Chapter 8  Tactics

• 8.1  Invocation of tactics
• 8.2  Applying theorems
• 8.3  Managing the local context
• 8.4  Controlling the proof flow
• 8.5  Case analysis and induction
• 8.6  Rewriting expressions
• 8.7  Performing computations
• 8.8  Automation
• 8.9  Controlling automation
• 8.10  Decision procedures
• 8.11  Everything after this point has yet to be sorted
• 8.12  Equality
• 8.13  Equality and inductive sets
• 8.14  Inversion
• 8.15  Classical tactics
• 8.16  Automatizing
• 8.17  Non-logical tactics
• 8.18  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. A deduction rule can be read in two ways. The first one says: “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 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 built 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.

8.1  Invocation of tactics

A tactic is applied as an ordinary command. It may be preceded by a goal selector (see Section 9.2). If no selector is specified, the default selector (see Section 8.1.1) is used.

tactic_invocation	::=	toplevel_selector : tactic .
	|	tactic .

8.1.1  Set Default Goal Selector ‘‘toplevel_selector’’.

After using this command, the default selector – used when no selector is specified when applying a tactic – is set to the chosen value. The initial value is 1, hence the tactics are, by default, applied to the first goal. Using Set Default Goal Selector ‘‘all’’ will make is so that tactics are, by default, applied to every goal simultaneously. Then, to apply a tactic tac to the first goal only, you can write 1:tac. Although more selectors are available, only ‘‘all’’ or a single natural number are valid default goal selectors.

8.1.2  Test Default Goal Selector.

This command displays the current default selector.

8.1.3  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 (ref₁ := term₁) … (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 n-th non dependent premise of the term, as determined by the type of term.

  
  Error message: No such binder

• A bindings list can also be a simple list of terms term₁ … 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 9) 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 that are not bound in the conclusion of the type are required.

  
  Error message: Not the right number of missing arguments

8.1.4  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	::=	[ident1 [at_occurrences] ,
		… ,
		identm [at_occurrences]]
		[|- [* [at_occurrences]]]
	|	* |- [* [at_occurrences]]
	|	*
at_occurrences	::=	at occurrences
occurrences	::=	[-] num1 … numn

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 num₁ⁱ … num_niⁱ] 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 the 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 left-to-right 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, 8.5.2, 2.9.

8.2  Applying theorems

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:

1. Not an exact proof

Variants:

1. eexact term

   This tactic behaves like exact but is able to handle terms and goals with meta-variables.

8.2.2  assumption

This tactic 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:

1. No such assumption

Variants:

1. eassumption

   This tactic behaves like assumption but is able to handle goals with meta-variables.

8.2.3  refine term

This tactic applies to any goal. It behaves like exact with a big difference: the user can leave some holes (denoted by _ or (_:type)) in the term. refine will generate as many subgoals as there are holes in the term. The type of holes must be either synthesized by the system or declared by an explicit cast like (_:nat->Prop). Any subgoal that occurs in other subgoals is automatically shelved, as if calling shelve_unifiable (see Section 8.17.4). This low-level tactic can be useful to advanced users.

Example:

Error messages:

1. invalid argument: the tactic refine does not know what to do with the term you gave.
2. Refine passed ill-formed term: the term you gave is not a valid proof (not easy to debug in general). This message may also occur in higher-level tactics that call refine internally.
3. 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.

Variants:

1. simple refine term

   This tactic behaves like refine, but it does not shelve any subgoal. It does not perform any beta-reduction either.

2. notypeclasses refine term

   This tactic behaves like refine except it performs typechecking without resolution of typeclasses.

3. simple notypeclasses refine term

   This tactic behaves like simple refine except it performs typechecking without resolution of typeclasses.

8.2.4  apply term

This tactic applies to any goal. The argument term is a term well-formed 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 left-to-right order.

The tactic apply relies on first-order unification with dependent types unless the conclusion of the type of term is of the form (P t₁ … 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) u₁ … 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 t₁ … t_n in the goal. See pattern in Section 8.7.7 to transform the goal so that it gets the form (fun x => Q) u₁ … u_n.

Error messages:

1. Unable 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 8.7.7, 8.6.5).

2. 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:

1. apply term with term₁ … term_n

   Provides apply with explicit instantiations for all dependent premises of the type of term that do not occur in the conclusion and consequently cannot be found by unification. Notice that term₁ … term_n must be given according to the order of these dependent premises of the type of term.

   
   Error message: Not the right number of missing arguments

2. apply term with (ref₁ := term₁) … (ref_n := term_n)

   This also provides apply with values for instantiating premises. Here, variables are referred by names and non-dependent products by increasing numbers (see syntax in Section 8.1.3).

3. apply term₁ , … , term_n

   This is a shortcut for apply term₁ ; [ .. | … ; [ .. | 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 term₁.

4. eapply term

   The tactic eapply behaves like apply but it does not fail when no instantiations are deducible for some variables in the premises. Rather, it turns these variables into existential variables which are variables still to instantiate (see Section 2.11). The instantiation is intended to be found later in the proof.

5. simple apply term

   This behaves like apply but it reasons modulo conversion only on subterms that contain no variables to instantiate. For instance, the following example does not succeed because it would require the conversion of id ?foo and O.

   Coq < Definition id (x : nat) := x.
   
   Coq < Hypothesis H : forall y, id y = y.
   
   Coq < Goal O = O.
   
   Coq < Fail simple apply H.
   The command has indeed failed with message:
   Unable to unify "id ?M158 = ?M158" with "0 = 0".
   1 subgoal
     
     ============================
     0 = 0
   

   Because it reasons modulo a limited amount of conversion, simple apply fails quicker than apply and it is then well-suited for uses in used-defined tactics that backtrack often. Moreover, it does not traverse tuples as apply does.

6. [simple] apply term₁ [with bindings_list₁] , …, term_n [with bindings_list_n]
   [simple] eapply term₁ [with bindings_list₁] , …, term_n [with bindings_list_n]

   This summarizes the different syntaxes for apply and eapply.

7. lapply term

   This tactic applies to any goal, say G. The argument term has to be well-formed in the current context, its type being reducible to a non-dependent 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.

Example: Assume we have a transitive relation R on nat:

Consider the goal (R n p) provable using the transitivity of R:

The direct application of Rtrans with apply fails because no value for y in Rtrans is found by apply:

A solution is to apply (Rtrans n m p) or (Rtrans n m).

Note that n can be inferred from the goal, so the following would work too.

More elegantly, apply Rtrans with (y:=m) allows only mentioning the unknown m:

Another solution is to mention the proof of (R x y) in Rtrans …

…or the proof of (R y z).

On the opposite, one can use eapply which postpones the problem of finding m. Then one can apply the hypotheses Rnm and Rmp. This instantiates the existential variable and completes the proof.

Remark: When the conclusion of the type of the term to apply is an inductive type isomorphic to a tuple type and apply looks recursively whether a component of the tuple matches the goal, it excludes components whose statement would result in applying an universal lemma of the form forall A, ... -> A. Excluding this kind of lemma can be avoided by setting the following option:

Set Universal Lemma Under Conjunction

This option, which preserves compatibility with versions of Coq prior to 8.4 is also available for apply term in ident (see Section 8.2.5).

8.2.5  apply term in ident

This tactic applies to any goal. The argument term is a term well-formed 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 width-first left-to-right 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 first-order pattern-matching with dependent types.

Error messages:

1. Statement without assumptions

   This happens if the type of term has no non dependent premise.

2. 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:

1. apply term , … , term in ident

   This applies each of term in sequence in ident.

2. 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 8.1.3).

3. 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.

4. apply term with bindings_list , … , term with bindings_list in ident as intro_pattern

   This works as apply term with bindings_list , … , term with bindings_list in ident then applies the intro_pattern to the hypothesis ident.

5. eapply term with bindings_list , … , term with bindings_list in ident as intro_pattern

   This works as apply term with bindings_list , … , term with bindings_list in ident as intro_pattern but using eapply.

6. 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 id ?1234 and O where ?1234 is a variable to instantiate. Tactic simple apply term in ident does not either traverse tuples as apply term in ident does.

7. [simple] apply term [with bindings_list] , … , term [with bindings_list] in ident [as intro_pattern]
   [simple] eapply term [with bindings_list] , … , term [with bindings_list] in ident [as intro_pattern]

   This summarizes the different syntactic variants of apply term in ident and eapply term in ident.

8.2.6  constructor num

This tactic applies to a goal such that its conclusion is an inductive type (say I). The argument num must be less or equal to the numbers of constructor(s) of I. Let ci be the i-th constructor of I, then constructor i is equivalent to intros; apply ci.

Error messages:

1. Not an inductive product
2. Not enough constructors

Variants:

1. constructor

   This tries constructor 1 then constructor 2, … , then constructor n where n is the number of constructors of the head of the goal.

2. constructor num with bindings_list

   Let ci be the i-th 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).

3. split

   This applies only if I has a single constructor. It is then equivalent to constructor 1. It is typically used in the case of a conjunction A∧ B.

   
   Error message: Not an inductive goal with 1 constructor

4. exists bindings_list

   This applies only if I has a single constructor. It is then equivalent to intros; constructor 1 with bindings_list. It is typically used in the case of an existential quantification ∃ x, P(x).

   
   Error message: Not an inductive goal with 1 constructor

5. exists bindings_list , … , bindings_list

   This iteratively applies exists bindings_list.

6. left
   right

   These tactics apply only if I has two constructors, for instance in the case of a disjunction A∨ B. Then, they are respectively equivalent to constructor 1 and constructor 2.

