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

Tactics¶

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 Requesting information).

Since not every rule applies to a given statement, not every tactic can be used to reduce a given 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 The tactic language.

Invocation of tactics¶

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

tactic_invocation ::=  toplevel_selector : tactic.
                       |tactic .

Option Default Goal Selector "toplevel_selector"¶: This option controls 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 value 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.

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:

bindings_list ::=  (ref := term) ... (ref := term)
                   term ... term

In a bindings list of the form (ref:= term)*, ref is either an ident or a num. The references are determined according to the type of term. If ref 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 is some number n, this number denotes the n-th non dependent premise of the term, as determined by the type of term.

Error No such binder.¶
A bindings list can also be a simple list of terms term*. In that case the references to which these terms correspond are determined by the tactic. In case of induction, destruct, elim and case, 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 Not the right number of missing arguments.¶

Occurrence sets and occurrence clauses¶

An occurrence clause is a modifier to some tactics that obeys the following syntax:

occurrence_clause ::=  in goal_occurrences
goal_occurrences  ::=  [ident [at_occurrences], ... , ident [at_occurrences] [|- [* [at_occurrences]]]]
                      | * |- [* [at_occurrences]]
                      | *
at_occurrences    ::=  at occurrences
occurrences       ::=  [-] num ... num

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 at num*? parts indicate that occurrences have to be selected in the hypotheses named ident. If no numbers are given for hypothesis ident, 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 Printing All, counting from left to right. In particular, occurrences of term in implicit arguments (see Implicit arguments) or coercions (see コアーション) 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. If 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 occurrence clauses: set, remember, induction, destruct.

参考

Managing the local context, Case analysis and induction, 全ての構成を表示する.

Applying theorems¶

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 Conversion rules).

Error Not an exact proof.¶

Variant eexact term.¶: This tactic behaves like exact but is able to handle terms and goals with existential variables.

assumption¶

This tactic looks in the local context for a hypothesis whose type is convertible to the goal. If it is the case, the subgoal is proved. Otherwise, it fails.

Error No such assumption.¶

Variant eassumption¶: This tactic behaves like assumption but is able to handle goals with existential variables.

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. This low-level tactic can be useful to advanced users.

Example

Inductive Option : Set := | Fail : Option | Ok : bool -> Option.: Option is defined Option_rect is defined Option_ind is defined Option_rec is defined
Definition get : forall x:Option, x <> Fail -> bool.: 1 subgoal ============================ forall x : Option, x <> Fail -> bool
refine (fun x:Option => match x return x <> Fail -> bool with | Fail => _ | Ok b => fun _ => b end).: 1 subgoal x : Option ============================ Fail <> Fail -> bool
intros; absurd (Fail = Fail); trivial.: No more subgoals.
Defined.: get is defined

Error Invalid argument.¶: The tactic refine does not know what to do with the term you gave.

Error 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.

Error Cannot infer a term for this placeholder.¶: There is a hole in the term you gave whose type cannot be inferred. Put a cast around it.

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

Variant notypeclasses refine term¶: This tactic behaves like refine except it performs type checking without resolution of typeclasses.

Variant simple notypeclasses refine term¶: This tactic behaves like simple refine except it performs type checking without resolution of typeclasses.

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 unify, then P is taken to be (fun x => Q). Otherwise, apply tries to define P by abstracting over t_1 ... t__n in the goal. See pattern to transform the goal so that it gets the form (fun x => Q) u₁ ... u_n.

Error Unable to unify term with term.¶: 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.

Error Unable to find an instance for the variables 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:

Variant apply term with term+

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 the collection term+ must be given according to the order of these dependent premises of the type of term.

Error Not the right number of missing arguments.

Variant apply term with bindings_list: This also provides apply with values for instantiating premises. Here, variables are referred by names and non-dependent products by increasing numbers (see bindings list).

Variant apply term+,: 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.

Variant 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 Existential variables). The instantiation is intended to be found later in the proof.

Variant 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.

Example

Definition id (x : nat) := x.: id is defined
Parameter H : forall y, id y = y.: H is declared
Goal O = O.: 1 subgoal ============================ 0 = 0
Fail simple apply H.: The command has indeed failed with message: Unable to unify "id ?M160 = ?M160" with "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 user-defined tactics that backtrack often. Moreover, it does not traverse tuples as apply does.

Variant simple? apply term with bindings_list?+,¶
Variant simple? eapply term with bindings_list?+,¶: This summarizes the different syntaxes for apply and eapply.

Variant 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:

Variable R : nat -> nat -> Prop.: Toplevel input, characters 0-32: > Variable R : nat -> nat -> Prop. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: R is declared as a local axiom [local-declaration,scope] R is declared
Hypothesis Rtrans : forall x y z:nat, R x y -> R y z -> R x z.: Toplevel input, characters 0-62: > Hypothesis Rtrans : forall x y z:nat, R x y -> R y z -> R x z. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Rtrans is declared as a local axiom [local-declaration,scope] Rtrans is declared
Variables n m p : nat.: Toplevel input, characters 0-22: > Variables n m p : nat. > ^^^^^^^^^^^^^^^^^^^^^^ Warning: n is declared as a local axiom [local-declaration,scope] n is declared Toplevel input, characters 0-22: > Variables n m p : nat. > ^^^^^^^^^^^^^^^^^^^^^^ Warning: m is declared as a local axiom [local-declaration,scope] m is declared Toplevel input, characters 0-22: > Variables n m p : nat. > ^^^^^^^^^^^^^^^^^^^^^^ Warning: p is declared as a local axiom [local-declaration,scope] p is declared
Hypothesis Rnm : R n m.: Toplevel input, characters 0-23: > Hypothesis Rnm : R n m. > ^^^^^^^^^^^^^^^^^^^^^^^ Warning: Rnm is declared as a local axiom [local-declaration,scope] Rnm is declared
Hypothesis Rmp : R m p.: Toplevel input, characters 0-23: > Hypothesis Rmp : R m p. > ^^^^^^^^^^^^^^^^^^^^^^^ Warning: Rmp is declared as a local axiom [local-declaration,scope] Rmp is declared

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

Goal R n p.: 1 subgoal ============================ R n p

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

Fail apply Rtrans.: The command has indeed failed with message: Unable to find an instance for the variable y.

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

apply (Rtrans n m p).: 2 subgoals ============================ R n m subgoal 2 is: R m p

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

apply (Rtrans _ m).: 2 subgoals ============================ R n m subgoal 2 is: R m p

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

apply Rtrans with (y := m).: 2 subgoals ============================ R n m subgoal 2 is: R m p

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

apply Rtrans with (1 := Rnm).: 1 subgoal ============================ R m p

... or the proof of (R y z).

apply Rtrans with (2 := Rmp).: 1 subgoal ============================ R n m

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.

eapply Rtrans.: 2 focused subgoals (shelved: 1) ============================ R n ?y subgoal 2 is: R ?y p
apply Rnm.: 1 subgoal ============================ R m p
apply Rmp.: No more subgoals.

注釈

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:

Flag 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 apply ... in).

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 <-> B 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 Statement without assumptions.¶: This happens if the type of term has no non-dependent premise.

Error Unable to apply.¶: This happens if the conclusion of ident does not match any of the non-dependent premises of the type of term.

Variant apply term+, in ident: This applies each term in sequence in ident.

Variant apply term with bindings_list+, in ident: This does the same but uses the bindings in each (ident := term) to instantiate the parameters of the corresponding type of term (see bindings list).

Variant eapply term with bindings_list?+, in ident: This works as apply ... in but turns unresolved bindings into existential variables, if any, instead of failing.

Variant apply term with bindings_list?+, in ident as intro_pattern¶: This works as apply ... in then applies the intro_pattern to the hypothesis ident.

Variant simple apply term in ident: This behaves like apply ... in 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 ?x and O where ?x is an existential variable to instantiate. Tactic simple apply term in ident does not either traverse tuples as apply term in ident does.

Variant simple? apply term with bindings_list?+, in ident as intro_pattern?
Variant simple? eapply 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.

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 c_i be the i-th constructor of I, then constructor i is equivalent to intros; apply c_i.

Error Not an inductive product.¶

Error Not enough constructors.¶

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

Variant constructor num with bindings_list: Let c be the i-th constructor of I, then constructor i with bindings_list is equivalent to intros; apply c with bindings_list.

警告

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).

Variant split with bindings_list?¶

This applies only if I has a single constructor. It is then equivalent to constructor 1 with bindings_list?. It is typically used in the case of a conjunction $A \wedge B$.

Variant 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 $\exists x, P(x).$

Variant exists bindings_list+,: This iteratively applies exists bindings_list.

Error Not an inductive goal with 1 constructor.¶

Variant left with bindings_list?¶

Variant right with bindings_list?¶

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

Error Not an inductive goal with 2 constructors.¶

Variant econstructor¶
Variant eexists¶
Variant esplit¶
Variant eleft¶
Variant 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 apply).

