
ident as intro_pattern in ident_{1} …
ident_{n}
This allows to name the hypotheses introduced in the context by inversion ident in ident_{1} … ident_{n}.
 inversion_clear
ident in ident_{1} …ident_{n}
Let ident_{1} … ident_{n}, be identifiers in the local context. This tactic behaves as generalizing ident_{1} … ident_{n}, and then performing inversion_clear.

inversion_clear ident as intro_pattern
in ident_{1} …ident_{n}
This allows to name the hypotheses introduced in the context by inversion_clear ident in ident_{1} …ident_{n}.
 dependent inversion
ident
That must be used when ident appears in the current goal. It acts like inversion and then substitutes ident for the corresponding term in the goal.
 dependent
inversion ident as intro_pattern
This allows to name the hypotheses introduced in the context by dependent inversion ident.
 dependent
inversion_clear ident
Like dependent inversion, except that ident is cleared from the local context.

dependent inversion_clear identas intro_pattern
This allows to name the hypotheses introduced in the context by dependent inversion_clear ident.
 dependent
inversion ident with term
This variant allows you to specify the generalization of the goal. It is useful when the system fails to generalize the goal automatically. If ident has type (I t) and I has type forall (x:T), s, then term must be of type I:forall (x:T), I x→ s′ where s′ is the type of the goal.

dependent inversion ident as intro_pattern
with term
This allows to name the hypotheses introduced in the context by dependent inversion ident with term.

dependent inversion_clear ident with term
Like dependent inversion … with but clears ident from the local context.

dependent inversion_clear ident as
intro_pattern with term
This allows to name the hypotheses introduced in the context by dependent inversion_clear ident with term.
 simple inversion ident
It is a very primitive inversion tactic that derives all the necessary equalities but it does not simplify the constraints as inversion does.
 simple inversion
ident as intro_pattern
This allows to name the hypotheses introduced in the context by simple inversion.
 inversion ident using ident′
Let ident have type (I t) (I an inductive predicate) in the local context, and ident′ be a (dependent) inversion lemma. Then, this tactic refines the current goal with the specified lemma.
 inversion
ident using ident′ in ident_{1}… ident_{n}
This tactic behaves as generalizing ident_{1}… ident_{n}, then doing inversion ident using ident′.
See also: 10.5 for detailed examples
8.10.2 Derive Inversion ident with forall (x:T), I t Sort sort
This command generates an inversion principle for the inversion … using tactic. Let I be an inductive predicate and x the variables occurring in t. This command generates and stocks the inversion lemma for the sort sort corresponding to the instance forall (x:T), I t with the name ident in the global environment. When applied it is equivalent to have inverted the instance with the tactic inversion.
Variants:

Derive Inversion_clear ident with
forall (x:T), I t Sort sort
When applied it is equivalent to having inverted the instance with the tactic inversion replaced by the tactic inversion_clear.  Derive Dependent Inversion ident with
forall (x:T), I t Sort sort
When applied it is equivalent to having inverted the instance with the tactic dependent inversion.  Derive Dependent Inversion_clear ident with
forall (x:T), I t Sort sort
When applied it is equivalent to having inverted the instance with the tactic dependent inversion_clear.
See also: 10.5 for examples
8.10.3 functional inversion ident
functional inversion is a highly experimental tactic which performs inversion on hypothesis ident of the form qualid term_{1}…term_{n} = term or term = qualid term_{1}…term_{n} where qualid must have been defined using Function (see Section 2.3).
Error messages:
 Hypothesis ident must contain at least one Function
 Cannot find inversion information for hypothesis ident This error may be raised when some inversion lemma failed to be generated by Function.
Variants:

functional inversion num
This does the same thing as intros until num then functional inversion ident where ident is the identifier for the last introduced hypothesis.
 functional inversion ident qualid
functional inversion num qualidIn case the hypothesis ident (or num) has a type of the form qualid_{1} term_{1}…term_{n} = qualid_{2} term_{n+1}…term_{n+m} where qualid_{1} and qualid_{2} are valid candidates to functional inversion, this variant allows to choose which must be inverted.
8.10.4 quote ident
This kind of inversion has nothing to do with the tactic inversion above. This tactic does change (ident t), where t is a term built in order to ensure the convertibility. In other words, it does inversion of the function ident. This function must be a fixpoint on a simple recursive datatype: see 10.8 for the full details.
Error messages:
Variants:

quote ident [ ident_{1} …ident_{n} ]
All terms that are built only with ident_{1} …ident_{n} will be considered by quote as constants rather than variables.
8.11 Classical tactics
In order to ease the proving process, when the Classical module is loaded. A few more tactics are available. Make sure to load the module using the Require Import command.
8.11.1 classical_left, classical_right
The tactics classical_left and classical_right are the analog of the left and right but using classical logic. They can only be used for disjunctions. Use classical_left to prove the left part of the disjunction with the assumption that the negation of right part holds. Use classical_right to prove the right part of the disjunction with the assumption that the negation of left part holds.
8.12 Automatizing
8.12.1 auto
This tactic implements a Prologlike resolution procedure to solve the current goal. It first tries to solve the goal using the assumption tactic, then it reduces the goal to an atomic one using intros and introducing the newly generated hypotheses as hints. Then it looks at the list of tactics associated to the head symbol of the goal and tries to apply one of them (starting from the tactics with lower cost). This process is recursively applied to the generated subgoals.
By default, auto only uses the hypotheses of the current goal and the hints of the database named core.
Variants:
 auto num
Forces the search depth to be num. The maximal search depth is 5 by default.
 auto with ident_{1} … ident_{n}
Uses the hint databases ident_{1} … ident_{n} in addition to the database core. See Section 8.13.1 for the list of predefined databases and the way to create or extend a database. This option can be combined with the previous one.
 auto with *
Uses all existing hint databases, minus the special database v62. See Section 8.13.1
 auto using lemma_{1} , …, lemma_{n}
Uses lemma_{1}, …, lemma_{n} in addition to hints (can be combined with the with ident option). If lemma_{i} is an inductive type, it is the collection of its constructors which is added as hints.
 auto using lemma_{1} , …, lemma_{n} with ident_{1} … ident_{n}
This combines the effects of the using and with options.
 trivial
This tactic is a restriction of auto that is not recursive and tries only hints which cost 0. Typically it solves trivial equalities like X=X.
 trivial with ident_{1} … ident_{n}
 trivial with *
Remark: auto either solves completely the goal or else leaves it
intact. auto and trivial never fail.
See also: Section 8.13.1
8.12.2 eauto
This tactic generalizes auto. In contrast with the latter, eauto uses unification of the goal against the hints rather than patternmatching (in other words, it uses eapply instead of apply). As a consequence, eauto can solve such a goal:
Warning: the hint: eapply ex_intro will only be used by eauto
Coq < Goal forall P:nat > Prop, P 0 > exists n, P n.
1 subgoal
============================
forall P0 : nat > Prop, P0 0 > exists n : nat, P0 n
Coq < eauto.
Proof completed.
Note that ex_intro should be declared as an hint.
See also: Section 8.13.1
8.12.3 autounfold with ident_{1} … ident_{n}
This tactic unfolds constants that were declared through a Hint Unfold in the given databases.
Variants:

autounfold with ident_{1} … ident_{n} in clause
Perform the unfolding in the given clause.
 autounfold with *
Uses the unfold hints declared in all the hint databases.
8.12.4 tauto
This tactic implements a decision procedure for intuitionistic propositional calculus based on the contractionfree sequent calculi LJT* of Roy Dyckhoff [54]. Note that tauto succeeds on any instance of an intuitionistic tautological proposition. tauto unfolds negations and logical equivalence but does not unfold any other definition.
The following goal can be proved by tauto whereas auto would fail:
1 subgoal
============================
forall (x : nat) (P0 : nat > Prop), x = 0 \/ P0 x > x <> 0 > P0 x
Coq < intros.
1 subgoal
x : nat
P0 : nat > Prop
H : x = 0 \/ P0 x
H0 : x <> 0
============================
P0 x
Coq < tauto.
Proof completed.
Moreover, if it has nothing else to do, tauto performs introductions. Therefore, the use of intros in the previous proof is unnecessary. tauto can for instance prove the following:
Coq < Goal forall (A:Prop) (P:nat > Prop),
Coq < A \/ (forall x:nat, ~ A > P x) > forall x:nat, ~ A > P x.
1 subgoal
============================
forall (A : Prop) (P0 : nat > Prop),
A \/ (forall x : nat, ~ A > P0 x) > forall x : nat, ~ A > P0 x
Coq <
Coq < tauto.
Proof completed.
Remark: In contrast, tauto cannot solve the following goal
Coq < A \/ (forall x:nat, ~ A > P x) > forall x:nat, ~ ~ (A \/ P x).
because (forall x:nat, ~ A > P x)
cannot be treated as atomic and an
instantiation of x
is necessary.
8.12.5 intuition tactic
The tactic intuition takes advantage of the searchtree built by the decision procedure involved in the tactic tauto. It uses this information to generate a set of subgoals equivalent to the original one (but simpler than it) and applies the tactic tactic to them [104]. If this tactic fails on some goals then intuition fails. In fact, tauto is simply intuition fail.
For instance, the tactic intuition auto applied to the goal
(forall (x:nat), P x)/\B > (forall (y:nat),P y)/\ P O \/B/\ P O
internally replaces it by the equivalent one:
(forall (x:nat), P x), B  P O
and then uses auto which completes the proof.
Originally due to César Muñoz, these tactics (tauto and intuition) have been completely reengineered by David Delahaye using mainly the tactic language (see Chapter 9). The code is now much shorter and a significant increase in performance has been noticed. The general behavior with respect to dependent types, unfolding and introductions has slightly changed to get clearer semantics. This may lead to some incompatibilities.
Variants:

intuition
Is equivalent to intuition auto with *.
8.12.6 rtauto
The rtauto tactic solves propositional tautologies similarly to what tauto does. The main difference is that the proof term is built using a reflection scheme applied to a sequent calculus proof of the goal. The search procedure is also implemented using a different technique.
Users should be aware that this difference may result in faster proofsearch but slower proofchecking, and rtauto might not solve goals that tauto would be able to solve (e.g. goals involving universal quantifiers).
8.12.7 firstorder
The tactic firstorder is an experimental extension of tauto to firstorder reasoning, written by Pierre Corbineau. It is not restricted to usual logical connectives but instead may reason about any firstorder class inductive definition.
Variants:

firstorder tactic
Tries to solve the goal with tactic when no logical rule may apply.
 firstorder with ident_{1} … ident_{n}
Adds lemmas ident_{1} … ident_{n} to the proofsearch environment.
 firstorder using qualid_{1} , … , qualid_{n}
Adds lemmas in auto hints bases qualid_{1} … qualid_{n} to the proofsearch environment. If qualid_{i} refers to an inductive type, it is the collection of its constructors which is added as hints.
 firstorder using qualid_{1} , … , qualid_{n} with ident_{1} … ident_{n}
This combines the effects of the using and with options.
Proofsearch is bounded by a depth parameter which can be set by typing the Set Firstorder Depth n vernacular command.
8.12.8 congruence
The tactic congruence, by Pierre Corbineau, implements the standard Nelson and Oppen congruence closure algorithm, which is a decision procedure for ground equalities with uninterpreted symbols. It also include the constructor theory (see 8.9.4 and 8.9.3). If the goal is a nonquantified equality, congruence tries to prove it with nonquantified equalities in the context. Otherwise it tries to infer a discriminable equality from those in the context. Alternatively, congruence tries to prove that a hypothesis is equal to the goal or to the negation of another hypothesis.
congruence is also able to take advantage of hypotheses stating quantified equalities, you have to provide a bound for the number of extra equalities generated that way. Please note that one of the members of the equality must contain all the quantified variables in order for congruence to match against it.
Coq < a=(f a) > (g b (f a))=(f (f a)) > (g a b)=(f (g b a)) > (g a b)=a.
1 subgoal
============================
a = f a > g b (f a) = f (f a) > g a b = f (g b a) > g a b = a
Coq < intros.
1 subgoal
H : a = f a
H0 : g b (f a) = f (f a)
H1 : g a b = f (g b a)
============================
g a b = a
Coq < congruence.
Proof completed.
1 subgoal
============================
f = pair a > Some (f c) = Some (f d) > c = d
Coq < intros.
1 subgoal
H : f = pair a
H0 : Some (f c) = Some (f d)
============================
c = d
Coq < congruence.
Proof completed.
Variants:

congruence n
Tries to add at most n instances of hypotheses stating quantified equalities to the problem in order to solve it. A bigger value of n does not make success slower, only failure. You might consider adding some lemmas as hypotheses using assert in order for congruence to use them.
Variants:

congruence with term_{1} … term_{n}
Adds term_{1} … term_{n} to the pool of terms used by congruence. This helps in case you have partially applied constructors in your goal.
Error messages:

I don’t know how to handle dependent equality
The decision procedure managed to find a proof of the goal or of a discriminable equality but this proof couldn’t be built in Coq because of dependentlytyped functions.  I couldn’t solve goal
The decision procedure didn’t find any way to solve the goal.  Goal is solvable by congruence but some arguments are missing. Try "congruence with …", replacing metavariables by arbitrary terms.
The decision procedure could solve the goal with the provision that additional arguments are supplied for some partially applied constructors. Any term of an appropriate type will allow the tactic to successfully solve the goal. Those additional arguments can be given to congruence by filling in the holes in the terms given in the error message, using the with variant described above.
8.12.9 omega
The tactic omega, due to Pierre Crégut,
is an automatic decision procedure for Presburger
arithmetic. It solves quantifierfree
formulas built with ~
, \/
, /\
,
>
on top of equalities, inequalities and disequalities on
both the type nat of natural numbers and Z of binary
integers. This tactic must be loaded by the command Require Import
Omega. See the additional documentation about omega
(see Chapter 19).
8.12.10 ring and ring_simplify term_{1} … term_{n}
The ring tactic solves equations upon polynomial expressions of a ring (or semiring) structure. It proceeds by normalizing both hand sides of the equation (w.r.t. associativity, commutativity and distributivity, constant propagation) and comparing syntactically the results.
ring_simplify applies the normalization procedure described above to the terms given. The tactic then replaces all occurrences of the terms given in the conclusion of the goal by their normal forms. If no term is given, then the conclusion should be an equation and both hand sides are normalized.
See Chapter 23 for more information on the tactic and how to declare new ring structures.
8.12.11 field, field_simplify term_{1}… term_{n} and field_simplify_eq
The field tactic is built on the same ideas as ring: this is a reflexive tactic that solves or simplifies equations in a field structure. The main idea is to reduce a field expression (which is an extension of ring expressions with the inverse and division operations) to a fraction made of two polynomial expressions.
Tactic field is used to solve subgoals, whereas field_simplify term_{1}…term_{n} replaces the provided terms by their reduced fraction. field_simplify_eq applies when the conclusion is an equation: it simplifies both hand sides and multiplies so as to cancel denominators. So it produces an equation without division nor inverse.
All of these 3 tactics may generate a subgoal in order to prove that denominators are different from zero.
See Chapter 23 for more information on the tactic and how to declare new field structures.
Example:
Coq < Goal forall x y:R,
Coq < (x * y > 0)%R >
Coq < (x * (1 / x + x / (x + y)))%R =
Coq < (( 1 / y) * y * ( x * (x / (x + y))  1))%R.
1 subgoal
x : R
y : R
H : (x * y > 0)%R
============================
(x + y)%R <> 0%R /\ y <> 0%R /\ x <> 0%R
See also: file plugins/setoid_ring/RealField.v for an example of instantiation,
theory theories/Reals for many examples of use of field.
8.12.12 fourier
This tactic written by Loïc Pottier solves linear inequalities on real numbers using Fourier’s method [63]. This tactic must be loaded by Require Import Fourier.
Example:
Coq < Require Import Fourier.
Coq < Goal forall x y:R, (x < y)%R > (y + 1 >= x  1)%R.
Proof completed.
8.12.13 autorewrite with ident_{1} …ident_{n}.
This tactic ^{5} carries out rewritings according the rewriting rule bases ident_{1} …ident_{n}.
Each rewriting rule of a base ident_{i} is applied to the main subgoal until it fails. Once all the rules have been processed, if the main subgoal has progressed (e.g., if it is distinct from the initial main goal) then the rules of this base are processed again. If the main subgoal has not progressed then the next base is processed. For the bases, the behavior is exactly similar to the processing of the rewriting rules.
The rewriting rule bases are built with the Hint Rewrite vernacular command.
Warning: This tactic may loop if you build non terminating rewriting systems.
Variant:

autorewrite with ident_{1} …ident_{n} using tactic
Performs, in the same way, all the rewritings of the bases ident_{1} ... ident_{n} applying tactic to the main subgoal after each rewriting step.  autorewrite with ident_{1} …ident_{n} in qualid
Performs all the rewritings in hypothesis qualid.
 autorewrite with ident_{1} …ident_{n} in qualid using tactic
Performs all the rewritings in hypothesis qualid applying tactic to the main subgoal after each rewriting step.
 autorewrite with ident_{1} …ident_{n} in clause
Performs all the rewritings in the clause clause.
The clause argument must not contain any type of nor value of.
See also: Section 8.13.4 for feeding the database of lemmas used by autorewrite.
See also: Section 10.7 for examples showing the use of
this tactic.
8.13 Controlling automation
8.13.1 The hints databases for auto and eauto
The hints for auto and eauto are stored in databases. Each database maps head symbols to a list of hints. One can use the command Print Hint ident to display the hints associated to the head symbol ident (see 8.13.3). Each hint has a cost that is an nonnegative integer, and an optional pattern. The hints with lower cost are tried first. A hint is tried by auto when the conclusion of the current goal matches its pattern or when it has no pattern.
Creating Hint databases
One can optionally declare a hint database using the command Create HintDb. If a hint is added to an unknown database, it will be automatically created.
Create HintDb ident [discriminated]
This command creates a new database named ident. The database is implemented by a Discrimination Tree (DT) that serves as an index of all the lemmas. The DT can use transparency information to decide if a constant should be indexed or not (c.f. 8.13.1), making the retrieval more efficient. The legacy implementation (the default one for new databases) uses the DT only on goals without existentials (i.e., auto goals), for nonImmediate hints and do not make use of transparency hints, putting more work on the unification that is run after retrieval (it keeps a list of the lemmas in case the DT is not used). The new implementation enabled by the discriminated option makes use of DTs in all cases and takes transparency information into account. However, the order in which hints are retrieved from the DT may differ from the order in which they were inserted, making this implementation observationaly different from the legacy one.
Variants:

Local Hint hint_definition :
ident_{1} … ident_{n}
This is used to declare a hint database that must not be exported to the other modules that require and import the current module. Inside a section, the option Local is useless since hints do not survive anyway to the closure of sections.
The general command to add a hint to some database ident_{1}, …, ident_{n} is:
Hint hint_definition : ident_{1} … ident_{n} 
where hint_definition is one of the following expressions:

Resolve term
This command adds apply term to the hint list with the head symbol of the type of term. The cost of that hint is the number of subgoals generated by apply term.
In case the inferred type of term does not start with a product the tactic added in the hint list is exact term. In case this type can be reduced to a type starting with a product, the tactic apply term is also stored in the hints list.
If the inferred type of term contains a dependent quantification on a predicate, it is added to the hint list of eapply instead of the hint list of apply. In this case, a warning is printed since the hint is only used by the tactic eauto (see 8.12.2). A typical example of a hint that is used only by eauto is a transitivity lemma.
Error messages:
Bound head variable
The head symbol of the type of term is a bound variable such that this tactic cannot be associated to a constant.
 term cannot be used as a hint
The type of term contains products over variables which do not appear in the conclusion. A typical example is a transitivity axiom. In that case the apply tactic fails, and thus is useless.
Variants: Resolve term_{1} …term_{m}
Adds each Resolve term_{i}.