   
   Error message: Not an inductive goal with 2 constructors

7. left with bindings_list
   right with bindings_list
   split with bindings_list

   As 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.

8. 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 8.2.4).

8.3  Managing the local context

8.3.1  intro

This tactic applies to a goal that 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 4.2¹. 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 ∀ 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 hnf until the tactic intro can be applied or the goal is not head-reducible.

Error messages:

1. No product even after head-reduction
2. ident is already used

Variants:

1. intros

   This repeats intro until it meets the head-constant. It never reduces head-constants and it never fails.

2. intro ident

   This 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).

3. intros ident₁ … ident_n

   This is equivalent to the composed tactic intro ident₁; … ; 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 8.3.2.

4. intros until ident

   This repeats intro until it meets a premise of the goal having form ( ident : term ) and discharges the variable named ident of the current goal.

   
   Error message: No such hypothesis in current goal

5. intros until num

   This repeats intro until the num-th non-dependent 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, as x=y -> y=x is the first non-dependent product. And on the subgoal forall 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 non-dependent product: forall x y:nat, nat -> x=y -> y=x

   
   Error message: No such hypothesis in current goal

   This happens when num is 0 or is greater than the number of non-dependent products of the goal.

6. intro after ident
   intro before ident
   intro at top
   intro at bottom

   These tactics apply intro and move 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 message: No such hypothesis : ident

7. intro ident₁ after ident₂
   intro ident₁ before ident₂
   intro ident₁ at top
   intro ident₁ at bottom

   These tactics behave as previously but naming the introduced hypothesis ident₁. It is equivalent to intro ident₁ followed by the appropriate call to move (see Section 8.3.5).

8.3.2  intros intro_pattern_list

This extension of the tactic intros allows to apply tactics on the fly on the variables or hypotheses which have been introduced. An introduction pattern list intro_pattern_list is a list of introduction patterns possibly containing the filling introduction patterns * and **. An introduction pattern is either:

• a naming introduction pattern, i.e. either one of:
  • the pattern ?
  • the pattern ?ident
  • an identifier
• an action introduction pattern which itself classifies into:
  • a disjunctive/conjunctive introduction pattern, i.e. either one of:
    • a disjunction of lists of patterns: [intro_pattern_list₁ | … | intro_pattern_list_n]
    • a conjunction of patterns: (p₁ , … , p_n)
    • a list of patterns (p₁ & … & p_n) for sequence of right-associative binary constructs
  • an equality introduction pattern, i.e. either one of:
    • a pattern for decomposing an equality: [= p₁ … p_n]
    • the rewriting orientations: -> or <-
  • the on-the-fly application of lemmas: p%term₁ …%term_n where p itself is not a pattern for on-the-fly application of lemmas (note: syntax is in experimental stage)
• the wildcard: _

Assuming a goal of type Q → P (non-dependent product), or of type ∀ 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 8.3.1;
• introduction over a disjunction of list of patterns [intro_pattern_list₁ | … | intro_pattern_list_n] 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 with n=2 since A\/B has 2 constructors): it destructs the introduced hypothesis as destruct (see Section 8.5.1) would and applies on each generated subgoal the corresponding tactic; intros intro_pattern_list_i. The introduction patterns in intro_pattern_list_i are expected to consume no more than the number of arguments of the i^th constructor. If it consumes less, then Coq completes the pattern so that all the arguments 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 (p₁, …, 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 p₁ … p_n to the arguments of the constructor of I (observe that (p₁, …, p_n) is an alternative notation for [p₁ … p_n]);
• introduction via (p₁ & … & p_n) is a shortcut for introduction via (p₁,(…,(…,p_n)…)); it expects the hypothesis to be a sequence of right-associative 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);
• if the product is over an equality type, then a pattern of the form [= p₁ … p_n] applies either injection (see Section 8.5.7) or discriminate (see Section 8.5.6) instead of destruct; if injection is applicable, the patterns p₁, …, p_n are used on the hypotheses generated by injection; if the number of patterns is smaller than the number of hypotheses generated, the pattern ? is used to complete the list;
• introduction over -> (respectively <-) expects the hypothesis to be an equality and the right-hand-side (respectively the left-hand-side) is replaced by the left-hand-side (respectively the right-hand-side) in the conclusion of the goal; the hypothesis itself is erased; if the term to substitute is a variable, it is substituted also in the context of goal and the variable is removed too;
• introduction over a pattern p%term₁ …%term_n first applies term₁,…, term_n on the hypothesis to be introduced (as in apply term₁, …, term_n in) prior to the application of the introduction pattern p;
• 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 8.3.3) 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 * introduces all forthcoming quantified variables appearing in a row; introduction over ** introduces all forthcoming quantified variables or hypotheses until the goal is not any more a quantification or an implication.

Example:

Remark: intros p₁ … p_n is not equivalent to intros p₁;…; intros p_n for the following reason: If one of the p_i is a wildcard pattern, he might succeed in the first case because the further hypotheses it depends in are eventually erased too while it might fail in the second case because of dependencies in hypotheses which are not yet introduced (and a fortiori not yet erased).

Remark: In intros intro_pattern_list, if the last introduction pattern is a disjunctive or conjunctive pattern [intro_pattern_list₁ | … | intro_pattern_list_n], the completion of intro_pattern_list_i so that all the arguments of the i^th constructors of the corresponding inductive type are introduced can be controlled with the following option:

Set Bracketing Last Introduction Pattern

Force completion, if needed, when the last introduction pattern is a disjunctive or conjunctive pattern (this is the default).

Unset Bracketing Last Introduction Pattern

Deactivate completion when the last introduction pattern is a disjunctive or conjunctive pattern.

8.3.3  clear ident

This tactic erases the hypothesis named ident in the local context of the current goal. As a consequence, ident is no more displayed and no more usable in the proof development.

Error messages:

1. No such hypothesis
2. ident is used in the conclusion
3. ident is used in the hypothesis ident’

Variants:

1. clear ident₁ … ident_n

   This is equivalent to clear ident₁. … clear ident_n.

2. 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.

   
   Error message: ident is not a local definition

3. clear - ident₁ … ident_n

   This tactic clears all the hypotheses except the ones depending in the hypotheses named ident₁ … ident_n and in the goal.

4. clear

   This tactic clears all the hypotheses except the ones the goal depends on.

5. clear dependent ident

   This clears the hypothesis ident and all the hypotheses that depend on it.

8.3.4  revert ident₁ … ident_n

This applies to any goal with variables ident₁ … ident_n. It moves the hypotheses (possibly defined) to the goal, if this respects dependencies. This tactic is the inverse of intro.

Error messages:

1. No such hypothesis
2. ident is used in the hypothesis ident’

Variants:

1. revert dependent ident

   This moves to the goal the hypothesis ident and all the hypotheses that depend on it.

8.3.5  move ident₁ after ident₂

This moves the hypothesis named ident₁ in the local context after the hypothesis named ident₂. The proof term is not changed.

If ident₁ comes before ident₂ in the order of dependencies, then all the hypotheses between ident₁ and ident₂ that (possibly indirectly) depend on ident₁ are moved too.

If ident₁ comes after ident₂ in the order of dependencies, then all the hypotheses between ident₁ and ident₂ that (possibly indirectly) occur in ident₁ are moved too.

Variants:

1. move ident₁ before ident₂

   This moves ident₁ towards and just before the hypothesis named ident₂.

2. move ident at top

   This moves ident at the top of the local context (at the beginning of the context).

3. move ident at bottom

   This moves ident at the bottom of the local context (at the end of the context).

Error messages:

1. No such hypothesis
2. Cannot move ident₁ after ident₂: it occurs in ident₂
3. Cannot move ident₁ after ident₂: it depends on ident₂

8.3.6  rename ident₁ into ident₂

This renames hypothesis ident₁ into ident₂ in the current context. The name of the hypothesis in the proof-term, however, is left unchanged.

Variants:

1. rename ident₁ into ident₂, …, ident_2k−1 into ident_2k

   This renames the variables ident₁ …ident₂k−1 into respectively ident₂ …ident₂k in parallel. In particular, the target identifiers may contain identifiers that exist in the source context, as long as the latter are also renamed by the same tactic.

Error messages:

1. No such hypothesis
2. ident₂ is already used

8.3.7  set ( ident := term )

This replaces term by ident in the conclusion of the current goal and adds the new definition ident := term to the local context.

If term has holes (i.e. subexpressions of the form “_”), the tactic first checks that all subterms matching the pattern are compatible before doing the replacement using the leftmost subterm matching the pattern.

Error messages:

1. The variable ident is already defined

Variants:

1. set ( ident := term ) in goal_occurrences

   This notation allows specifying which occurrences of term have to be substituted in the context. The in goal_occurrences clause is an occurrence clause whose syntax and behavior are described in Section 8.1.4.

2. set ( ident binder … binder := term )

   This is equivalent to set ( ident := fun binder … binder => term ).

3. set term

   This behaves as set ( ident := term ) but ident is generated by Coq. This variant also supports an occurrence clause.

4. set ( ident₀ binder … binder := term ) in goal_occurrences
   set term in goal_occurrences

   These are the general forms that combine the previous possibilities.

5. remember term as ident

   This behaves as set ( ident := term ) in * and using a logical (Leibniz’s) equality instead of a local definition.

6. remember term as ident eqn:ident

   This behaves as remember term as ident, except that the name of the generated equality is also given.

7. 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.

8. 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 |-.

9. pose ( ident binder … binder := term )

   This is equivalent to pose ( ident := fun binder … binder => term ).

10. pose term

    This behaves as pose ( ident := term ) but ident is generated by Coq.