Managing the local context¶

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 Typing rules 1. If the goal starts with a let-binder, then the tactic implements a mix of the "Let" and "Conv".

If the current goal is a dependent product forall x:T, U (resp let x:=t in U) then intro puts x:T (resp x:=t) in the local context. The new subgoal is U.

If the goal is a non-dependent product $T \rightarrow U$, 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 No product even after head-reduction.¶

Variant intro ident

This applies intro but forces ident to be the name of the introduced hypothesis.

Error ident is already used.¶

注釈

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 Qualified names).

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

Variant intros ident+.: This is equivalent to the composed tactic intro ident; ... ; intro ident.

Variant intros until ident

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

Error No such hypothesis in current goal.¶

Variant intros until num

This repeats intro until the num-th non-dependent product.

Example

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.

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 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.

Variant intro ident₁? after ident₂

Variant intro ident₁? before ident₂

Variant intro ident₁? at top

Variant intro ident₁? at bottom

These tactics apply intro ident₁? and move the freshly introduced hypothesis respectively after the hypothesis ident₂, before the hypothesis ident₂, 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. It is equivalent to intro ident₁? followed by the appropriate call to move ... after ..., move ... before ..., move ... at top, or move ... at bottom.

注釈

intro at bottom is a synonym for intro with no argument.

Error No such hypothesis: ident.¶

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_list | ... | intro_pattern_list]
  - a conjunction of patterns: (p+,)
  - a list of patterns (p+&) for sequence of right-associative binary constructs
- an equality introduction pattern, i.e. either one of:
  - a pattern for decomposing an equality: [= p+]
  - the rewriting orientations: -> or <-
- the on-the-fly application of lemmas: p%term+ 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 $\forall$x:T, P (dependent product), the behavior of intros p is defined inductively over the structure of the introduction pattern p:

Introduction on ? performs the introduction, and lets Coq choose a fresh name for the variable;

Introduction on ?ident performs the introduction, and lets Coq choose a fresh name for the variable based on ident;

Introduction on ident behaves as described in intro

Introduction over a disjunction of list of patterns [intro_pattern_list | ... | intro_pattern_list ] 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 destruct) would and applies on each generated subgoal the corresponding tactic;

The introduction patterns in intro_pattern_list 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 (p+,) expects the goal to be a product over an inductive type I with a single constructor that itself has at least n arguments: It performs a case analysis over the hypothesis, as destruct would, and applies the patterns p+ to the arguments of the constructor of I (observe that (p+) is an alternative notation for [p+]);

Introduction via (p+&) is a shortcut for introduction via (p,( ... ,( ..., p ) ... )); it expects the hypothesis to be a sequence of right-associative binary inductive constructors such as conj or ex_intro; for instance, a 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 [= p+] applies either injection or discriminate instead of destruct; if injection is applicable, the patterns p+, 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 over <-) 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%term+ first applies term+ on the hypothesis to be introduced (as in apply term+,) 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) 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

Goal forall A B C:Prop, A \/ B /\ C -> (A -> C) -> C.: 1 subgoal ============================ forall A B C : Prop, A \/ B /\ C -> (A -> C) -> C
intros * [a | (_,c)] f.: 2 subgoals A, B, C : Prop a : A f : A -> C ============================ C subgoal 2 is: C

注釈

intros p+ is not equivalent to intros p; ... ; intros p for the following reason: If one of the p is a wildcard pattern, it might succeed in the first case because the further hypotheses it depends on 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).

注釈

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

Flag Bracketing Last Introduction Pattern¶: Force completion, if needed, when the last introduction pattern is a disjunctive or conjunctive pattern (on by default).

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 No such hypothesis.¶

Error ident is used in the conclusion.¶

Error ident is used in the hypothesis ident.¶

Variant clear ident+: This is equivalent to clear ident. ... clear ident.

Variant clear - ident+: This variant clears all the hypotheses except the ones depending in the hypotheses named ident+ and in the goal.

Variant clear: This variants clears all the hypotheses except the ones the goal depends on.

Variant clear dependent ident: This clears the hypothesis ident and all the hypotheses that depend on it.

Variant clearbody ident+¶

This tactic expects ident+ to be local definitions and clears their respective bodies. In other words, it turns the given definitions into assumptions.

Error ident is not a local definition.¶

revert ident+¶

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

Error No such hypothesis.

Error ident₁ is used in the hypothesis ident₂.

Variant revert dependent ident¶: This moves to the goal the hypothesis ident and all the hypotheses that depend on it.

move ident₁ after ident₂¶

This moves the hypothesis named ident₁ in the local context after the hypothesis named ident₂, where “after” is in reference to the direction of the move. 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, and all of them are thus moved after ident₂ in the order of dependencies.

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

Variant move ident₁ before ident₂¶: This moves ident₁ towards and just before the hypothesis named ident₂. As for move ... after ..., dependencies over ident₁ (when ident₁ comes before ident₂ in the order of dependencies) or in the type of ident₁ (when ident₁ comes after ident₂ in the order of dependencies) are moved too.

Variant move ident at top¶: This moves ident at the top of the local context (at the beginning of the context).

Variant move ident at bottom¶: This moves ident at the bottom of the local context (at the end of the context).

Error No such hypothesis.

Error Cannot move ident₁ after ident₂: it occurs in the type of ident₂.¶

Error Cannot move ident₁ after ident₂: it depends on ident₂.¶

Example

Goal forall x :nat, x = 0 -> forall z y:nat, y=y-> 0=x.: 1 subgoal ============================ forall x : nat, x = 0 -> nat -> forall y : nat, y = y -> 0 = x
intros x H z y H0.: 1 subgoal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
move x after H0.: 1 subgoal z, y : nat H0 : y = y x : nat H : x = 0 ============================ 0 = x
Undo.: 1 subgoal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
move x before H0.: 1 subgoal z, y, x : nat H : x = 0 H0 : y = y ============================ 0 = x
Undo.: 1 subgoal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
move H0 after H.: 1 subgoal x, y : nat H0 : y = y H : x = 0 z : nat ============================ 0 = x
Undo.: 1 subgoal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
move H0 before H.: 1 subgoal x : nat H : x = 0 y : nat H0 : y = y z : nat ============================ 0 = x

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.

Variant rename ident_i into ident_j+,: This renames the variables ident_i into ident_j 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 No such hypothesis.

Error ident is already used.

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 The variable ident is already defined.¶

Variant 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 goal occurences.

Variant set (ident binders := term) in goal_occurrences?: This is equivalent to set (ident := fun binders => term) in goal_occurrences?.

Variant set term in goal_occurrences?: This behaves as set (ident := term) in goal_occurrences? but ident is generated by Coq.

Variant eset (ident binders? := term) in goal_occurrences?¶
Variant eset term in goal_occurrences?¶: While the different variants of set expect that no existential variables are generated by the tactic, eset removes this constraint. In practice, this is relevant only when eset is used as a synonym of epose, i.e. when the term does not occur in the goal.

remember term as ident₁ eqn:ident₂?¶

This behaves as set (ident₁ := term) in *, using a logical (Leibniz’s) equality instead of a local definition. If ident₂ is provided, it will be the name of the new equation.

Variant remember term as ident₁ eqn:ident₂? in goal_occurrences: This is a more general form of remember that remembers the occurrences of term specified by an occurrence set.

Variant eremember term as ident₁ eqn:ident₂? in goal_occurrences?¶: While the different variants of remember expect that no existential variables are generated by the tactic, eremember removes this constraint.

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 |-.

Variant pose (ident binders := term): This is equivalent to pose (ident := fun binders => term).

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

Variant epose (ident binders? := term)

Variant epose term¶: While the different variants of pose expect that no existential variables are generated by the tactic, epose removes this constraint.

decompose [qualid+] term¶

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

Example

Goal forall A B C:Prop, A /\ B /\ C \/ B /\ C \/ C /\ A -> C.: 1 subgoal ============================ forall A B C : Prop, A /\ B /\ C \/ B /\ C \/ C /\ A -> C
intros A B C H; decompose [and or] H.: 3 subgoals A, B, C : Prop H : A /\ B /\ C \/ B /\ C \/ C /\ A H1 : A H0 : B H3 : C ============================ C subgoal 2 is: C subgoal 3 is: C
all: assumption.: No more subgoals.
Qed.: Unnamed_thm is defined

注釈

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

Variant decompose sum term: This decomposes sum types (like or).

Variant decompose record term: This decomposes record types (inductive types with one constructor, like and and exists and those defined with the Record command.

Controlling the proof flow¶

assert (ident : form)¶: This tactic applies to any goal. assert (H : U) adds a new hypothesis of name H asserting U to the current goal and opens a new subgoal U 2. The subgoal U comes first in the list of subgoals remaining to prove.