Bound head variable
 Immediate term
This command adds apply term; trivial to the hint list associated with the head symbol of the type of ident in the given database. This tactic will fail if all the subgoals generated by apply term are not solved immediately by the trivial tactic (which only tries tactics with cost 0).
This command is useful for theorems such as the symmetry of equality or n+1=m+1 → n=m that we may like to introduce with a limited use in order to avoid useless proofsearch.
The cost of this tactic (which never generates subgoals) is always 1, so that it is not used by trivial itself.
Error messages:
Variants: Immediate term_{1} …term_{m}
Adds each Immediate term_{i}.
 Immediate term_{1} …term_{m}
 Constructors ident
If ident is an inductive type, this command adds all its constructors as hints of type Resolve. Then, when the conclusion of current goal has the form (ident …), auto will try to apply each constructor.
Error messages:
Variants: Constructors ident_{1} …ident_{m}
Adds each Constructors ident_{i}.
 Constructors ident_{1} …ident_{m}
 Unfold qualid
This adds the tactic unfold qualid to the hint list that will only be used when the head constant of the goal is ident. Its cost is 4.
Variants: Unfold ident_{1} …ident_{m}
Adds each Unfold ident_{i}.
 Unfold ident_{1} …ident_{m}
 Transparent, Opaque qualid
This adds a transparency hint to the database, making qualid a transparent or opaque constant during resolution. This information is used during unification of the goal with any lemma in the database and inside the discrimination network to relax or constrain it in the case of discriminated databases.
Variants: Transparent, Opaque ident_{1} …ident_{m}
Declares each ident_{i} as a transparent or opaque constant.
 Transparent, Opaque ident_{1} …ident_{m}
 Extern num [pattern] => tactic
This hint type is to extend auto with tactics other than apply and unfold. For that, we must specify a cost, an optional pattern and a tactic to execute. Here is an example:
Hint Extern 4 (~(_ = _)) => discriminate.
Now, when the head of the goal is a disequality, auto will try discriminate if it does not manage to solve the goal with hints with a cost less than 4.
One can even use some subpatterns of the pattern in the tactic script. A subpattern is a question mark followed by an ident, like ?X1 or ?X2. Here is an example:
Coq < Require Import List.
Coq < Hint Extern 5 ({?X1 = ?X2} + {?X1 <> ?X2}) =>
Coq < generalize X1, X2; decide equality : eqdec.
Coq < Goal
Coq < forall a b:list (nat * nat), {a = b} + {a <> b}.
1 subgoal
============================
forall a b : list (nat * nat), {a = b} + {a <> b}
Coq < info auto with eqdec.
== intro a; intro b; generalize a, b; decide equality;
generalize a1, p; decide equality.
generalize b1, n0; decide equality.
generalize a3, n; decide equality.
Proof completed.
Remark: One can use an Extern hint with no pattern to do
patternmatching on hypotheses using match goal with inside
the tactic.
Variants:

Hint hint_definition
No database name is given: the hint is registered in the core database.
 Hint Local hint_definition :
ident_{1} … ident_{n}
This is used to declare hints that must not be exported to the other modules that require and import the current module. Inside a section, the option Local is useless since hints do not survive anyway to the closure of sections.
 Hint Local hint_definition
Idem for the core database.
8.13.2 Hint databases defined in the Coq standard library
Several hint databases are defined in the Coq standard library. The actual content of a database is the collection of the hints declared to belong to this database in each of the various modules currently loaded. Especially, requiring new modules potentially extend a database. At Coq startup, only the core and v62 databases are non empty and can be used.
 core
 This special database is automatically used by auto. It contains only basic lemmas about negation, conjunction, and so on from. Most of the hints in this database come from the Init and Logic directories.
 arith
 This database contains all lemmas about Peano’s arithmetic proved in the directories Init and Arith
 zarith
 contains lemmas about binary signed integers from the directories theories/ZArith. When required, the module Omega also extends the database zarith with a highcost hint that calls omega on equations and inequalities in nat or Z.
 bool
 contains lemmas about booleans, mostly from directory theories/Bool.
 datatypes
 is for lemmas about lists, streams and so on that are mainly proved in the Lists subdirectory.
 sets
 contains lemmas about sets and relations from the directories Sets and Relations.
 typeclass_instances
 contains all the type class instances declared in the environment, including those used for setoid_rewrite, from the Classes directory.
There is also a special database called v62. It collects all hints that were declared in the versions of Coq prior to version 6.2.4 when the databases core, arith, and so on were introduced. The purpose of the database v62 is to ensure compatibility with further versions of Coq for developments done in versions prior to 6.2.4 (auto being replaced by auto with v62). The database v62 is intended not to be extended (!). It is not included in the hint databases list used in the auto with * tactic.
Furthermore, you are advised not to put your own hints in the core database, but use one or several databases specific to your development.
8.13.3 Print Hint
This command displays all hints that apply to the current goal. It fails if no proof is being edited, while the two variants can be used at every moment.
Variants:
 Print Hint ident
This command displays only tactics associated with ident in the hints list. This is independent of the goal being edited, so this command will not fail if no goal is being edited.
 Print Hint *
This command displays all declared hints.
 Print HintDb ident
This command displays all hints from database ident.
8.13.4 Hint Rewrite term_{1} …term_{n} : ident
This vernacular command adds the terms term_{1} …term_{n} (their types must be equalities) in the rewriting base ident with the default orientation (left to right). Notice that the rewriting bases are distinct from the auto hint bases and that auto does not take them into account.
This command is synchronous with the section mechanism (see 2.4): when closing a section, all aliases created by Hint Rewrite in that section are lost. Conversely, when loading a module, all Hint Rewrite declarations at the global level of that module are loaded.
Variants:

Hint Rewrite > term_{1} …term_{n} : ident
This is strictly equivalent to the command above (we only make explicit the orientation which otherwise defaults to >).  Hint Rewrite < term_{1} …term_{n} : ident
Adds the rewriting rules term_{1} …term_{n} with a righttoleft orientation in the base ident.  Hint Rewrite term_{1} …term_{n} using tactic : ident
When the rewriting rules term_{1} …term_{n} in ident will be used, the tactic tactic will be applied to the generated subgoals, the main subgoal excluded.  Print Rewrite HintDb ident
This command displays all rewrite hints contained in ident.
8.13.5 Hints and sections
Hints provided by the Hint commands are erased when closing a section. Conversely, all hints of a module A that are not defined inside a section (and not defined with option Local) become available when the module A is imported (using e.g. Require Import A.).
8.13.6 Setting implicit automation tactics
Proof with tactic.
This command may be used to start a proof. It defines a default
tactic to be used each time a tactic command tactic_{1} is ended by
“...
”. In this case the tactic command typed by the user is
equivalent to tactic_{1};tactic.
See also: Proof. in Section 7.1.4.
Declare Implicit Tactic tactic.
This command declares a tactic to be used to solve implicit arguments that Coq does not know how to solve by unification. It is used every time the term argument of a tactic has one of its holes not fully resolved.
Here is an example:
quo is assumed
Coq < Notation "x // y" := (quo x y _) (at level 40).
Coq <
Coq < Declare Implicit Tactic assumption.
Coq < Goal forall n m, m<>0 > { q:nat & { r  q * m + r = n } }.
1 subgoal
============================
forall n m : nat, m <> 0 > {q : nat & {r : nat  q * m + r = n}}
Coq < intros.
1 subgoal
n : nat
m : nat
H : m <> 0
============================
{q : nat & {r : nat  q * m + r = n}}
Coq < exists (n // m).
1 subgoal
n : nat
m : nat
H : m <> 0
============================
{r : nat  n // m * m + r = n}
The tactic exists (n // m) did not fail. The hole was solved by assumption so that it behaved as exists (quo n m H).
8.14 Generation of induction principles with Scheme
The Scheme command is a highlevel tool for generating automatically (possibly mutual) induction principles for given types and sorts. Its syntax follows the schema:
Scheme ident_{1} := Induction for ident’_{1} Sort sort_{1}
with
…
with ident_{m} := Induction for ident’_{m} Sort sort_{m}
where ident’_{1} … ident’_{m} are different inductive type identifiers belonging to the same package of mutual inductive definitions. This command generates ident_{1}… ident_{m} to be mutually recursive definitions. Each term ident_{i} proves a general principle of mutual induction for objects in type term_{i}.
Variants:

Scheme ident_{1} := Minimality for ident’_{1} Sort sort_{1}
with
…
with ident_{m} := Minimality for ident’_{m} Sort sort_{m}Same as before but defines a nondependent elimination principle more natural in case of inductively defined relations.
 Scheme Equality for ident_{1}
Tries to generate a boolean equality and a proof of the decidability of the usual equality.
 Scheme Induction for ident_{1} Sort sort_{1}
with
…
with Induction for ident_{m} Sort sort_{m}If you do not provide the name of the schemes, they will be automatically computed from the sorts involved (works also with Minimality).
See also: Section 10.3
8.14.1 Automatic declaration of schemes
It is possible to deactivate the automatic declaration of the induction
principles when defining a new inductive type with the
Unset Elimination Schemes command. It may be
reactivated at any time with Set Elimination Schemes.
You can also activate the automatic declaration of those boolean equalities
(see the second variant of Scheme) with the Set Equality Schemes
command. However you have to be careful with this option since
Coq may now reject welldefined inductive types because it cannot compute
a boolean equality for them.
8.14.2 Combined Scheme
The Combined Scheme command is a tool for combining induction principles generated by the Scheme command. Its syntax follows the schema :
Combined Scheme ident_{0} from ident_{1}, .., ident_{n}
ident_{1} …ident_{n} are different inductive principles that must belong to
the same package of mutual inductive principle definitions. This command
generates ident_{0} to be the conjunction of the principles: it is
built from the common premises of the principles and concluded by the
conjunction of their conclusions.
See also: Section 10.3.1
8.15 Generation of induction principles with Functional Scheme
The Functional Scheme command is a highlevel experimental tool for generating automatically induction principles corresponding to (possibly mutually recursive) functions. Its syntax follows the schema:
Functional Scheme ident_{1} := Induction for ident’_{1} Sort sort_{1}
with
…
with ident_{m} := Induction for ident’_{m} Sort sort_{m}
where ident’_{1} … ident’_{m} are different mutually defined function names (they must be in the same order as when they were defined). This command generates the induction principles ident_{1}…ident_{m}, following the recursive structure and case analyses of the functions ident’_{1} … ident’_{m}.
Functional Scheme
There is a difference between obtaining an induction scheme by using Functional Scheme on a function defined by Function or not. Indeed Function generally produces smaller principles, closer to the definition written by the user.
See also: Section 10.4
8.16 Simple tactic macros
A simple example has more value than a long explanation:
Solve is defined
Coq < Ltac ElimBoolRewrite b H1 H2 :=
Coq < elim b; [ intros; rewrite H1; eauto  intros; rewrite H2; eauto ].
ElimBoolRewrite is defined
The tactics macros are synchronous with the Coq section mechanism: a tactic definition is deleted from the current environment when you close the section (see also 2.4) where it was defined. If you want that a tactic macro defined in a module is usable in the modules that require it, you should put it outside of any section.
Chapter 9 gives examples of more complex userdefined tactics.
 1
 but it does not rename the hypothesis in the proofterm...
 2
 Actually, only the second subgoal will be generated since the other one can be automatically checked.
 3
 This corresponds to the cut rule of sequent calculus.
 4
 Reminder: opaque constants will not be expanded by δ reductions
 5
 The behavior of this tactic has much changed compared to the versions available in the previous distributions (V6). This may cause significant changes in your theories to obtain the same result. As a drawback of the reengineering of the code, this tactic has also been completely revised to get a very compact and readable version.