8.3.8  decompose [ qualid₁ … qualid_n ] term

This tactic recursively decomposes a complex proposition in order to obtain atomic ones.

Example:

decompose does not work on right-hand sides of implications or products.

Variants:

1. decompose sum term

   This decomposes sum types (like or).

2. 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).

8.4  Controlling the proof flow

8.4.1  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 U². The subgoal U comes first in the list of subgoals remaining to prove.

Error messages:

1. Not a proposition or a type

   Arises when the argument form is neither of type Prop, Set nor Type.

Variants:

1. assert form

   This behaves as assert ( ident : form ) but ident is generated by Coq.

2. assert form by tactic

   This tactic behaves like assert but applies tactic to solve the subgoals generated by assert.

   
   Error message: Proof is not complete

3. assert form as intro_pattern

   If intro_pattern is a naming introduction pattern (see Section 8.3.2), 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 an action introduction pattern, the tactic behaves like assert form followed by the action done by this introduction pattern.

4. assert form as intro_pattern by tactic

   This combines the two previous variants of assert.

5. assert ( ident := term )

   This behaves as assert (ident : type);[exact term|idtac] where type is the type of term. This is deprecated in favor of pose proof.

   
   Error message: Variable ident is already declared

6. 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 := term) and pose proof term as intro_pattern is the same as applying the intro_pattern to term.

7. enough (ident : form)

   This adds a new hypothesis of name ident asserting form to the goal the tactic enough is applied to. A new subgoal stating form is inserted after the initial goal rather than before it as assert would do.

8. enough form

   This behaves like enough (ident : form) with the name ident of the hypothesis generated by Coq.

9. enough form as intro_pattern

   This behaves like enough form using intro_pattern to name or destruct the new hypothesis.

10. enough (ident : form) by tactic
    enough form by tactic
    enough form as intro_pattern by tactic

    This behaves as above but with tactic expected to solve the initial goal after the extra assumption form is added and possibly destructed. If the as intro_pattern clause generates more than one subgoal, tactic is applied to all of them.

11. 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.

12. specialize (ident term₁ … term_n)
    specialize ident with bindings_list

    The tactic specialize works on local hypothesis ident. The premises of this hypothesis (either universal quantifications or non-dependent implications) are instantiated by concrete terms coming either from arguments term₁ … term_n or from a bindings list (see Section 8.1.3 for more about bindings lists). In the second form, all instantiation elements must be given, whereas in the first form the application to term₁ … term_n can be partial. The first form is equivalent to assert (ident’ := ident term₁ … 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.

    
    Error messages:

    1. ident is used in hypothesis ident’
    2. ident is used in conclusion

8.4.2  generalize term

This tactic applies to any goal. It generalizes the conclusion with respect to some term.

Example:

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:

1. generalize term₁ , … , term_n

   This is equivalent to generalize term_n; … ; generalize term₁. Note that the sequence of term_i’s are processed from n to 1.

2. generalize term at num₁ … num_i

   This is equivalent to generalize term but it generalizes only over the specified occurrences of term (counting from left to right on the expression printed using option Set Printing All).

3. generalize term as ident

   This is equivalent to generalize term but it uses ident to name the generalized hypothesis.

4. generalize term₁ at num₁₁ … num_1i1 as ident₁ , … , term_n at num_n1 … num_nin as ident₂

   This is the most general form of generalize that combines the previous behaviors.

5. generalize dependent term

   This generalizes term but also all hypotheses that depend on term. It clears the generalized hypotheses.

8.4.3  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.

8.4.4  instantiate ( ident := term )

The instantiate tactic refines (see Section 8.2.3) an existential variable ident with the term term. It is equivalent to only [ident]: refine term (preferred alternative).

Remarks:

1. To be able to refer to an existential variable by name, the user must have given the name explicitly (see 2.11).
2. When you are referring to hypotheses which you did not name explicitly, be aware that Coq may make a different decision on how to name the variable in the current goal and in the context of the existential variable. This can lead to surprising behaviors.

Variants:

1. instantiate ( num := term ) This variant allows to refer to an existential variable which was not named by the user. The num argument is the position of the existential variable from right to left in the goal. Because this variant is not robust to slight changes in the goal, its use is strongly discouraged.
2. instantiate ( num := term ) in ident
3. instantiate ( num := term ) in ( Value of ident )
4. 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.

5. 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.

8.4.5  admit

The admit tactic allows temporarily skipping a subgoal so as to progress further in the rest of the proof. A proof containing admitted goals cannot be closed with Qed but only with Admitted.

Variants:

1. give_up

   Synonym of admit.

8.4.6  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.

8.4.7  contradiction

This tactic applies to any goal. The contradiction tactic attempts to find in the current context (after all intros) an hypothesis that is equivalent to an empty inductive type (e.g. False), to the negation of a singleton inductive type (e.g. True or x=x), or two contradictory hypotheses.

Error messages:

1. No such assumption

Variants:

1. contradiction ident

   The proof of False is searched in the hypothesis named ident.

8.4.8  contradict ident

This tactic allows manipulating 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

8.4.9  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  Case analysis and induction

The tactics presented in this section implement induction or case analysis on inductive or co-inductive objects (see Section 4.5).

8.5.1  destruct term

This tactic applies to any goal. The argument term must be of inductive or co-inductive type and the tactic generates subgoals, one for each possible form of term, i.e. one for each constructor of the inductive or co-inductive type. Unlike induction, no induction hypothesis is generated by destruct.

There are special 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 ident is not anymore dependent in the goal after application of destruct, it is erased (to avoid erasure, use parentheses, as in 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).
• In case term is an hypothesis ident of the context, and ident is not anymore dependent in the goal after application of destruct, it is erased (to avoid erasure, use parentheses, as in destruct (ident)).
• The argument term can also be a pattern of which holes are denoted by “_”. In this case, the tactic checks that all subterms matching the pattern in the conclusion and the hypotheses are compatible and performs case analysis using this subterm.

Variants:

1. destruct term₁, …, term_n

   This is a shortcut for destruct term₁; …; destruct term_n.

2. 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 [ p₁₁ …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 introduction pattern (see Section 8.3.2). This provides a concise notation for chaining destruction of an hypothesis.

3. destruct term eqn:naming_intro_pattern

   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 specified by naming_intro_pattern (see Section 8.3.2), in particular ? can be used to let Coq generate a fresh name.

4. 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 8.1.3).

5. edestruct term

   This tactic behaves like destruct term except 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 8.2.4).

6. destruct term₁ using term₂
   destruct term₁ using term₂ with bindings_list

   These are synonyms of induction term₁ using term₂ and induction term₁ using term₂ with bindings_list.

7. 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 8.1.4.

8. destruct term₁ with bindings_list₁ as disj_conj_intro_pattern eqn:naming_intro_pattern using term₂ with bindings_list₂ in goal_occurrences
   edestruct term₁ with bindings_list₁ as disj_conj_intro_pattern eqn:naming_intro_pattern using term₂ with bindings_list₂ in goal_occurrences

   These are the general forms of destruct and edestruct. They combine the effects of the with, as, eqn:, using, and in clauses.

9. case term

   The tactic case is a more basic tactic to perform case analysis without recursion. It behaves as elim term but using a case-analysis elimination principle and not a recursive one.

10. case term with bindings_list

    Analogous to elim term with bindings_list above.

11. ecase term
    ecase term with bindings_list

    In 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.

12. simple destruct ident

    This tactic behaves as intros until ident; case ident when ident is a quantified variable of the goal.

13. simple destruct num

    This tactic behaves as intros until num; case ident where ident is the name given by intros until num to the num-th non-dependent premise of the goal.

14. 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.

8.5.2  induction term

This tactic applies to any goal. The argument term must be of inductive type and the tactic induction generates subgoals, one for each possible form of term, i.e. one for each constructor of the inductive type.

If the argument is dependent in either the conclusion or some hypotheses of the goal, the argument is replaced by the appropriate constructor form in each of the resulting subgoals and induction hypotheses are added to the local context using names whose prefix is IH.

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 ident is not anymore dependent in the goal after application of induction, it is erased (to avoid erasure, use parentheses, as in 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).
• In case term is an hypothesis ident of the context, and ident is not anymore dependent in the goal after application of induction, it is erased (to avoid erasure, use parentheses, as in induction (ident)).
• The argument term can also be a pattern of which holes are denoted by “_”. In this case, the tactic checks that all subterms matching the pattern in the conclusion and the hypotheses are compatible and performs induction using this subterm.

Example:

Error messages:

1. Not an inductive product
2. Unable to find an instance for the variables ident …ident

   Use in this case the variant elim … with … below.

Variants:

1. 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 [ p₁₁ … 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 8.3.2). For instance, for an inductive type with one constructor, the pattern notation (p₁ , … , p_n) can be used instead of [ p₁ … p_n ].

2. 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 8.1.3).

3. 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 8.2.4).

4. induction term₁ using term₂

   This behaves as induction term₁ but using term₂ as induction scheme. It does not expect the conclusion of the type of term₁ to be inductive.

5. induction term₁ using term₂ with bindings_list

   This behaves as induction term₁ using term₂ but also providing instances for the premises of the type of term₂.

6. induction term₁, …, 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.

7. induction term in goal_occurrences

   This syntax is used for selecting which occurrences of term the induction has to be carried on. The in goal_occurrences clause is an occurrence clause whose syntax and behavior is described in Section 8.1.4. If variables or hypotheses not mentioning term in their type are listed in goal_occurrences, those are generalized as well in the statement to prove.