Error Not a proposition or a type.¶: Arises when the argument form is neither of type Prop, Set nor Type.

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

Variant assert form by tactic

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

Error Proof is not complete.¶

Variant assert form as intro_pattern: If intro_pattern is a naming introduction pattern (see intro), 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.

Variant assert form as intro_pattern by tactic: This combines the two previous variants of assert.

Variant assert (ident := term )

This behaves as assert (ident : type) by exact term where type is the type of term. This is deprecated in favor of pose proof. If the head of term is ident, the tactic behaves as specialize term.

Error Variable ident is already declared.¶

Variant eassert form as intro_pattern by tactic¶

Variant assert (ident := term): While the different variants of assert expect that no existential variables are generated by the tactic, eassert removes this constraint. This allows not to specify the asserted statement completeley before starting to prove it.

Variant 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.

Variant epose proof term as intro_pattern?: While pose proof expects that no existential variables are generated by the tactic, epose proof removes this constraint.

Variant 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.

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

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

Variant enough (ident : form) by tactic

Variant enough form by tactic

Variant 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.

Variant eenough (ident : form) by tactic¶

Variant eenough form by tactic

Variant eenough form as intro_pattern by tactic: While the different variants of enough expect that no existential variables are generated by the tactic, eenough removes this constraint.

Variant cut form¶: This tactic applies to any goal. It implements the non-dependent case of the “App” rule given in Typing rules. (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.

Variant specialize (ident term*) as intro_pattern?

Variant specialize ident with bindings_list as intro_pattern?¶

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 term* or from a bindings list. In the first form the application to term* can be partial. The first form is equivalent to assert (ident := ident term*). In the second form, instantiation elements can also be partial. In this case the uninstantiated arguments are inferred by unification if possible or left quantified in the hypothesis otherwise. With the as clause, the local hypothesis ident is left unchanged and instead, the modified hypothesis is introduced as specified by the intro_pattern. 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. The as clause is especially useful in this case to immediately introduce the instantiated statement as a local hypothesis.

Error ident is used in hypothesis ident.¶

Error ident is used in conclusion.¶

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

Example

Goal forall x y:nat, 0 <= x + y + y.: 1 subgoal ============================ forall x y : nat, 0 <= x + y + y
Proof.
intros *.: 1 subgoal x, y : nat ============================ 0 <= x + y + y

Show.: 1 subgoal x, y : nat ============================ 0 <= x + y + y
generalize (x + y + y).: 1 subgoal x, y : nat ============================ forall n : nat, 0 <= n

If the goal is G and t is a subterm of type T in the goal, then generalize t replaces the goal by forall (x:T), G′ where G′ is obtained from G by replacing all occurrences of t by x. The name of the variable (here n) is chosen based on T.

Variant generalize term+: This is equivalent to generalize term; ... ; generalize term. Note that the sequence of term _i 's are processed from n to 1.

Variant generalize term at num+: 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 Printing All).

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

Variant generalize term at num+ as ident+,: This is the most general form of generalize that combines the previous behaviors.

Variant generalize dependent term: This generalizes term but also all hypotheses that depend on term. It clears the generalized hypotheses.

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.

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

注釈

To be able to refer to an existential variable by name, the user must have given the name explicitly (see Existential variables).

注釈

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.

Variant 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.

Variant instantiate ( num := term ) in ident

Variant instantiate ( num := term ) in ( Value of ident )

Variant 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.

Variant 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.

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.

Variant give_up: Synonym of admit.

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.

contradiction¶: This tactic applies to any goal. The contradiction tactic attempts to find in the current context (after all intros) a 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 No such assumption.

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

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

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.

Case analysis and induction¶

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

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.

Variant destruct ident

If ident denotes 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 ident is a hypothesis 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)).

Variant destruct num: destruct num behaves as intros until num followed by destruct applied to the last introduced hypothesis.

注釈

For destruction of a numeral, use syntax destruct (num) (not very interesting anyway).

Variant destruct pattern: The argument of destruct 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.

Variant destruct term+,: This is a shortcut for destruct term; ...; destruct term.

Variant destruct term as disj_conj_intro_pattern: This behaves as destruct term but uses the names in disj_conj_intro_pattern to name the variables introduced in the context. The disj_conj_intro_pattern must have the form [p11 ... p1n | ... | pm1 ... pmn ] 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 pi1 ... pin in order. If there are not enough names, destruct invents names for the remaining variables to introduce. More generally, the pij can be any introduction pattern (see intros). This provides a concise notation for chaining destruction of a hypothesis.

Variant destruct term eqn:naming_intro_pattern¶: This behaves as destruct term but adds an equation between term and the value that it takes in each of the possible cases. The name of the equation is specified by naming_intro_pattern (see intros), in particular ? can be used to let Coq generate a fresh name.

Variant destruct term with bindings_list: This behaves like destruct term providing explicit instances for the dependent premises of the type of term.

Variant 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.

Variant destruct term using term with bindings_list?: This is synonym of induction term using term with bindings_list?.

Variant 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 occurences sets.

Variant destruct term with bindings_list? as disj_conj_intro_pattern? eqn:naming_intro_pattern? using term with bindings_list?? in goal_occurrences?
Variant edestruct term with bindings_list? as disj_conj_intro_pattern? eqn:naming_intro_pattern? using term with bindings_list?? 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.

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.

Variant case term with bindings_list: Analogous to elim term with bindings_list above.

Variant 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.

Variant simple destruct ident¶: This tactic behaves as intros until ident; case ident when ident is a quantified variable of the goal.

Variant 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.

Variant case_eq term: The tactic case_eq is a variant of the case tactic that allows 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.

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 inductionident 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.

注釈

For simple induction on a numeral, use syntax induction (num) (not very interesting anyway).
In case term is a 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

Lemma induction_test : forall n:nat, n = n -> n <= n.: 1 subgoal ============================ forall n : nat, n = n -> n <= n
intros n H.: 1 subgoal n : nat H : n = n ============================ n <= n
induction n.: 2 subgoals H : 0 = 0 ============================ 0 <= 0 subgoal 2 is: S n <= S n

Error Not an inductive product.

Error Unable to find an instance for the variables ident ... ident.: Use in this case the variant elim ... with below.

Variant induction term as disj_conj_intro_pattern: This behaves as induction but uses the names in disj_conj_intro_pattern to name the variables introduced in the context. The disj_conj_intro_pattern must typically be of the form [ p ₁₁ ... p _1n | ... | p_m1 ... p_mn ] 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_in 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 intros ...). For instance, for an inductive type with one constructor, the pattern notation (p₁ , ... , p_n ) can be used instead of [ p₁ ... p_n ].

Variant induction term with bindings_list: This behaves like induction providing explicit instances for the premises of the type of term (see bindings list).

Variant einduction term¶: This tactic behaves like induction except 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.

Variant induction term using term¶: This behaves as induction but using term as induction scheme. It does not expect the conclusion of the type of the first term to be inductive.

Variant induction term using term with bindings_list: This behaves as induction ... using ... but also providing instances for the premises of the type of the second term.

Variant induction term+, 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.

Variant 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 occurences sets. 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

Lemma comm x y : x + y = y + x.: 1 subgoal x, y : nat ============================ x + y = y + x
induction y in x |- *.: 2 subgoals x : nat ============================ x + 0 = 0 + x subgoal 2 is: x + S y = S y + x
Show 2.: subgoal 2 is: x, y : nat IHy : forall x : nat, x + y = y + x ============================ x + S y = S y + x

Variant induction term with bindings_list as disj_conj_intro_pattern using term with bindings_list in goal_occurrences

Variant einduction term with bindings_list as disj_conj_intro_pattern using term with bindings_list 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.

Variant 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.

Variant elim term with bindings_list¶: Allows to give explicit instances to the premises of the type of term (see bindings list).

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

Variant elim term using term

Variant elim term using term with bindings_list: Allows the user to give explicitly an induction principle term that is not the standard one for the underlying inductive type of term. The bindings_list clause allows instantiating premises of the type of term.

Variant elim term with bindings_list using term with bindings_list

Variant eelim term with bindings_list using term with bindings_list: 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.

Variant 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.

Variant simple induction ident¶: This tactic behaves as intros until ident; elim ident when ident is a quantified variable of the goal.

Variant 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.

double induction ident ident¶: This tactic is deprecated and should be replaced by induction ident; induction ident (or induction ident ; destruct ident depending on the exact needs).

Variant 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

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 [McB00] and the work of Cristina Cornes around inversion [CT95]. 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

Lemma le_minus : forall n:nat, n < 1 -> n = 0.: 1 subgoal ============================ forall n : nat, n < 1 -> n = 0
intros n H ; induction H.: 2 subgoals n : nat ============================ n = 0 subgoal 2 is: n = 0

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.

Require Import Coq.Program.Equality.
Lemma le_minus : forall n:nat, n < 1 -> n = 0.: 1 subgoal ============================ forall n : nat, n < 1 -> n = 0
intros n H ; dependent induction H.: 2 subgoals ============================ 0 = 0 subgoal 2 is: n = 0

The subgoal is cleaned up as the tactic tries to automatically simplify the subgoals with respect to the generated equalities. In this enriched context, it becomes possible to solve this subgoal.

reflexivity.: 1 subgoal n : nat H : S n <= 0 IHle : 0 = 1 -> n = 0 ============================ n = 0

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

inversion H.: No more subgoals.

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.

Variant dependent induction ident generalizing ident+: 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.

Variant 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 the larger example of dependent induction and an explanation of the underlying technique.

function induction (qualid term+)¶: 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 高度な再帰関数) or Functional Scheme (see Generation of induction principles with Functional Scheme). Note that this tactic is only available after a Require Import FunInd.

Example

Require Import FunInd.: [Loading ML file extraction_plugin.cmxs ... done] [Loading ML file recdef_plugin.cmxs ... done]
Functional Scheme minus_ind := Induction for minus Sort Prop.: sub_equation is defined minus_ind is defined
Check minus_ind.: minus_ind : forall P : nat -> nat -> nat -> Prop, (forall n m : nat, n = 0 -> P 0 m n) -> (forall n m k : nat, n = S k -> m = 0 -> P (S k) 0 n) -> (forall n m k : nat, n = S k -> forall l : nat, m = S l -> P k l (k - l) -> P (S k) (S l) (k - l)) -> forall n m : nat, P n m (n - m)
Lemma le_minus (n m:nat) : n - m <= n.: 1 subgoal n, m : nat ============================ n - m <= n
functional induction (minus n m) using minus_ind; simpl; auto.: No more subgoals.
Qed.: le_minus is defined

注釈

(qualid term+) 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.

注釈

Parentheses around qualid term+ are not mandatory and can be skipped.

注釈

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 高度な再帰関数) or Functional Scheme (see Generation of induction principles with Functional Scheme) command) corresponding to the sort of the goal. Therefore functional induction may fail if the induction scheme qualid is not defined. See also 高度な再帰関数 for the function terms accepted by Function.

注釈

There is a difference between obtaining an induction scheme for a function by using Function (see 高度な再帰関数) and by using Functional Scheme after a normal definition using Fixpoint or Definition. See 高度な再帰関数 for details.

参考

高度な再帰関数, Generation of induction principles with Functional Scheme and inversion

Error Cannot find induction information on qualid.¶

Error Not the right number of induction arguments.¶

Variant functional induction (qualid term+) as disj_conj_intro_pattern using term with bindings_list: Similarly to induction and elim, 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.

discriminate term¶

This tactic proves any goal from an assumption stating that two structurally different terms of an inductive set are equal. For example, from (S (S O))=(S O) we can derive by absurdity any proposition.

The argument term is assumed to be a proof of a statement of conclusion term = term with the two terms being elements of an inductive set. To build the proof, the tactic traverses the normal forms 3 of the terms looking for a couple of subterms u and w (u subterm of the normal form of term and w subterm of the normal form of term), 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.

注釈

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 No primitive equality found.¶

Error Not a discriminable equality.¶

Variant 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.

Variant 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.

Variant ediscriminate num

Variant 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.

Variant discriminate

This behaves like discriminate ident if ident is the name of an hypothesis to which discriminate is applicable; if the current goal is of the form term <> term, this behaves as intro ident; discriminate ident.

Error No discriminable equalities.¶

injection term¶

The injection tactic exploits the property that constructors of inductive types are injective, i.e. that if c is a constructor of an inductive type and c t₁ and c t₂ are equal then t₁ and t₂ are equal too.

If term is a proof of a statement of conclusion term = term, then injection applies the injectivity of constructors as deep as possible to derive the equality of all the subterms of term and term at positions where the terms start to differ. For example, from (S p, S n) = (q, S (S m)) we may derive S p = q and n = S m. For this tactic to work, the terms should be typed with an inductive type and they should be neither convertible, nor having a different head constructor. If these conditions are satisfied, the tactic derives the equality of all the subterms at positions where they differ and adds them as antecedents to the conclusion of the current goal.

Example

Consider the following goal:

Inductive list : Set := | nil : list | cons : nat -> list -> list.: list is defined list_rect is defined list_ind is defined list_rec is defined
Parameter P : list -> Prop.: P is declared
Goal forall l n, P nil -> cons n l = cons 0 nil -> P l.: 1 subgoal ============================ forall (l : list) (n : nat), P nil -> cons n l = cons 0 nil -> P l

intros.: 1 subgoal l : list n : nat H : P nil H0 : cons n l = cons 0 nil ============================ P l
injection H0.: 1 subgoal l : list n : nat H : P nil H0 : cons n l = cons 0 nil ============================ l = nil -> n = 0 -> P l

Beware that injection yields 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 Generation of induction principles with Scheme), the use of a sigma type is avoided.

注釈

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 Not a projectable equality but a discriminable one.¶

Error Nothing to do, it is an equality between convertible terms.¶

Error Not a primitive equality.¶

Error Nothing to inject.¶

Variant 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.

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

Variant einjection num¶

Variant 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.

Variant injection

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

Error goal does not satisfy the expected preconditions.¶

Variant injection term with bindings_list? as intro_pattern+

Variant injection num as intro_pattern+

Variant injection as intro_pattern+

Variant einjection term with bindings_list? as intro_pattern+

Variant einjection num as intro_pattern+

Variant einjection as intro_pattern+

These variants apply intros 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 a hypothesis.

Flag Structural Injection¶: This option 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). This option is off by default.

Flag Keep Proof Equalities¶: 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 option controls this behavior.

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.

注釈

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.

注釈

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 Generation of inversion principles with Derive Inversion.

注釈

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 :flag`Keep Proof Equalities`.

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

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

Variant 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_1n | ... | p_m1 ... p_mn ] 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 = 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_in 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. これが例です。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.

Example

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
Goal forall l:list nat, contains0 (1 :: l) -> contains0 l.: 1 subgoal ============================ forall l : list nat, contains0 (1 :: l) -> contains0 l
intros l H; inversion H as [ | l' p Hl' [Heqp Heql'] ].: 1 subgoal l : list nat H : contains0 (1 :: l) l' : list nat p : nat Hl' : contains0 l Heqp : p = 1 Heql' : l' = l ============================ contains0 l

Variant inversion num as intro_pattern: This allows naming the hypotheses introduced by inversion num in the context.

Variant 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.

Variant inversion ident in ident+: Let ident+ be identifiers in the local context. This tactic behaves as generalizing ident+, and then performing inversion.

Variant inversion ident as intro_pattern in ident+: This allows naming the hypotheses introduced in the context by inversion ident in ident+.

Variant inversion_clear ident in ident+: Let ident+ be identifiers in the local context. This tactic behaves as generalizing ident+, and then performing inversion_clear.

Variant inversion_clear ident as intro_pattern in ident+: This allows naming the hypotheses introduced in the context by inversion_clear ident in ident+.

Variant 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.

Variant dependent inversion ident as intro_pattern: This allows naming the hypotheses introduced in the context by dependent inversion ident.

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

Variant dependent inversion_clear ident as intro_pattern: This allows naming the hypotheses introduced in the context by dependent inversion_clear ident.

Variant dependent inversion ident with term¶: This variant allows you to specify the generalization of the goal. It is useful when the system fails to generalize the goal automatically. If ident has type (I t) and I has type $\forall$ (x:T), s, then term must be of type I:$\forall$ (x:T), I x -> s' where s' is the type of the goal.

Variant dependent inversion ident as intro_pattern with term: This allows naming the hypotheses introduced in the context by dependent inversion ident with term.

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

Variant 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.

Variant 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.

Variant simple inversion ident as intro_pattern: This allows naming the hypotheses introduced in the context by simple inversion.

Variant 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.

Variant inversion ident using ident in ident+: This tactic behaves as generalizing ident+, then doing inversion ident using ident.

Variant inversion_sigma: This tactic turns equalities of dependent pairs (e.g., existT P x p = existT P y q, frequently left over by inversion on a dependent type family) into pairs of equalities (e.g., a hypothesis H : x = y and a hypothesis of type rew H in p = q); these hypotheses can subsequently be simplified using subst, without ever invoking any kind of axiom asserting uniqueness of identity proofs. If you want to explicitly specify the hypothesis to be inverted, or name the generated hypotheses, you can invoke induction H as [H1 H2] using eq_sigT_rect. This tactic also works for sig, sigT2, and sig2, and there are similar eq_sig ***_rect induction lemmas.

Example

Non-dependent inversion.