   
   Example:

   Coq < Lemma comm x y : x + y = y + x.
   1 subgoal
     
     x, y : nat
     ============================
     x + y = y + x
   
   Coq < induction y in x |- *.
   2 subgoals
     
     x : nat
     ============================
     x + 0 = 0 + x
   subgoal 2 is:
    x + S y = S y + x
   
   Coq < Show 2.
   subgoal 2 is:
     
     x, y : nat
     IHy : forall x : nat, x + y = y + x
     ============================
     x + S y = S y + x
   
8. induction term₁ with bindings_list₁ as disj_conj_intro_pattern using term₂ with bindings_list₂ in goal_occurrences
   einduction term₁ with bindings_list₁ as disj_conj_intro_pattern using term₂ with bindings_list₂ in goal_occurrences

   These are the most general forms of induction and einduction. It combines the effects of the with, as, using, and in clauses.

9. 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 non-dependent 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.

10. elim term with bindings_list

    Allows to give explicit instances to the premises of the type of term (see Section 8.1.3).

11. eelim term

    In case the type of term has dependent premises, this turns them into existential variables to be resolved later on.

12. elim term₁ using term₂
    elim term₁ using term₂ with bindings_list

    Allows the user to give explicitly an elimination predicate term₂ that is not the standard one for the underlying inductive type of term₁. The bindings_list clause allows instantiating premises of the type of term₂.

13. elim term₁ with bindings_list₁ using term₂ with bindings_list₂
    eelim term₁ with bindings_list₁ using term₂ with bindings_list₂

    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.

14. 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 that does not occur in the goal, then elim t is equivalent to elimtype I; 2: exact t.

15. simple induction ident

    This tactic behaves as intros until ident; elim ident when ident is a quantified variable of the goal.

16. simple induction num

    This tactic behaves as intros until num; elim ident where ident is the name given by intros until num to the num-th non-dependent premise of the goal.

8.5.3  double induction ident₁ ident₂

This tactic is deprecated and should be replaced by induction ident₁; induction ident₂ (or induction ident₁; destruct ident₂ depending on the exact needs).

Variant:

1. double induction num₁ num₂

   This tactic is deprecated and should be replaced by induction num₁; induction num₃ where num₃ is the result of num₂-num₁.

8.5.4  dependent induction ident

The experimental tactic dependent induction performs induction-inversion 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 [107] and the work of Cristina Cornes around inversion [36]. 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.

Example:

Here we did not 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.

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.

Now we are in a contradictory context and the proof can be solved.

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.

Variants:

1. dependent induction ident generalizing ident₁ …ident_n

   This performs 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.

2. dependent destruction ident

   This performs 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.

See also: 10.1 for a larger example of dependent induction and an explanation of the underlying technique.

8.5.5  functional induction (qualid term₁ … term_n)

The 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 13.2).

Remark: (qualid term₁ … 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: Parentheses 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 13.2) 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: 2.3,13.2,13.2, 8.14.1

Error messages:

1. Cannot find induction information on qualid
2. Not the right number of induction arguments

Variants:

1. functional induction (qualid term₁ … term_n) as disj_conj_intro_pattern using term_m+1 with bindings_list

   Similarly to Induction and elim (see Section 8.5.2), this allows giving explicitly the name of the introduced variables, 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.

8.5.6  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 term₁ = term₂ with term₁ and term₂ being elements of an inductive set. To build the proof, the tactic traverses the normal forms³ of term₁ and term₂ looking for a couple of subterms u and w (u subterm of the normal form of term₁ and w subterm of the normal form of term₂), 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:

1. No primitive equality found
2. Not a discriminable equality

Variants:

1. 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.

2. 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.

3. 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.

4. 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 term₁ <> term₂, this behaves as intro ident; discriminate ident.

   
   Error message: No discriminable equalities

8.5.7  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 t₁) and (c t₂) are two terms that are equal then  t₁ and  t₂ are equal too.

If term is a proof of a statement of conclusion term₁ = term₂, then injection applies injectivity as deep as possible to derive the equality of all the subterms of term₁ and term₂ 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 term₁ and term₂ should be elements of an inductive set and they should be neither explicitly equal, nor structurally different. We mean by this that, if n₁ and n₂ are their respective normal forms, then:

• n₁ and n₂ should not be syntactically equal,
• there must not exist any pair of subterms u and w, u subterm of n₁ and w subterm of n₂ , 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 term₁ and term₂ placed in the same positions and puts them as antecedents of the current goal.

Example: Consider the following goal:

Beware that injection yields an equality in a sigma type whenever the injected object has a dependent type P with its two instances in different types (P t₁ ... t_n) and (P u₁ ... u_n). If t₁ and u₁ are the same and have for type an inductive type for which a decidable equality has been declared using the command Scheme Equality (see 13.1), the use of a sigma type is avoided.

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:

1. Not a projectable equality but a discriminable one
2. Nothing to do, it is an equality between convertible terms
3. Not a primitive equality

Variants:

1. 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.

2. injection term with bindings_list

   This does the same as injection term but using the given bindings to instantiate parameters or hypotheses of term.

3. 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.

4. injection

   If the current goal is of the form term₁ <> term₂, this behaves as intro ident; injection ident.

   
   Error message: goal does not satisfy the expected preconditions

5. 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 so that all equalities generated are moved in the context of hypotheses. The number of intro_pattern must not exceed the number of equalities newly generated. If it is smaller, fresh names are automatically generated to adjust the list of intro_pattern to the number of new equalities. The original equality is erased if it corresponds to an hypothesis.

It is possible to ensure that injection term erases the original hypothesis and leaves the generated equalities in the context rather than putting them as antecedents of the current goal, as if giving injection term as (with an empty list of names). To obtain this behavior, the option Set Structural Injection must be activated. This option is off by default.

By default, injection only creates new equalities between terms whose type is in sort Type or Set, thus implementing a special behavior for objects that are proofs of a statement in Prop. This behavior can be turned off by setting the option Set Keep Proof Equalities.

8.5.8  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.

Remark: As inversion proofs may be large in size, we recommend the user to stock the lemmas whenever the same instance needs to be inverted several times. See Section 13.3.

Remark: Part of the behavior of the inversion tactic is to generate equalities between expressions that appeared in the hypothesis that is being processed. By default, no equalities are generated if they relate two proofs (i.e. equalities between terms whose type is in sort Prop). This behavior can be turned off by using the option Set Keep Proof Equalities.

Variants:

1. inversion num

   This does the same thing as intros until num then inversion ident where ident is the identifier for the last introduced hypothesis.

2. inversion_clear ident

   This behaves as inversion and then erases ident  from the context.

3. inversion ident as intro_pattern

   This generally behaves as inversion but using names in intro_pattern for naming hypotheses. The intro_pattern must have the form [ p₁₁ … 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, inversion 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.

   The inversion … as variant of inversion generally behaves in a slightly more expectable way than inversion (no artificial duplication of some hypotheses referring to other hypotheses) To take benefit of these improvements, it is enough to use inversion … as [], letting the names being finally chosen by Coq.

   Coq < Inductive contains0 : list nat -> Prop :=
           | in_hd : forall l, contains0 (0 :: l)
           | 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
   
4. inversion num as intro_pattern

   This allows naming the hypotheses introduced by inversion num in the context.

5. inversion_clear ident as intro_pattern

   This allows naming the hypotheses introduced by inversion_clear in the context. Notice that hypothesis names can be provided as if inversion were called, even though the inversion_clear will eventually erase the hypotheses.

6. inversion ident in ident₁ … ident_n

   Let ident₁ … ident_n, be identifiers in the local context. This tactic behaves as generalizing ident₁ … ident_n, and then performing inversion.

7. inversion ident as intro_pattern in ident₁ … ident_n

   This allows naming the hypotheses introduced in the context by inversion ident in ident₁ … ident_n.

8. inversion_clear ident in ident₁ … ident_n

   Let ident₁ … ident_n, be identifiers in the local context. This tactic behaves as generalizing ident₁ … ident_n, and then performing inversion_clear.

9. inversion_clear ident as intro_pattern in ident₁ … ident_n

   This allows naming the hypotheses introduced in the context by inversion_clear ident in ident₁ … ident_n.

10. 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.

11. dependent inversion ident as intro_pattern

    This allows naming the hypotheses introduced in the context by dependent inversion ident.

12. dependent inversion_clear ident

    Like dependent inversion, except that ident is cleared from the local context.

13. dependent inversion_clear ident as intro_pattern

    This allows naming the hypotheses introduced in the context by dependent inversion_clear ident.

14. 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 ∀ (x:T), s, then term  must be of type I:∀ (x:T), I x→ s′ where s′ is the type of the goal.

15. dependent inversion ident as intro_pattern with term

    This allows naming the hypotheses introduced in the context by dependent inversion ident with term.

16. dependent inversion_clear ident with term

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

17. dependent inversion_clear ident as intro_pattern with term

    This allows naming the hypotheses introduced in the context by dependent inversion_clear ident with term.

18. 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.

19. simple inversion ident as intro_pattern

    This allows naming the hypotheses introduced in the context by simple inversion.

20. 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.

21. inversion ident using ident′ in ident₁… ident_n

    This tactic behaves as generalizing ident₁… ident_n, then doing inversion ident using ident′.

Example 1: Non-dependent inversion

Let us consider the relation Le over natural numbers and the following variables:

Let us consider the following goal:

To prove the goal, we may need to reason by cases on H and to derive that m is necessarily of the form (S m₀) for certain m₀ and that (Le n m₀). Deriving these conditions corresponds to prove that the only possible constructor of (Le (S n) m) is LeS and that we can invert the -> in the type of LeS. This inversion is possible because Le is the smallest set closed by the constructors LeO and LeS.

Note that m has been substituted in the goal for (S m0) and that the hypothesis (Le n m0) has been added to the context.

Sometimes it is interesting to have the equality m=(S m0) in the context to use it after. In that case we can use inversion that does not clear the equalities:

Example 2: Dependent inversion

Let us consider the following goal:

As H occurs in the goal, we may want to reason by cases on its structure and so, we would like inversion tactics to substitute H by the corresponding term in constructor form. Neither Inversion nor Inversion_clear make such a substitution. To have such a behavior we use the dependent inversion tactics:

Note that H has been substituted by (LeS n m0 l) and m by (S m0).

8.5.9  fix ident num

This tactic is a primitive tactic to start a proof by induction. In general, it is easier to rely on higher-level induction tactics such as the ones described in Section 8.5.2.

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:

1. fix ident₁ num with ( ident₂ binder₂ … binder₂ [{ struct ident′₂ }] : type₂ ) … ( ident_n 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 binder₂ … binder₂, type₂, …, forall binder_n … binder_n, type_n. The identifiers ident₁ … 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).

8.5.10  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 co-inductive 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:

1. cofix ident₁ with ( ident₂ binder₂ … binder₂ : type₂ ) … ( ident_n binder_n … binder_n : type_n )

   This starts a proof by mutual coinduction. The statements to be simultaneously proved are respectively forall binder₂ … binder₂, type₂, …, forall binder_n … binder_n, type_n. The identifiers ident₁ … ident_n are the names of the coinduction hypotheses.

8.6  Rewriting expressions

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.

8.6.1  rewrite term

This tactic applies to any goal. The type of term must have the form

forall (x₁:A₁) … (x_n:A_n)eq term₁ term₂.

where eq is the Leibniz equality or a registered setoid equality.

Then rewrite term finds the first subterm matching term₁ in the goal, resulting in instances term₁′ and term₂′ and then replaces every occurrence of term₁′ by term₂′. Hence, some of the variables x_i are solved by unification, and some of the types A₁, …, A_n become new subgoals.

Error messages:

1. The term provided does not end with an equation
2. Tactic generated a subgoal identical to the original goal

   This happens if term₁ does not occur in the goal.

Variants:

1. rewrite -> term

   Is equivalent to rewrite term

2. rewrite <- term

   Uses the equality term₁=term₂ from right to left

3. rewrite term in clause

   Analogous to rewrite term but rewriting is done following clause (similarly to 8.7). 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.

   Orientation -> or <- can be inserted before the term to rewrite.

4. rewrite term at occurrences

   Rewrite only the given occurrences of term₁′. Occurrences are specified from left to right as for pattern (§8.7.7). The rewrite is always performed using setoid rewriting, even for Leibniz’s equality, so one has to Import Setoid to use this variant.

5. rewrite term by tactic

   Use tactic to completely solve the side-conditions arising from the rewrite.

6. rewrite term₁ , … , term_n

   Is equivalent to the n successive tactics rewrite term₁ 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.

7. 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.
8. 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.

8.6.2  replace term₁ with term₂

This tactic applies to any goal. It replaces all free occurrences of term₁ in the current goal with term₂ and generates the equality term₂=term₁ as a subgoal. This equality is automatically solved if it occurs among the assumption, or if its symmetric form occurs. It is equivalent to cut term₂=term₁; [intro Hn; rewrite <- Hn; clear Hn| assumption || symmetry; try assumption].

Error messages:

1. terms do not have convertible types

Variants:

1. replace term₁ with term₂ by tactic

   This acts as replace term₁ with term₂ but applies tactic to solve the generated subgoal term₂=term₁.

2. replace term

   Replaces term with term’ using the first assumption whose type has the form term=term’ or term’=term.

3. replace -> term

   Replaces term with term’ using the first assumption whose type has the form term=term’

4. replace <- term

   Replaces term with term’ using the first assumption whose type has the form term’=term

5. replace term₁ with term₂ in clause
   replace term₁ with term₂ in clause by tactic
   replace term in clause
   replace -> term in clause
   replace <- term in clause

   Acts as before but the replacements take place in clause (see Section 8.7) and not only in the conclusion of the goal. The clause argument must not contain any type of nor value of.

6. cutrewrite <- (term₁ = term₂)

   This tactic is deprecated. It acts like replace term₂ with term₁, or, equivalently as enough (term₁ = term₂) as <-.

7. cutrewrite -> (term₁ = term₂)

   This tactic is deprecated. It can be replaced by enough (term₁ = term₂) as ->.

8.6.3  subst ident

This tactic applies to a goal that has ident in its context and (at least) one hypothesis, say H, of type ident = t or t = ident with ident not occurring in t. 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.

If ident is a local definition of the form ident := t, it is also unfolded and cleared.

Remark: When several hypotheses have the form ident = t or t = ident, the first one is used.

Remark: If H is itself dependent in the goal, it is replaced by the proof of reflexivity of equality.

Variants:

1. subst ident₁ … ident_n

   This is equivalent to subst ident₁; …; subst ident_n.

2. subst

   This applies subst repeatedly from top to bottom to all identifiers of the context for which an equality of the form ident = t or t = ident or ident := t exists, with ident not occurring in t.

   Remark: The behavior of subst can be controlled using option Set Regular Subst Tactic. When this option is activated, subst also deals with the following corner cases:

   • A context with ordered hypotheses ident₁ = ident₂ and ident₁ = t, or t′ = ident₁ with t′ not a variable, and no other hypotheses of the form ident₂ = u or u = ident₂; without the option, a second call to subst would be necessary to replace ident₂ by t or t′ respectively.
   • The presence of a recursive equation which without the option would be a cause of failure of subst.
   • A context with cyclic dependencies as with hypotheses ident₁ = f ident₂ and ident₂ = g ident₁ which without the option would be a cause of failure of subst.

   Additionally, it prevents a local definition such as ident := t to be unfolded which otherwise it would exceptionally unfold in configurations containing hypotheses of the form ident = u, or u′ = ident with u′ not a variable.

   Finally, it preserves the initial order of hypotheses, which without the option it may break.

   The option is on by default.

8.6.4  stepl term

This tactic is for chaining rewriting steps. It assumes a goal of the form “R term₁ term₂” 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 term₂” and adds a new goal stating “eq term term₁”.

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 27).

Variants:

1. stepl term by tactic

   This applies stepl term then applies tactic to the second goal.

2. stepr term
   stepr term by tactic

   This behaves as stepl but on the right-hand-side 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 command

   Declare Right Step term.

8.6.5  change term

This tactic applies to any goal. It implements the rule “Conv” given in Section 4.4. change U replaces the current goal T with U providing that U is well-formed and that T and U are convertible.

Error messages:

1. Not convertible

Variants:

1. change term₁ with term₂

   This replaces the occurrences of term₁ by term₂ in the current goal. The terms term₁ and term₂ must be convertible.

2. change term₁ at num₁ … num_i with term₂

   This replaces the occurrences numbered num₁ … num_i of term₁ by term₂ in the current goal. The terms term₁ and term₂ must be convertible.

   
   Error message: Too few occurrences

3. change term in ident
4. change term₁ with term₂ in ident
5. change term₁ at num₁ … num_i with term₂ in ident

   This applies the change tactic not to the goal but to the hypothesis ident.

See also: 8.7

8.7  Performing computations

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 8.6.1, replace 8.6.2 and autorewrite 8.8.4 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 ident₁ … ident_n |- only in hypotheses ident₁ …ident_n

in ident₁ … ident_n |- * in hypotheses ident₁ …ident_n and in the conclusion

in * |- in every hypothesis

in * (equivalent to in * |- *) everywhere

in (type of ident₁) (value of ident₂) … |- in type part of ident₁, in the value part of ident₂, etc.

For backward compatibility, the notation in ident₁…ident_n performs the conversion in hypotheses ident₁…ident_n.

8.7.1  cbv flag₁ … flag_n, lazy flag₁ … 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 pattern-matching over a constructed term, and unfolding of fix and cofix expressions) and ζ (contraction of local definitions), the flags are either beta, delta, match, fix, cofix, iota or zeta. The iota flag is a shorthand for match, fix and cofix. The delta flag itself can be refined into delta [qualid₁…qualid_k] or delta -[qualid₁…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 let-in construction inside the term itself (use here the zeta flag). In any cases, opaque constants are not unfolded (see Section 6.10.1).

Normalization according to the flags is done by first evaluating the head of the expression into a weak-head normal form, i.e. until the evaluation is bloked by a variable (or an opaque constant, or an axiom), as e.g. in x u₁ ... u_n, or match x with ... end, or (fix f x {struct x} := ...) x, or is a constructed form (a λ-expression, a constructor, a cofixpoint, an inductive type, a product type, a sort), or is a redex that the flags prevent to reduce. Once a weak-head normal form is obtained, subterms are recursively reduced using the same strategy.

Reduction to weak-head normal form can be done using two strategies: lazy (lazy tactic), or call-by-value (cbv tactic). The lazy strategy is a call-by-need strategy, with sharing of reductions: the arguments of a function call are weakly evaluated only when necessary, and if an argument is used several times then it is weakly 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 call-by-value strategy is the one used in ML languages: the arguments of a function call are systematically weakly evaluated first. Despite the lazy strategy always performs fewer reductions than the call-by-value strategy, the latter is generally more efficient for evaluating purely computational expressions (i.e. with few dead code).

Variants:

1. compute
   cbv

   These are synonyms for cbv beta delta iota zeta.

2. lazy

   This is a synonym for lazy beta delta iota zeta.

3. compute [qualid₁…qualid_k]
   cbv [qualid₁…qualid_k]

   These are synonyms of cbv beta delta [qualid₁…qualid_k] iota zeta.