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

Inductive Le : nat -> nat -> Set := | LeO : forall n:nat, Le 0 n | LeS : forall n m:nat, Le n m -> Le (S n) (S m).: Le is defined Le_rect is defined Le_ind is defined Le_rec is defined
Variable P : nat -> nat -> Prop.: Toplevel input, characters 0-32: > Variable P : nat -> nat -> Prop. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: P is declared as a local axiom [local-declaration,scope] P is declared
Variable Q : forall n m:nat, Le n m -> Prop.: Toplevel input, characters 0-44: > Variable Q : forall n m:nat, Le n m -> Prop. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Q is declared as a local axiom [local-declaration,scope] Q is declared

Let us consider the following goal:

Goal forall n m, Le (S n) m -> P n m.: Le is defined Le_rect is defined Le_ind is defined Le_rec is defined Toplevel input, characters -78--46: Warning: P is declared as a local axiom [local-declaration,scope] P is declared Toplevel input, characters -45--1: Warning: Q is declared as a local axiom [local-declaration,scope] Q is declared 1 subgoal ============================ forall n m : nat, Le (S n) m -> P n m
intros.: 1 subgoal n, m : nat H : Le (S n) m ============================ P n m

Show.: 1 subgoal n, m : nat H : Le (S n) m ============================ P n m

To prove the goal, we may need to reason by cases on H and to derive that m is necessarily of the form (S m 0 ) for certain m 0 and that (Le n m 0 ). Deriving these conditions corresponds to proving that the only possible constructor of (Le (S n) m) isLeS 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.

inversion_clear H.: 1 subgoal n, m, m0 : nat H0 : Le n m0 ============================ P n (S m0)

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:

inversion H.: 1 subgoal n, m : nat H : Le (S n) m n0, m0 : nat H1 : Le n m0 H0 : n0 = n H2 : S m0 = m ============================ P n (S m0)

Example

Dependent inversion.

Let us consider the following goal:

Inductive Le : nat -> nat -> Set := | LeO : forall n:nat, Le 0 n | LeS : forall n m:nat, Le n m -> Le (S n) (S m).: Le is defined Le_rect is defined Le_ind is defined Le_rec is defined
Variable P : nat -> nat -> Prop.: Toplevel input, characters 0-32: > Variable P : nat -> nat -> Prop. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: P is declared as a local axiom [local-declaration,scope] P is declared
Variable Q : forall n m:nat, Le n m -> Prop.: Toplevel input, characters 0-44: > Variable Q : forall n m:nat, Le n m -> Prop. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Q is declared as a local axiom [local-declaration,scope] Q is declared
Goal forall n m (H:Le (S n) m), Q (S n) m H.: 1 subgoal ============================ forall (n m : nat) (H : Le (S n) m), Q (S n) m H
intros.: 1 subgoal n, m : nat H : Le (S n) m ============================ Q (S n) m H

Show.: 1 subgoal n, m : nat H : Le (S n) m ============================ Q (S n) m H

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 do such a substitution. To have such a behavior we use the dependent inversion tactics:

dependent inversion_clear H.: 1 subgoal n, m, m0 : nat l : Le n m0 ============================ Q (S n) (S m0) (LeS n m0 l)

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

Example

Using inversion_sigma.

Let us consider the following inductive type of length-indexed lists, and a lemma about inverting equality of cons:

Require Import Coq.Logic.Eqdep_dec.
Inductive vec A : nat -> Type := | nil : vec A O | cons {n} (x : A) (xs : vec A n) : vec A (S n).: vec is defined vec_rect is defined vec_ind is defined vec_rec is defined
Lemma invert_cons : forall A n x xs y ys, @cons A n x xs = @cons A n y ys -> xs = ys.: 1 subgoal ============================ forall (A : Type) (n : nat) (x : A) (xs : vec A n) (y : A) (ys : vec A n), cons A x xs = cons A y ys -> xs = ys
Proof.
intros A n x xs y ys H.: 1 subgoal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys ============================ xs = ys

After performing inversion, we are left with an equality of existTs:

inversion H.: 1 subgoal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H2 : existT (fun n : nat => vec A n) n xs = existT (fun n : nat => vec A n) n ys ============================ xs = ys

We can turn this equality into a usable form with inversion_sigma:

inversion_sigma.: 1 subgoal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H0 : n = n H3 : eq_rect n (fun a : nat => vec A a) xs n H0 = ys ============================ xs = ys

To finish cleaning up the proof, we will need to use the fact that that all proofs of n = n for n a nat are eq_refl:

let H := match goal with H : n = n |- _ => H end in pose proof (Eqdep_dec.UIP_refl_nat _ H); subst H.: 1 subgoal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H3 : eq_rect n (fun a : nat => vec A a) xs n eq_refl = ys ============================ xs = ys
simpl in *.: 1 subgoal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H3 : xs = ys ============================ xs = ys

Finally, we can finish the proof:

assumption.: No more subgoals.
Qed.: invert_cons is defined

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 induction.

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, counting both dependent and non dependent products, but skipping local definitions. 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 Requesting information).

Variant fix ident num with (ident binder+ [{struct ident}] : type)+: This starts a proof by mutual induction. The statements to be simultaneously proved are respectively forall binder ... binder, type. The identifiers ident are the names of the induction hypotheses. The identifiers ident 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).

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 Requesting information).

Variant cofix ident with (ident binder+ : type)+: This starts a proof by mutual coinduction. The statements to be simultaneously proved are respectively forall binder ... binder, type The identifiers ident are the names of the coinduction hypotheses.

Rewriting expressions¶

These tactics use the equality eq:forall A:Type, A->A->Prop defined in file Logic.v (see Logic). The notation for eq T t u is simply t=u dropping the implicit type of t and u.

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 The term provided does not end with an equation.¶

Error Tactic generated a subgoal identical to the original goal. This happens if term does not occur in the goal.¶

Variant rewrite -> term: Is equivalent to rewrite term

Variant rewrite <- term: Uses the equality term₁ = term ₂ from right to left

Variant rewrite term in clause

Analogous to rewrite term but rewriting is done following clause (similarly to performing computations). 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 hypotheses H_i different from 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.

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

Variant rewrite term by tactic: Use tactic to completely solve the side-conditions arising from the rewrite.

Variant rewrite term+,: Is equivalent to the n successive tactics rewrite term+;, each one working on the first subgoal generated by the previous one. Orientation -> or <- can be inserted before each term to rewrite. One unique clause can be added at the end after the keyword in; it will then affect all rewrite operations.

In all forms of rewrite described above, a term to rewrite can be immediately prefixed by one of the following modifiers:

? : the tactic rewrite ?term performs the rewrite of term as many times as possible (perhaps zero time). This form never fails.
num? : works similarly, except that it will do at most num rewrites.
! : works as ?, except that at least one rewrite should succeed, otherwise the tactic fails.
num! (or simply num) : precisely num rewrites of term will be done, leading to failure if these num rewrites are not possible.

Variant 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.

replace term with term’¶

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

Error Terms do not have convertible types.¶

Variant replace term with term’ by tactic: This acts as replace term with term’ but applies tactic to solve the generated subgoal term = term’.

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

Variant replace -> term: Replaces term with term’ using the first assumption whose type has the form term = term’

Variant replace <- term: Replaces term with term’ using the first assumption whose type has the form term’ = term

Variant replace term with term? in clause by tactic?

Variant replace -> term in clause

Variant replace <- term in clause: Acts as before but the replacements take place in the specified clause (see Performing computations) and not only in the conclusion of the goal. The clause argument must not contain any type of nor value of.

Variant cutrewrite <- (term = term’)¶: This tactic is deprecated. It can be replaced by enough (term = term’) as <-.

Variant cutrewrite -> (term = term’): This tactic is deprecated. It can be replaced by enough (term = term’) as ->.

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.

注釈

When several hypotheses have the form ident = t or t = ident, the first one is used.
If H is itself dependent in the goal, it is replaced by the proof of reflexivity of equality.

Variant subst ident+: This is equivalent to subst ident₁; ...; subst ident_n.

Variant 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.

Flag Regular Subst Tactic¶

This option controls the behavior of subst. When it is activated (it is by default), subst also deals with the following corner cases:

A context with ordered hypotheses ident₁ = ident₂ and ident₁ = t, or t′ = ident_1` 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. default.

stepl term¶

This tactic is for chaining rewriting steps. It assumes a goal of the form R term term where R is a binary relation and relies on a database of lemmas of the form forall x y z, R x y -> eq x z -> R z y where eq is typically a setoid equality. The application of stepl term then replaces the goal by R term term and adds a new goal stating eq term term.

Command Declare Left Step term¶: Adds term to the database used by stepl.

This tactic is especially useful for parametric setoids which are not accepted as regular setoids for rewrite and setoid_replace (see Generalized rewriting).