4. compute -[qualid₁…qualid_k]
   cbv -[qualid₁…qualid_k]

   These are synonyms of cbv beta delta -[qualid₁…qualid_k] iota zeta.

5. lazy [qualid₁…qualid_k]
   lazy -[qualid₁…qualid_k]

   These are respectively synonyms of lazy beta delta [qualid₁…qualid_k] iota zeta and lazy beta delta -[qualid₁…qualid_k] iota zeta.

6. vm_compute

   This tactic evaluates the goal using the optimized call-by-value evaluation bytecode-based virtual machine described in [77]. This algorithm is dramatically more efficient than the algorithm used for the cbv tactic, but it cannot be fine-tuned. It is specially interesting for full evaluation of algebraic objects. This includes the case of reflection-based tactics.

7. native_compute

   This tactic evaluates the goal by compilation to Objective Caml as described in [16]. If Coq is running in native code, it can be typically two to five times faster than vm_compute. Note however that the compilation cost is higher, so it is worth using only for intensive computations.

8.7.2  red

This tactic applies to a goal that has the form forall (x:T1)…(xk:Tk), t with t βιζ-reducing to c t1 … tn and c a constant. If c is transparent then it replaces c with its definition (say t) and then reduces (t t1 … tn) according to βιζ-reduction rules.

Error messages:

1. Not reducible

8.7.3  hnf

This tactic applies to any goal. It replaces the current goal with its head normal form according to the βδιζ-reduction rules, i.e. it reduces the head of the goal until it becomes a product or an irreducible term. All inner βι-redexes are also reduced.

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).

8.7.4  cbn and simpl

These tactics apply to any goal. They try to reduce a term to something still readable instead of fully normalizing it. They perform a sort of strong normalization with two key differences:

• They unfold a constant if and only if it leads to a ι-reduction, i.e. reducing a match or unfolding a fixpoint.
• While reducing a constant unfolding to (co)fixpoints, the tactics use the name of the constant the (co)fixpoint comes from instead of the (co)fixpoint definition in recursive calls.

The cbn tactic is claimed to be a more principled, faster and more predictable replacement for simpl.

The cbn tactic accepts the same flags as cbv and lazy. The behavior of both simpl and cbn can be tuned using the Arguments vernacular command as follows:

• A constant can be marked to be never unfolded by cbn or simpl:
  Coq < Arguments minus n m : simpl never.
  
  After that command an expression like (minus (S x) y) is left untouched by the tactics cbn and simpl.
• A constant can be marked to be unfolded only if applied to enough arguments. The number of arguments required can be specified using the / symbol in the arguments list of the Arguments vernacular command.
  Coq < Definition fcomp A B C f (g : A -> B) (x : A) : C := f (g x).
  
  Coq < Notation "f \o g" := (fcomp f g) (at level 50).
  
  Coq < Arguments fcomp {A B C} f g x /.
  
  After that command the expression (f \o g) is left untouched by simpl while ((f \o g) t) is reduced to (f (g t)). The same mechanism can be used to make a constant volatile, i.e. always unfolded.
  Coq < Definition volatile := fun x : nat => x.
  
  Coq < Arguments volatile / x.
  
• A constant can be marked to be unfolded only if an entire set of arguments evaluates to a constructor. The ! symbol can be used to mark such arguments.
  Coq < Arguments minus !n !m.
  
  After that command, the expression (minus (S x) y) is left untouched by simpl, while (minus (S x) (S y)) is reduced to (minus x y).
• A special heuristic to determine if a constant has to be unfolded can be activated with the following command:
  Coq < Arguments minus n m : simpl nomatch.
  
  The heuristic avoids to perform a simplification step that would expose a match construct in head position. For example the expression (minus (S (S x)) (S y)) is simplified to (minus (S x) y) even if an extra simplification is possible.

In detail, the tactic simpl first applies βι-reduction. Then, it expands transparent constants and tries to reduce further using βι-reduction. But, when no ι rule is applied after unfolding then δ-reductions 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 in the recursive calls are possibly unfolded by simpl. 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. This is the main difference between simpl and cbn. The tactic cbn reduces whenever it will be able to reuse it or not: succ t is reduced to S t.

Variants:

1. cbn [qualid₁…qualid_k]
   cbn -[qualid₁…qualid_k]

   These are respectively synonyms of cbn beta delta [qualid₁…qualid_k] iota zeta and cbn beta delta -[qualid₁…qualid_k] iota zeta (see 8.7.1).

2. simpl pattern

   This applies simpl only to the subterms matching pattern in the current goal.

3. simpl pattern at num₁ … num_i

   This applies simpl only to the num₁, …, num_i occurrences of the subterms matching pattern in the current goal.

   
   Error message: Too few occurrences

4. simpl qualid
   simpl string

   This applies simpl only to the applicative subterms whose head occurrence is the unfoldable constant qualid (the constant can be referred to by its notation using string if such a notation exists).

5. simpl qualid at num₁ … num_i
   simpl string at num₁ … num_i
   This applies simpl only to the num₁, …, num_i applicative subterms whose head occurrence is qualid (or string).

Refolding Reduction

This option (off by default) controls the use of the refolding strategy of cbn while doing reductions in unification, type inference and tactic applications. It can result in expensive unifications, as refolding currently uses a potentially exponential heuristic.

8.7.5  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 βι-normal form.

Error messages:

1. qualid does not denote an evaluable constant

Variants:

1. unfold qualid₁, …, qualid_n

   Replaces simultaneously qualid₁, …, qualid_n with their definitions and replaces the current goal with its βι normal form.

2. unfold qualid₁ at num₁¹, …, num_i¹, …, qualid_n at num₁ⁿ … num_jⁿ

   The lists num₁¹, …, num_i¹ and num₁ⁿ, …, num_jⁿ specify the occurrences of qualid₁, …, 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

3. 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.

4. 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).

5. unfold qualid_or_string₁ at num₁¹, …, num_i¹, …, qualid_or_string_n at num₁ⁿ … num_jⁿ

   This is the most general form, where qualid_or_string is either a qualid or a string referring to a notation.

8.7.6  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:

1. fold term₁ … term_n

   Equivalent to fold term₁;…; fold term_n.

8.7.7  pattern term

This command applies to any goal. The argument term must be a free subterm of the current goal. The command pattern performs β-expansion (the inverse of β-reduction) of the current goal (say T) by

1. replacing all occurrences of term in T with a fresh variable
2. abstracting this variable
3. 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:

1. pattern term at num₁ … num_n

   Only the occurrences num₁ … num_n of term are considered for β-expansion. Occurrences are located from left to right.

2. pattern term at - num₁ … num_n

   All occurrences except the occurrences of indexes num₁ … num_n of term are considered for β-expansion. Occurrences are located from left to right.

3. pattern term₁, …, term_m

   Starting from a goal φ(t₁ … t_m), the tactic pattern t₁, …, t_m generates the equivalent goal (fun (x₁:A₁) … (x_m:A_m) => φ(x₁… x_m)) t₁ … t_m. If t_i occurs in one of the generated types A_j these occurrences will also be considered and possibly abstracted.

4. pattern term₁ at num₁¹ … num_n1¹, …, term_m at num₁^m … num_nm^m

   This behaves as above but processing only the occurrences num₁¹, …, num_i¹ of term₁, …, num₁^m, …, num_j^m of term_m starting from term_m.

5. pattern term₁ [at [-] num₁¹ … num_n1¹] , …, term_m [at [-] num₁^m … num_nm^m]

   This is the most general syntax that combines the different variants.

8.7.8  Conversion tactics applied to hypotheses

conv_tactic in ident₁ … ident_n

Applies the conversion tactic conv_tactic to the hypotheses ident₁, …, 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:

1. No such hypothesis : ident.

8.8  Automation

8.8.1  auto

This tactic implements a Prolog-like 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 introduces 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:

1. auto num

   Forces the search depth to be num. The maximal search depth is 5 by default.

2. auto with ident₁ … ident_n

   Uses the hint databases ident₁ … ident_n in addition to the database core. See Section 8.9.1 for the list of pre-defined databases and the way to create or extend a database.

3. auto with *

   Uses all existing hint databases. See Section 8.9.1

4. auto using lemma₁ , … , lemma_n

   Uses lemma₁, …, 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.

5. info_auto

   Behaves like auto but shows the tactics it uses to solve the goal. This variant is very useful for getting a better understanding of automation, or to know what lemmas/assumptions were used.

6. [info_]auto [num] [using lemma₁ , … , lemma_n] [with ident₁ … ident_n]

   This is the most general form, combining the various options.

7. trivial

   This tactic is a restriction of auto that is not recursive and tries only hints that cost 0. Typically it solves trivial equalities like X=X.

8. trivial with ident₁ … ident_n
9. trivial with *
10. trivial using lemma₁ , … , lemma_n
11. info_trivial
12. [info_]trivial [using lemma₁ , … , lemma_n] [with ident₁ … ident_n]

Remark: auto either solves completely the goal or else leaves it intact. auto and trivial never fail.

See also: Section 8.9.1

8.8.2  eauto

This tactic generalizes auto. While auto does not try resolution hints which would leave existential variables in the goal, eauto does try them (informally speaking, it uses simple eapply where auto uses simple apply). As a consequence, eauto can solve such a goal:

Note that ex_intro should be declared as a hint.

Variants:

1. [info_]eauto [num] [using lemma₁ , … , lemma_n] [with ident₁ … ident_n]

   The various options for eauto are the same as for auto.

See also: Section 8.9.1

8.8.3  autounfold with ident₁ … ident_n

This tactic unfolds constants that were declared through a Hint Unfold in the given databases.