Variant stepl term by tactic: This applies stepl term then applies tactic to the second goal.

Variant 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.

Command Declare Right Step term¶: Adds term to the database used by stepr.

change term¶

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

Error Not convertible.¶

Variant change term with term’: This replaces the occurrences of term by term’ in the current goal. The term term and term’ must be convertible.

Variant change term at num+ with term’

This replaces the occurrences numbered num+ of term by term’ in the current goal. The terms term and term’ must be convertible.

Error Too few occurrences.¶

Variant change term at num+? with term? in ident: This applies the change tactic not to the goal but to the hypothesis ident.

参考

Performing computations

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, rewrite, replace and autorewrite also use goal clauses) and are introduced by the keyword in. If no goal clause is provided, the default is to perform the conversion only in the conclusion.

The syntax and description of the various goal clauses is the following:

in ident+ |- only in hypotheses ident+
in ident+ |- * in hypotheses ident+ and in the conclusion
in * |- in every hypothesis
in * (equivalent to in * |- *) everywhere
in (type of ident) (value of ident) ... |- in type part of ident, in the value part of ident, etc.

For backward compatibility, the notation in ident+ performs the conversion in hypotheses ident+.

cbv flag*¶

lazy flag*¶

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 $\beta$ (reduction of functional application), $\delta$ (unfolding of transparent constants, see Controlling the reduction strategies and the conversion algorithm), $\iota$ (reduction of pattern matching over a constructed term, and unfolding of fix and cofix expressions) and $\zeta$ (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 qualid+ or delta -qualid+, 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 Controlling the reduction strategies and the conversion algorithm).

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 blocked by a variable (or an opaque constant, or an axiom), as e.g. in x u1 ... un , or match x with ... end, or (fix f x {struct x} := ...) x, or is a constructed form (a $\lambda$-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 little dead code).

Variant compute

Variant cbv: These are synonyms for cbv beta delta iota zeta.

Variant lazy: This is a synonym for lazy beta delta iota zeta.

Variant compute qualid+

Variant cbv qualid+: These are synonyms of cbv beta delta qualid+ iota zeta.

Variant compute -qualid+

Variant cbv -qualid+: These are synonyms of cbv beta delta -qualid+ iota zeta.

Variant lazy qualid+

Variant lazy -qualid+: These are respectively synonyms of lazy beta delta qualid+ iota zeta and lazy beta delta -qualid+ iota zeta.

Variant vm_compute¶: This tactic evaluates the goal using the optimized call-by-value evaluation bytecode-based virtual machine described in [GregoireL02]. 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.

Variant native_compute¶

This tactic evaluates the goal by compilation to Objective Caml as described in [BDenesGregoire11]. 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.

Flag NativeCompute Profiling¶: On Linux, if you have the perf profiler installed, this option makes it possible to profile native_compute evaluations.

Option NativeCompute Profile Filename string¶: This option specifies the profile output; the default is native_compute_profile.data. The actual filename used will contain extra characters to avoid overwriting an existing file; that filename is reported to the user. That means you can individually profile multiple uses of native_compute in a script. From the Linux command line, run perf report on the profile file to see the results. Consult the perf documentation for more details.

Flag Debug Cbv¶: This option makes cbv (and its derivative compute) print information about the constants it encounters and the unfolding decisions it makes.

red¶

This tactic applies to a goal that has the form:

forall (x:T1) ... (xk:Tk), T

with T $\beta$$\iota$$\zeta$-reducing to c t₁ ... t_n and c a constant. If c is transparent then it replaces c with its definition (say t) and then reduces (t t₁ ... t_n ) according to $\beta$$\iota$$\zeta$-reduction rules.

Error Not reducible.¶

Error No head constant to reduce.¶

hnf¶

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

Example: The term fun n : nat => S n + S n is not reduced by hnf.

注釈

The $\delta$ rule only applies to transparent constants (see Controlling the reduction strategies and the conversion algorithm on transparency and opacity).

cbn¶

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 $\iota$-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:

Example

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 argument list of the Arguments vernacular command.

Example

Definition fcomp A B C f (g : A -> B) (x : A) : C := f (g x).
fcomp is defined

Arguments fcomp {A B C} f g x /.

Notation "f \o g" := (fcomp f g) (at level 50).

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.

Example

Definition volatile := fun x : nat => x.
volatile is defined

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.

Example

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:

Example

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 $\beta$$\iota$-reduction. Then, it expands transparent constants and tries to reduce further using $\beta$$\iota$- reduction. But, when no $\iota$ rule is applied after unfolding then $\delta$-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.

Variant cbn qualid+

Variant cbn -qualid+: These are respectively synonyms of cbn beta delta qualid+ iota zeta and cbn beta delta -qualid+ iota zeta (see cbn).

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

Variant simpl pattern at num+

This applies simpl only to the num+ occurrences of the subterms matching pattern in the current goal.

Error Too few occurrences.

Variant simpl qualid

Variant 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).

Variant simpl qualid at num+

Variant simpl string at num+: This applies simpl only to the num+ applicative subterms whose head occurrence is qualid (or string).

Flag Debug RAKAM¶: This option makes cbn print various debugging information. RAKAM is the Refolding Algebraic Krivine Abstract Machine.

unfold qualid¶: This tactic applies to any goal. The argument qualid must denote a defined transparent constant or local definition (see Definitions and Controlling the reduction strategies and the conversion algorithm). The tactic unfold applies the $\delta$ rule to each occurrence of the constant to which qualid refers in the current goal and then replaces it with its $\beta$$\iota$-normal form.

Error qualid does not denote an evaluable constant.¶

Variant unfold qualid in ident: Replaces qualid in hypothesis ident with its definition and replaces the hypothesis with its $\beta$$\iota$ normal form.

Variant unfold qualid+,: Replaces simultaneously qualid+, with their definitions and replaces the current goal with its $\beta$$\iota$ normal form.

Variant unfold qualid at num+,+,

The lists num+, specify the occurrences of qualid to be unfolded. Occurrences are located from left to right.

Error Bad occurrence number of qualid.¶

Error qualid does not occur.¶

Variant 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.

Variant 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 Local interpretation rules for notations).

Variant unfold qualid_or_string at num+,+,: This is the most general form, where qualid_or_string is either a qualid or a string referring to a notation.

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.

Variant fold term+: Equivalent to fold term ; ... ; fold term.

pattern term¶

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

replacing all occurrences of term in T with a fresh variable
abstracting this variable
applying the abstracted goal to term

For instance, if the current goal T is expressible as $\varphi$(t) where the notation captures all the instances of t in $\varphi$(t), then pattern t transforms it into (fun x:A => $\varphi$(x)) t. This tactic can be used, for instance, when the tactic apply fails on matching.

Variant pattern term at num+: Only the occurrences num+ of term are considered for $\beta$-expansion. Occurrences are located from left to right.

Variant pattern term at - num+: All occurrences except the occurrences of indexes num+ of term are considered for $\beta$-expansion. Occurrences are located from left to right.

Variant pattern term+,: Starting from a goal $\varphi$(t₁ ... t_m), the tactic pattern t₁, ..., t_m generates the equivalent goal (fun (x₁:A₁) ... (x_m :A_m ) =>$\varphi$(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.

Variant pattern term at num++,: This behaves as above but processing only the occurrences num+ of term starting from term.

Variant pattern term at -? num+,?+,: This is the most general syntax that combines the different variants.

Conversion tactics applied to hypotheses¶

conv_tactic in ident+,

Applies the conversion tactic conv_tactic to the hypotheses ident+. The tactic conv_tactic is any of the conversion tactics listed in this section.

If ident is a local definition, then ident can be replaced by (type of ident) to address not the body but the type of the local definition.

Example: unfold not in (type of H1) (type of H3).

Error No such hypothesis: ident.

Automation¶

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.

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

Variant auto with ident+: Uses the hint databases ident+ in addition to the database core. See The Hints Databases for auto and eauto for the list of pre-defined databases and the way to create or extend a database.

Variant auto with *: Uses all existing hint databases. See The Hints Databases for auto and eauto

Variant auto using lemma+: Uses lemma+ in addition to hints (can be combined with the with ident option). If lemma is an inductive type, it is the collection of its constructors which is added as hints.

Variant 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.

Variant debug auto¶: Behaves like auto but shows the tactics it tries to solve the goal, including failing paths.

Variant info_?auto num? using lemma+? with ident+?: This is the most general form, combining the various options.

Variant 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.

Variant trivial with ident+

Variant trivial with *

Variant trivial using lemma+

Variant debug trivial¶

Variant info_trivial¶

Variant info_?trivial using lemma+? with ident+?

注釈

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

The following options enable printing of informative or debug information for the auto and trivial tactics:

Flag Info Auto¶
Flag Debug Auto¶
Flag Info Trivial¶
Flag Debug Trivial¶

参考

The Hints Databases for auto and eauto

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 usessimple eapply where auto uses simple apply). As a consequence, eauto can solve such a goal:

Example

Hint Resolve ex_intro.: The hint ex_intro will only be used by eauto, because applying ex_intro would leave variable x as unresolved existential variable.
Goal forall P:nat -> Prop, P 0 -> exists n, P n.: 1 subgoal ============================ forall P : nat -> Prop, P 0 -> exists n : nat, P n
eauto.: No more subgoals.

Note that ex_intro should be declared as a hint.

Variant info_?eauto num? using lemma+? with ident+?: The various options for eauto are the same as for auto.

eauto also obeys the following options:

Flag Info Eauto¶
Flag Debug Eauto¶

参考

The Hints Databases for auto and eauto

autounfold with ident+¶

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

Variant autounfold with ident+ in clause: Performs the unfolding in the given clause.

Variant autounfold with *: Uses the unfold hints declared in all the hint databases.

autorewrite with ident+¶

This tactic 4 carries out rewritings according to the rewriting rule bases ident+.

Each rewriting rule from the base ident 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.

警告

This tactic may loop if you build non terminating rewriting systems.

Variant autorewrite with ident+ using tactic: Performs, in the same way, all the rewritings of the bases ident+ applying tactic to the main subgoal after each rewriting step.

Variant autorewrite with ident+ in qualid: Performs all the rewritings in hypothesis qualid.

Variant autorewrite with ident+ in qualid using tactic: Performs all the rewritings in hypothesis qualid applying tactic to the main subgoal after each rewriting step.

Variant autorewrite with ident+ in clause: Performs all the rewriting in the clause clause. The clause argument must not contain any type of nor value of.

参考

Hint-Rewrite for feeding the database of lemmas used by autorewrite and autorewrite for examples showing the use of this tactic.

easy¶

This tactic tries to solve the current goal by a number of standard closing steps. In particular, it tries to close the current goal using the closing tactics trivial, reflexivity, symmetry, contradiction and inversion of hypothesis. If this fails, it tries introducing variables and splitting and-hypotheses, using the closing tactics afterwards, and splitting the goal using split and recursing.

This tactic solves goals that belong to many common classes; in particular, many cases of unsatisfiable hypotheses, and simple equality goals are usually solved by this tactic.

Variant now tactic¶: Run tactic followed by easy. This is a notation for tactic; easy.

Controlling automation¶

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.

Command Print Hint ident¶: Use this command to display the hints associated to the head symbol ident (see Print Hint). 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.

Command 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. The hints databases for auto and eauto), 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 does 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 ident+ is

Command Hint hint_definition : ident+¶

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

Variant Local Hint hint_definition : ident+: 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.

Variant Local Hint hint_definition: Idem for the core database.

Variant Hint 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 num if specified. The associated pattern is inferred from the conclusion of the type of term or the given pattern if 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 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 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.

Variant Hint Resolve term+: Adds each Hint Resolve term.

Variant Hint 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).

Variant Resolve <- term: Adds the right-to-left implication of an equivalence as a hint.

Variant 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 term cannot be used as a hint

Variant Immediate term+: Adds each Hint Immediate term.

Variant Hint 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 ident is not an inductive type¶

Variant Hint Constructors ident+: Adds each Hint Constructors ident.

Variant Hint 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.

Variant Hint Unfold ident+: Adds each Hint Unfold ident.

Variant Hint ( 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.

Variant Hint ( Transparent | Opaque ) ident+: Declares each ident as a transparent or opaque constant.

Variant Hint 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.

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:

Example

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

Variant Hint Cut regexp

警告

These hints currently only apply to typeclass proof search and the typeclasses eauto tactic.

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 that 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.

Variant Hint 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 term ... term, 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.

注釈

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

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 hints declared to belong to this database in each of the various modules currently loaded. Especially, requiring new modules may extend the database. At Coq startup, only the core database is nonempty 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. 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 typeclass 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.

Command Remove Hints term+ : ident+¶

This command removes the hints associated to terms term+ in databases ident+.

Command 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:

Command 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.

Command Print Hint *: This command displays all declared hints.

Command Print HintDb ident¶: This command displays all hints from database ident.

Command Hint Rewrite term+ : ident+¶

This vernacular command adds the terms term+ (their types must be equalities) in the rewriting bases ident+ with the default orientation (left to right). Notice that the rewriting bases are distinct from the auto hint bases and thatauto does not take them into account.

This command is synchronous with the section mechanism (see セクション機構): 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:

Command Hint Rewrite -> term+ : ident+: This is strictly equivalent to the command above (we only make explicit the orientation which otherwise defaults to ->).

Command Hint Rewrite <- term+ : ident+: Adds the rewriting rules term+ with a right-to-left orientation in the bases ident+.

Command Hint Rewrite term+ using tactic : ident+: When the rewriting rules term+ in ident+ will be used, the tactic tactic will be applied to the generated subgoals, the main subgoal excluded.

Command Print Rewrite HintDb ident¶: This command displays all rewrite hints contained in ident.

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, 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 be 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.

Option Loose Hint Behavior ( "Lax" | "Warn" | "Strict" )¶

This option accepts three values, which control the behavior of hints w.r.t. Import:

"Lax": this is the default, and corresponds to the historical behavior, that is, hints defined outside of a section have a global scope.
"Warn": 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.
"Strict": changes the behavior of an unloaded hint to a immediate fail tactic, allowing to emulate an import-scoped hint mechanism.

Setting implicit automation tactics¶

Command 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.

参考

Proof in Switching on/off the proof editing mode.

Variant Proof with tactic using ident+: Combines in a single line Proof with and Proof using, see Switching on/off the proof editing mode

Variant Proof using ident+ with tactic: Combines in a single line Proof with and Proof using, see Switching on/off the proof editing mode

Command 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.

Example

Parameter quo : nat -> forall n:nat, n<>0 -> nat.: quo is declared
Notation "x // y" := (quo x y _) (at level 40).
Declare Implicit Tactic assumption.
Goal forall n m, m<>0 -> { q:nat & { r | q * m + r = n } }.: quo is declared 1 subgoal ============================ forall n m : nat, m <> 0 -> {q : nat & {r : nat | q * m + r = n}}
intros.: 1 subgoal n, m : nat H : m <> 0 ============================ {q : nat & {r : nat | q * m + r = n}}
exists (n // m).: 1 subgoal n, m : nat H : m <> 0 ============================ {r : nat | n // m * m + r = n}

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

Decision procedures¶

tauto¶

This tactic implements a decision procedure for intuitionistic propositional calculus based on the contraction-free sequent calculi LJT* of Roy Dyckhoff [Dyc92]. 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:

Example

Goal forall (x:nat) (P:nat -> Prop), x = 0 \/ P x -> x <> 0 -> P x.: 1 subgoal ============================ forall (x : nat) (P : nat -> Prop), x = 0 \/ P x -> x <> 0 -> P x
intros.: 1 subgoal x : nat P : nat -> Prop H : x = 0 \/ P x H0 : x <> 0 ============================ P x
tauto.: No more subgoals.

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 for:

Example

Goal forall (A:Prop) (P:nat -> Prop), A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ A -> P x.: 1 subgoal ============================ forall (A : Prop) (P : nat -> Prop), A \/ (forall x : nat, ~ A -> P x) -> forall x : nat, ~ A -> P x
tauto.: No more subgoals.

注釈

In contrast, tauto cannot solve the following goal Goal forall (A:Prop) (P:nat -> Prop), A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ ~ (A \/ P x). because (forall x:nat, ~ A -> P x) cannot be treated as atomic and an instantiation of x is necessary.

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

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 [Mun94]. 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 The tactic language). 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.

Variant intuition: Is equivalent to intuition auto with *.

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

Flag Intuition Negation Unfolding¶: Controls whether intuition unfolds inner negations which do not need to be unfolded. This option is on by default.

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).

Note that this tactic is only available after a Require Import Rtauto.

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.

Option Firstorder Solver tactic¶

The default tactic used by firstorder when no rule applies is auto with *, it can be reset locally or globally using this option.

Command Print Firstorder Solver¶: Prints the default tactic used by firstorder when no rule applies.

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

Variant firstorder using qualid+: Adds lemmas qualid+ to the proof-search environment. If qualid refers to an inductive type, it is the collection of its constructors which are added to the proof-search environment.

Variant firstorder with ident+: Adds lemmas from auto hint bases ident+ to the proof-search environment.

Variant firstorder tactic using qualid+ with ident+: This combines the effects of the different variants of firstorder.

Option Firstorder Depth num¶: This option controls the proof-search depth bound.

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 includes constructor theory (see injection and discriminate). 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, but you have to provide a bound for the number of extra equalities generated that way. Please note that one of the sides of the equality must contain all the quantified variables in order for congruence to match against it.