Variants:

1. autounfold with ident₁ … ident_n in clause

   Performs the unfolding in the given clause.

2. autounfold with *

   Uses the unfold hints declared in all the hint databases.

8.8.4  autorewrite with ident₁ … ident_n

This tactic ⁴ carries out rewritings according the rewriting rule bases ident₁ …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:

1. autorewrite with ident₁ … ident_n using tactic

   Performs, in the same way, all the rewritings of the bases ident₁ … ident_n applying tactic to the main subgoal after each rewriting step.

2. autorewrite with ident₁ … ident_n in qualid

   Performs all the rewritings in hypothesis qualid.

3. autorewrite with ident₁ … ident_n in qualid using tactic

   Performs all the rewritings in hypothesis qualid applying tactic to the main subgoal after each rewriting step.

4. autorewrite with ident₁ … ident_n in clause

   Performs all the rewriting in the clause clause. The clause argument must not contain any type of nor value of.

See also: Section 8.9.5 for feeding the database of lemmas used by autorewrite.

See also: Section 10.2 for examples showing the use of this tactic.

8.9  Controlling automation

8.9.1  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 8.9.4). Each hint has a cost that is a 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. 8.9.1), 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 non-Immediate 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 observationally different from the legacy one.

The general command to add a hint to some databases ident₁, …, ident_n is

Hint hint_definition : ident1 … identn

Variants:

1. Hint hint_definition

   No database name is given: the hint is registered in the core database.

2. Local Hint hint_definition : ident₁ … 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.

3. Local Hint hint_definition

   Idem for the core database.

The hint_definition is one of the following expressions:

• Resolve term[| [num] [pattern]]

  This command adds simple 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 simple apply term or numif specified. The associated pattern is inferred from the conclusion of the type of termor the given patternif specified.

  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 however be reduced to a type starting with a product, the tactic simple apply term is also stored in the hints list.

  If the inferred type of term contains a dependent quantification on a variable which occurs only in the premisses of the type and not in its conclusion, no instance could be inferred for the variable by unification with the goal. In this case, the hint is added to the hint list of eauto (see 8.8.2) instead of the hint list of auto and a warning is printed. A typical example of a hint that is used only by eauto is a transitivity lemma.

  
  Error messages:

  1. term cannot be used as a hint

     The head symbol of the type of term is a bound variable such that this tactic cannot be associated to a constant.

  
  Variants:

  1. Resolve term₁ … term_m

     Adds each Resolve term_i.

  2. Resolve -> term

     Adds the left-to-right implication of an equivalence as a hint (informally the hint will be used as apply <- term, although as mentionned before, the tactic actually used is a restricted version of apply).

  3. Resolve <- term

     Adds the right-to-left implication of an equivalence as a hint.

• Immediate term

  This command adds simple 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 simple 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 proof-search.

  The cost of this tactic (which never generates subgoals) is always 1, so that it is not used by trivial itself.

  
  Error messages:

  1. term cannot be used as a hint

  
  Variants:

  1. Immediate term₁ … term_m

     Adds each Immediate term_i.

• 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:

  1. ident is not an inductive type

  
  Variants:

  1. Constructors ident₁ … ident_m

     Adds each Constructors ident_i.

• 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:

  1. Unfold ident₁ … ident_m

     Adds each Unfold ident_i.

• 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:

  1. Transparent, Opaque ident₁ … ident_m

     Declares each ident_i as a transparent or opaque constant.

• 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 sub-patterns of the pattern in the tactic script. A sub-pattern is a question mark followed by an identifier, like ?X1 or ?X2. Here is an example:

  Coq < Require Import List.
  
  Coq < Hint Extern 5   ({?X1 = ?X2} + {?X1 <> ?X2}) =>
         generalize X1, X2; decide equality : eqdec.
  
  Coq < Goal
        forall a b:list (nat * nat), {a = b} + {a <> b}.
  1 subgoal
    
    ============================
    forall a b : list (nat * nat), {a = b} + {a <> b}
  
  Coq < Info 1 auto with eqdec.
  <g_auto::auto@0> "eqdec"
  No more subgoals.
  
• Cut regexp

  Warning: these hints currently only apply to typeclass proof search and the typeclasses eauto tactic (20.6.5).

  This command can be used to cut the proof-search tree according to a regular expression matching paths to be cut. The grammar for regular expressions is the following. Beware, there is no operator precedence during parsing, one can check with Print HintDb to verify the current cut expression:

  e ::= ident hint or instance identifier       _ any hint       e | e′ disjunction       e e′ sequence       e * Kleene star       emp empty       eps epsilon       ( e )

  The emp regexp does not match any search path while eps matches the empty path. During proof search, the path of successive successful hints on a search branch is recorded, as a list of identifiers for the hints (note Hint Extern’s do not have an associated identifier). Before applying any hint ident the current path p extended with ident is matched against the current cut expression c associated to the hint database. If matching succeeds, the hint is not applied. The semantics of Hint Cut e is to set the cut expression to c | e, the initial cut expression being emp.

• Mode (+ | ! | -)^* qualid

  This sets an optional mode of use of the identifier qualid. When proof-search faces a goal that ends in an application of qualid to arguments term₁ … term_n, the mode tells if the hints associated to qualid can be applied or not. A mode specification is a list of n +, ! or - items that specify if an argument of the identifier is to be treated as an input (+), if its head only is an input (!) or an output (-) of the identifier. For a mode to match a list of arguments, input terms and input heads must not contain existential variables or be existential variables respectively, while outputs can be any term. Multiple modes can be declared for a single identifier, in that case only one mode needs to match the arguments for the hints to be applied.

  The head of a term is understood here as the applicative head, or the match or projection scrutinee’s head, recursively, casts being ignored.

  Hint Mode is especially useful for typeclasses, when one does not want to support default instances and avoid ambiguity in general. Setting a parameter of a class as an input forces proof-search to be driven by that index of the class, with ! giving more flexibility by allowing existentials to still appear deeper in the index but not at its head.

Remark: One can use an Extern hint with no pattern to do pattern-matching on hypotheses using match goal with inside the tactic.

8.9.2  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 database is non empty and can be used.

core

This special database is automatically used by auto, except when pseudo-database nocore is given to auto. The core database 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 high-cost 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.

You are advised not to put your own hints in the core database, but use one or several databases specific to your development.

8.9.3  Remove Hints term₁ … term_n :  ident₁ … ident_m

This command removes the hints associated to terms term₁ … term_n in databases ident₁ … ident_m.

8.9.4  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:

1. 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.

2. Print Hint *

   This command displays all declared hints.

3. Print HintDb ident

   This command displays all hints from database ident.

8.9.5  Hint Rewrite term₁ … term_n :  ident₁ … ident_m

This vernacular command adds the terms term₁ … term_n (their types must be equalities) in the rewriting bases ident₁, …, ident_m 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:

1. Hint Rewrite -> term₁ … term_n : ident₁ … ident_m

   This is strictly equivalent to the command above (we only make explicit the orientation which otherwise defaults to ->).

2. Hint Rewrite <- term₁ … term_n : ident₁ … ident_m

   Adds the rewriting rules term₁ … term_n with a right-to-left orientation in the bases ident₁, …, ident_m.

3. Hint Rewrite term₁ … term_n using tactic : ident₁ … ident_m

   When the rewriting rules term₁ … term_n in ident₁, …, ident_m will be used, the tactic tactic will be applied to the generated subgoals, the main subgoal excluded.

4. Print Rewrite HintDb ident

   This command displays all rewrite hints contained in ident.

8.9.6  Hint locality

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.).

As of today, hints only have a binary behavior regarding locality, as described above: either they disappear at the end of a section scope, or they remain global forever. This causes a scalability issue, because hints coming from an unrelated part of the code may badly influence another development. It can be mitigated to some extent thanks to the Remove Hints command (see  8.9.3), but this is a mere workaround and has some limitations (for instance, external hints cannot be removed).

A proper way to fix this issue is to bind the hints to their module scope, as for most of the other objects Coq uses. Hints should only made available when the module they are defined in is imported, not just required. It is very difficult to change the historical behavior, as it would break a lot of scripts. We propose a smooth transitional path by providing the Loose Hint Behavior option which accepts three flags allowing for a fine-grained handling of non-imported hints.

Variants:

1. Set Loose Hint Behavior "Lax"

   This is the default, and corresponds to the historical behavior, that is, hints defined outside of a section have a global scope.

2. Set Loose Hint Behavior "Warn"

   When set, it outputs a warning when a non-imported hint is used. Note that this is an over-approximation, because a hint may be triggered by a run that will eventually fail and backtrack, resulting in the hint not being actually useful for the proof.

3. Set Loose Hint Behavior "Strict"

   When set, it changes the behavior of an unloaded hint to a immediate fail tactic, allowing to emulate an import-scoped hint mechanism.

8.9.7  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 tactic₁ is ended by “...”. In this case the tactic command typed by the user is equivalent to tactic₁;tactic.

See also: Proof. in Section 7.1.4.

Variants:

1. Proof with tactic using ident₁ … ident_n

   Combines in a single line Proof with and Proof using, see 7.1.5

2. Proof using ident₁ … ident_n with tactic

   Combines in a single line Proof with and Proof using, see 7.1.5

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:

The tactic exists (n // m) did not fail. The hole was solved by assumption so that it behaved as exists (quo n m H).

8.10  Decision procedures

8.10.1  tauto

This tactic implements a decision procedure for intuitionistic propositional calculus based on the contraction-free sequent calculi LJT* of Roy Dyckhoff [56]. 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:

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:

Remark: In contrast, tauto cannot solve the following goal

because (forall x:nat, ~ A -> P x) cannot be treated as atomic and an instantiation of x is necessary.

Variants:

1. dtauto

   While tauto recognizes inductively defined connectives isomorphic to the standard connective and, prod, or, sum, False, Empty_set, unit, True, dtauto recognizes also all inductive types with one constructors and no indices, i.e. record-style connectives.

8.10.2  intuition tactic

The tactic intuition takes advantage of the search-tree 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 [113]. 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 re-engineered 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:

1. intuition

   Is equivalent to intuition auto with *.

2. dintuition

   While intuition recognizes inductively defined connectives isomorphic to the standard connective and, prod, or, sum, False, Empty_set, unit, True, dintuition recognizes also all inductive types with one constructors and no indices, i.e. record-style connectives.

Some aspects of the tactic intuition can be controlled using options. To avoid that inner negations which do not need to be unfolded are unfolded, use:

Unset Intuition Negation Unfolding

To do that all negations of the goal are unfolded even inner ones (this is the default), use:

Set Intuition Negation Unfolding

To avoid that inner occurrence of iff which do not need to be unfolded are unfolded (this is the default), use:

Unset Intuition Iff Unfolding

To do that all negations of the goal are unfolded even inner ones (this is the default), use:

Set Intuition Iff Unfolding

8.10.3  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 proof-search but slower proof-checking, and rtauto might not solve goals that tauto would be able to solve (e.g. goals involving universal quantifiers).

8.10.4  firstorder

The tactic firstorder is an experimental extension of tauto to first-order reasoning, written by Pierre Corbineau. It is not restricted to usual logical connectives but instead may reason about any first-order class inductive definition.

The default tactic used by firstorder when no rule applies is auto with *, it can be reset locally or globally using the Set Firstorder Solver tactic vernacular command and printed using Print Firstorder Solver.

Variants:

1. firstorder tactic

   Tries to solve the goal with tactic when no logical rule may apply.

2. firstorder using qualid₁ , … , qualid_n

   Adds lemmas qualid₁ … qualid_n to the proof-search environment. If qualid_i refers to an inductive type, it is the collection of its constructors which are added to the proof-search environment.

3. firstorder with ident₁ … ident_n

   Adds lemmas from auto hint bases ident₁ … ident_n to the proof-search environment.

4. firstorder tactic using qualid₁ , … , qualid_n with ident₁ … ident_n

   This combines the effects of the different variants of firstorder.

Proof-search is bounded by a depth parameter which can be set by typing the Set Firstorder Depth n vernacular command.

8.10.5  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 8.5.7 and 8.5.6). If the goal is a non-quantified equality, congruence tries to prove it with non-quantified 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.

Variants:

1. 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.

2. congruence with term₁ … term_n

   Adds term₁ … term_n to the pool of terms used by congruence. This helps in case you have partially applied constructors in your goal.

Error messages:

1. 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 could not be built in Coq because of dependently-typed functions.

2. 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.

8.11  Everything after this point has yet to be sorted

8.11.1  constr_eq term₁ term₂

This tactic applies to any goal. It checks whether its arguments are equal modulo alpha conversion and casts.

Error message: Not equal

8.11.2  unify term₁ term₂

This tactic applies to any goal. It checks whether its arguments are unifiable, potentially instantiating existential variables.

Error message: Not unifiable

Variants:

1. unify term₁ term₂ with ident

   Unification takes the transparency information defined in the hint database ident into account (see Section 8.9.1).

8.11.3  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 8.2.4).

Error message: Not an evar

8.11.4  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.

Error message: No evars

8.11.5  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

8.12  Equality

8.12.1  f_equal

This tactic applies to a goal of the form f a₁ … a_n = f′ a′₁ … a′_n. Using f_equal on such a goal leads to subgoals f=f′ and a₁=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.

8.12.2  reflexivity

This tactic applies to a goal that 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:

1. The conclusion is not a substitutive equation
2. Unable to unify … with …

8.12.3  symmetry

This tactic applies to a goal that has the form t=u and changes it into u=t.

Variants:

1. symmetry in ident

   If the statement of the hypothesis ident has the form t=u, the tactic changes it to u=t.

8.12.4  transitivity term

This tactic applies to a goal that has the form t=u and transforms it into the two subgoals t=term and term=u.

8.13  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 =.

8.13.1  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. It solves goals of the form {x=y}+{~x=y} as well.

8.13.2  compare term₁ term₂

This tactic compares two given objects term₁ and term₂ of an inductive datatype. If G is the current goal, it leaves the sub-goals term₁=term₂ -> G and ~term₁=term₂ -> G. The type of term₁ and term₂ must satisfy the same restrictions as in the tactic decide equality.

8.13.3  simplify_eq term

Let term be the proof of a statement of conclusion term₁=term₂. If term₁ and term₂ 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:

1. 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.

2. 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.

3. 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.

4. simplify_eq

   If the current goal has form t₁<>t₂, it behaves as intro ident; simplify_eq ident.

8.13.4  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:

1. dependent rewrite <- ident

   Analogous to dependent rewrite -> but uses the equality from right to left.

8.14  Inversion

8.14.1  functional inversion ident

functional inversion is a tactic that performs inversion on hypothesis ident of the form qualid term₁…term_n = term or term = qualid term₁…term_n where qualid must have been defined using Function (see Section 2.3).

Error messages:

1. Hypothesis ident must contain at least one Function
2. Cannot find inversion information for hypothesis ident

   This error may be raised when some inversion lemma failed to be generated by Function.

Variants:

1. 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.

2. functional inversion ident qualid
   functional inversion num qualid

   If the hypothesis ident (or num) has a type of the form qualid₁ term₁…term_n = qualid₂ term_n+1…term_n+m where qualid₁ and qualid₂ are valid candidates to functional inversion, this variant allows choosing which qualid is inverted.

8.14.2  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 10.3 for the full details.

Error messages:

1. quote: not a simple fixpoint

   Happens when quote is not able to perform inversion properly.

Variants:

1. quote ident [ ident₁ …ident_n ]

   All terms that are built only with ident₁ …ident_n will be considered by quote as constants rather than variables.

8.15  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.

8.15.1  classical_left and 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.

8.16  Automatizing

8.16.1  btauto

The tactic btauto implements a reflexive solver for boolean tautologies. It solves goals of the form t = u where t and u are constructed over the following grammar:

t ::= x ∣ true ∣ false∣ orb t1 t2 ∣ andb t1 t2 ∣xorb t1 t2 ∣negb t ∣if t1 then t2 else t3

Whenever the formula supplied is not a tautology, it also provides a counter-example.

Internally, it uses a system very similar to the one of the ring tactic.

8.16.2  omega

The tactic omega, due to Pierre Crégut, is an automatic decision procedure for Presburger arithmetic. It solves quantifier-free 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 21).

8.16.3  ring and ring_simplify term₁ … term_n

The ring tactic solves equations upon polynomial expressions of a ring (or semi-ring) 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 25 for more information on the tactic and how to declare new ring structures. All declared field structures can be printed with the Print Rings command.

8.16.4  field, field_simplify term₁ … 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 term₁…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 25 for more information on the tactic and how to declare new field structures. All declared field structures can be printed with the Print Fields command.

Example:

See also: file plugins/setoid_ring/RealField.v for an example of instantiation,
    theory theories/Reals for many examples of use of field.

8.16.5  fourier

This tactic written by Loïc Pottier solves linear inequalities on real numbers using Fourier’s method [65]. This tactic must be loaded by Require Import Fourier.

Example:

8.17  Non-logical tactics

8.17.1  cycle num

This tactic puts the num first goals at the end of the list of goals. If num is negative, it will put the last |num| goals at the beginning of the list.

Example:

8.17.2  swap num₁ num₂

This tactic switches the position of the goals of indices num₁ and num₂. If either num₁ or num₂ is negative then goals are counted from the end of the focused goal list. Goals are indexed from 1, there is no goal with position 0.

Example:

8.17.3  revgoals

This tactics reverses the list of the focused goals.

Example:

8.17.4  shelve

This tactic moves all goals under focus to a shelf. While on the shelf, goals will not be focused on. They can be solved by unification, or they can be called back into focus with the command Unshelve (Section 8.17.5).

Variants:

1. shelve_unifiable

   Shelves only the goals under focus that are mentioned in other goals. Goals that appear in the type of other goals can be solved by unification.

   
   Example:

   Coq < Goal exists n, n=0.
   1 subgoal
     
     ============================
     exists n : nat, n = 0
   
   Coq < refine (ex_intro _ _ _).
   1 focused subgoal
   (shelved: 1)
     
     ============================
     ?Goal = 0
   
   Coq < all:shelve_unifiable.
   1 focused subgoal
   (shelved: 1)
     
     ============================
     ?Goal = 0
   
   Coq < reflexivity.
   No more subgoals.
   

8.17.5  Unshelve

This command moves all the goals on the shelf (see Section 8.17.4) from the shelf into focus, by appending them to the end of the current list of focused goals.

8.17.6  give_up

This tactic removes the focused goals from the proof. They are not solved, and cannot be solved later in the proof. As the goals are not solved, the proof cannot be closed.

The give_up tactic can be used while editing a proof, to choose to write the proof script in a non-sequential order.

8.18  Simple tactic macros

A simple example has more value than a long explanation:

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 user-defined tactics.

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1

Actually, only the second subgoal will be generated since the other one can be automatically checked.

2

This corresponds to the cut rule of sequent calculus.

3

Reminder: opaque constants will not be expanded by δ reductions.

4

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 re-engineering of the code, this tactic has also been completely revised to get a very compact and readable version.