Example

Theorem T (A:Type) (f:A -> A) (g: A -> A -> A) a b: a=(f a) -> (g b (f a))=(f (f a)) -> (g a b)=(f (g b a)) -> (g a b)=a.: 1 subgoal A : Type f : A -> A g : A -> A -> A a, b : A ============================ a = f a -> g b (f a) = f (f a) -> g a b = f (g b a) -> g a b = a
intros.: 1 subgoal A : Type f : A -> A g : A -> A -> A a, b : A H : a = f a H0 : g b (f a) = f (f a) H1 : g a b = f (g b a) ============================ g a b = a
congruence.: No more subgoals.
Qed.: T is defined
Theorem inj (A:Type) (f:A -> A * A) (a c d: A) : f = pair a -> Some (f c) = Some (f d) -> c=d.: 1 subgoal A : Type f : A -> A * A a, c, d : A ============================ f = pair a -> Some (f c) = Some (f d) -> c = d
intros.: 1 subgoal A : Type f : A -> A * A a, c, d : A H : f = pair a H0 : Some (f c) = Some (f d) ============================ c = d
congruence.: No more subgoals.
Qed.: inj is defined

Variant 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.

Variant congruence with term+¶: Adds term+ to the pool of terms used by congruence. This helps in case you have partially applied constructors in your goal.

Error 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.

Error Goal is solvable by congruence but some arguments are missing. Try congruence with term+, 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 congruence with variant described above.

Flag Congruence Verbose¶: This option makes congruence print debug information.

Checking properties of terms¶

Each of the following tactics acts as the identity if the check succeeds, and results in an error otherwise.

constr_eq term term¶: This tactic checks whether its arguments are equal modulo alpha conversion and casts.

Error Not equal.¶

unify term term¶: This tactic checks whether its arguments are unifiable, potentially instantiating existential variables.

Error Unable to unify term with term.

Variant unify term term with ident: Unification takes the transparency information defined in the hint database ident into account (see the hints databases for auto and eauto).

is_evar term¶: This tactic checks whether its argument is a current existential variable. Existential variables are uninstantiated variables generated by eapply and some other tactics.

Error Not an evar.¶

has_evar term¶: This tactic 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 No evars.¶

is_var term¶: This tactic checks whether its argument is a variable or hypothesis in the current goal context or in the opened sections.

Error Not a variable or hypothesis.¶

Equality¶

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.

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 The conclusion is not a substitutive equation.¶

Error Unable to unify ... with ...¶

symmetry¶

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

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

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.

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 =.

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.

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

simplify_eq term¶: Let term be the proof of a statement of conclusion term = term. If term and term 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.

注釈

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.

Variant 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.

Variant 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.

Variant esimplify_eq num

Variant 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.

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

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.

Variant dependent rewrite <- ident¶: Analogous to dependent rewrite -> but uses the equality from right to left.

Inversion¶

functional inversion ident¶

functional inversion is a tactic that performs inversion on hypothesis ident of the form qualid term+ = term or term = qualid term+ where qualid must have been defined using Function (see 高度な再帰関数). Note that this tactic is only available after a Require Import FunInd.

Error Hypothesis ident must contain at least one Function.¶

Error Cannot find inversion information for hypothesis ident.¶: This error may be raised when some inversion lemma failed to be generated by Function.

Variant functional inversion num: This does the same thing as intros until num folowed by functional inversion ident where ident is the identifier for the last introduced hypothesis.

Variant functional inversion ident qualid

Variant 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.

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 quote for the full details.

Error quote: not a simple fixpoint.¶: Happens when quote is not able to perform inversion properly.

Variant quote ident ident*: All terms that are built only with ident* will be considered by quote as constants rather than variables.

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.

classical_left¶

Variant 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.

Automating¶

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.

Note that this tactic is only available after a Require Import Btauto.

Error Cannot recognize a boolean equality.¶: The goal is not of the form t = u. Especially note that btauto doesn't introduce variables into the context on its own.

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 Omega: a solver for quantifier-free problems in Presburger Arithmetic).

ring¶

ring_simplify term+¶

The ring tactic solves equations upon polynomial expressions of a ring (or semiring) structure. It proceeds by normalizing both hand sides of the equation (w.r.t. associativity, commutativity and distributivity, constant propagation) and comparing syntactically the results.

ring_simplify applies the normalization procedure described above to the given terms. 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 The ring and field tactic families 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.

field¶

field_simplify term+¶

field_simplify_eq¶

The field tactic is built on the same ideas as ring: this is a reflexive tactic that solves or simplifies equations in a field structure. The main idea is to reduce a field expression (which is an extension of ring expressions with the inverse and division operations) to a fraction made of two polynomial expressions.

Tactic field is used to solve subgoals, whereas field_simplify term+ 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 The ring and field tactic families 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

Require Import Reals.: [Loading ML file z_syntax_plugin.cmxs ... done] [Loading ML file r_syntax_plugin.cmxs ... done] [Loading ML file quote_plugin.cmxs ... done] [Loading ML file newring_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done] [Loading ML file fourier_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done]
Goal forall x y:R, (x * y > 0)%R -> (x * (1 / x + x / (x + y)))%R = ((- 1 / y) * y * (- x * (x / (x + y)) - 1))%R.: 1 subgoal ============================ forall x y : R, (x * y > 0)%R -> (x * (1 / x + x / (x + y)))%R = (-1 / y * y * (- x * (x / (x + y)) - 1))%R
intros; field.: 1 subgoal x, y : R H : (x * y > 0)%R ============================ (x + y)%R <> 0%R /\ y <> 0%R /\ x <> 0%R

参考

File plugins/setoid_ring/RealField.v for an example of instantiation, theory theories/Reals for many examples of use of field.

fourier¶

This tactic written by Loïc Pottier solves linear inequalities on real numbers using Fourier’s method [Fou90]. This tactic must be loaded by Require Import Fourier.

Example: .. coqtop:: reset all

Require Import Reals. Require Import Fourier. Goal forall x y:R, (x < y)%R -> (y + 1 >= x - 1)%R. intros; fourier.

Non-logical tactics¶

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

Parameter P : nat -> Prop.: P is declared
Goal P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5.: 1 subgoal ============================ P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5
repeat split.: 5 subgoals ============================ P 1 subgoal 2 is: P 2 subgoal 3 is: P 3 subgoal 4 is: P 4 subgoal 5 is: P 5
all: cycle 2.: 5 subgoals ============================ P 3 subgoal 2 is: P 4 subgoal 3 is: P 5 subgoal 4 is: P 1 subgoal 5 is: P 2
all: cycle -3.: 5 subgoals ============================ P 5 subgoal 2 is: P 1 subgoal 3 is: P 2 subgoal 4 is: P 3 subgoal 5 is: P 4

swap num num¶: This tactic switches the position of the goals of indices num and num. If either num or num 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

Parameter P : nat -> Prop.: P is declared
Goal P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5.: 1 subgoal ============================ P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5
repeat split.: 5 subgoals ============================ P 1 subgoal 2 is: P 2 subgoal 3 is: P 3 subgoal 4 is: P 4 subgoal 5 is: P 5
all: swap 1 3.: 5 subgoals ============================ P 3 subgoal 2 is: P 2 subgoal 3 is: P 1 subgoal 4 is: P 4 subgoal 5 is: P 5
all: swap 1 -1.: 5 subgoals ============================ P 5 subgoal 2 is: P 2 subgoal 3 is: P 1 subgoal 4 is: P 4 subgoal 5 is: P 3

revgoals¶

This tactics reverses the list of the focused goals.

Example

Parameter P : nat -> Prop.: P is declared
Goal P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5.: 1 subgoal ============================ P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5
repeat split.: 5 subgoals ============================ P 1 subgoal 2 is: P 2 subgoal 3 is: P 3 subgoal 4 is: P 4 subgoal 5 is: P 5
all: revgoals.: 5 subgoals ============================ P 5 subgoal 2 is: P 4 subgoal 3 is: P 3 subgoal 4 is: P 2 subgoal 5 is: P 1

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.

Variant 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

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

Command Unshelve¶: This command moves all the goals on the shelf (see shelve) from the shelf into focus, by appending them to the end of the current list of focused goals.

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.

Simple tactic macros¶

A simple example has more value than a long explanation:

Example

Ltac Solve := simpl; intros; auto.: Solve is defined
Ltac ElimBoolRewrite b H1 H2 := elim b; [ intros; rewrite H1; eauto | intros; rewrite H2; eauto ].: ElimBoolRewrite is defined

The tactics macros are synchronous with the Coq section mechanism: a tactic definition is deleted from the current environment when you close the section (see also セクション機構) 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.

The tactic language gives examples of more complex user-defined tactics.

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 changed a lot 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.