Reasoning with equalities¶
There are multiple notions of equality in Coq:
Leibniz equality is the standard way to define equality in Coq and the Calculus of Inductive Constructions, which is in terms of a binary relation, i.e. a binary function that returns a
Prop
. The standard library defineseq
similar to this:Inductive eq {A : Type} (x : A) : A > Prop := eq_refl : eq x x.The notation
x = y
represents the termeq x y
. The notationx = y :> A
gives the type of x and y explicitly.Setoid equality defines equality in terms of an equivalence relation. A setoid is a set that is equipped with an equivalence relation (see https://en.wikipedia.org/wiki/Setoid). These are needed to form a quotient set or quotient (see https://en.wikipedia.org/wiki/Equivalence_class). In Coq, users generally work with setoids rather than constructing quotients, for which there is no specific support.
Definitional equality is equality based on the conversion rules, which Coq can determine automatically. When two terms are definitionally equal, Coq knows it can replace one with the other, such as with
change
X with Y
, among many other advantages. "Convertible" is another way of saying that two terms are definitionally equal.
Tactics for dealing with equality of inductive types such as injection
and inversion
are described here.
Tactics for simple equalities¶
 Tactic reflexivity¶
For a goal with the form
forall open_binders ,? t = u
, verifies thatt
andu
are definitionally equal, and if so, solves the goal (by applyingeq_refl
). If not, it fails.The tactic may also be applied to goals with the form
forall open_binders ,? R term_{1} term_{2}
whereR
is a reflexive relation registered with theEquivalence
orReflexive
typeclasses. SeeClass
andInstance
.
 Tactic symmetry simple_occurrences?¶
Changes a goal that has the form
forall open_binders ,? t = u
intou = t
.simple_occurrences
may be used to apply the change in the selected hypotheses and/or the conclusion.The tactic may also be applied to goals with the form
forall open_binders ,? R term_{1} term_{2}
whereR
is a symmetric relation registered with theEquivalence
orSymmetric
typeclasses. SeeClass
andInstance
.
 Tactic transitivity one_term¶
Changes a goal that has the form
forall open_binders ,? t = u
into the two subgoalst = one_term
andone_term = u
.The tactic may also be applied to goals with the form
forall open_binders ,? R term_{1} term_{2}
whereR
is a transitive relation registered with theEquivalence
orTransitivity
typeclasses. SeeClass
andInstance
. Tactic etransitivity¶
This tactic behaves like
transitivity
, using a fresh evar instead of a concreteone_term
.
 Tactic f_equal¶
For a goal with the form
f a_{1} ... a_{n} = g b_{1} ... b_{n}
, creates subgoalsf = g
anda_{i} = b_{i}
for then
arguments. Subgoals that can be proven byreflexivity
orcongruence
are solved automatically.
Rewriting with Leibniz and setoid equality¶
 Tactic rewrite oriented_rewriter+, occurrences? by ltac_expr3?¶
 oriented_rewriter
::=
><? natural? ?!? one_term_with_bindingsReplaces subterms with other subterms that have been proven to be equal. The type of
one_term
must have the form:forall open_binders ,? EQ term_{1} term_{2}
where
EQ
is the Leibniz equalityeq
or a registered setoid equality. Note thateq term_{1} term_{2}
is typically written with the infix notationterm_{1} = term_{2}
. You mustRequire Setoid
to use the tactic with a setoid equality or with setoid rewriting.rewrite one_term
finds subterms matchingterm_{1}
in the goal, and replaces them withterm_{2}
(or the reverse if<
is given). Some of the variablesx
_{i} are solved by unification, and some of the typesA_{1}, …, A_{n}
may become new subgoals.rewrite
won't find occurrences insideforall
that refer to variables bound by theforall
; use the more advancedsetoid_rewrite
if you want to find such occurrences.oriented_rewriter+,
The
oriented_rewriter
s are applied sequentially to the first goal generated by the previousoriented_rewriter
. If any of them fail, the tactic fails.><?
For
>
(the default),term_{1}
is rewritten intoterm_{2}
. For<
,term_{2}
is rewritten intoterm_{1}
.natural? ?!?
natural
is the number of rewrites to perform. If?
is given,natural
is the maximum number of rewrites to perform; otherwisenatural
is the exact number of rewrites to perform.?
(withoutnatural
) performs the rewrite as many times as possible (possibly zero times). This form never fails.!
(withoutnatural
) performs the rewrite as many times as possible and at least once. The tactic fails if the requested number of rewrites can't be performed.natural !
is equivalent tonatural
.occurrences
If
occurrences
specifies multiple occurrences, the tactic succeeds if any of them can be rewritten. If not specified, only the first occurrence in the conclusion is replaced.Note
If
at occs_nums
is specified, rewriting is always done with setoid rewriting, even for Leibniz equality, which means that you mustRequire Setoid
to use that form. However, note thatrewrite
(even when using setoid rewriting) andsetoid_rewrite
don't behave identically (as is noted above and below).by ltac_expr3
If specified, is used to resolve all side conditions generated by the tactic.
Note
For each selected hypothesis and/or the conclusion,
rewrite
finds the first matching subterm in depthfirst search order. Only subterms identical to that first matched subterm are rewritten. If theat
clause is specified, only these subterms are considered when counting occurrences. To select a different set of matching subterms, you can specify how some or all of the free variables are bound by using awith
clause (seeone_term_with_bindings
).For instance, if we want to rewrite the righthand side in the following goal, this will not work:
 Require Import Arith.
 [Loading ML file ring_plugin.cmxs (using legacy method) ... done]
 Lemma example x y : x + y = y + x.
 1 goal x, y : nat ============================ x + y = y + x
 rewrite Nat.add_comm at 2.
 Toplevel input, characters 025: > rewrite Nat.add_comm at 2. > ^^^^^^^^^^^^^^^^^^^^^^^^^ Error: Invalid occurrence number: 2.
One can explicitly specify how some variables are bound to match a different subterm:
 rewrite Nat.add_comm with (m := x).
 1 goal x, y : nat ============================ x + y = x + y
Note that the more advanced
setoid_rewrite
tactic behaves differently, and thus the number of occurrences available to rewrite may differ between the two tactics. Error Tactic failure: Setoid library not loaded.¶
 Error Cannot find a relation to rewrite.¶
 Error Tactic generated a subgoal identical to the original goal.¶
 Error Found no subterm matching term in ident.¶
 Error Found no subterm matching term in the current goal.¶
This happens if
term
does not occur in, respectively, the named hypothesis or the goal.
 Tactic erewrite oriented_rewriter+, occurrences? by ltac_expr3?¶
Works like
rewrite
, but turns unresolved bindings, if any, into existential variables instead of failing. It has the same parameters asrewrite
.
 Flag Keyed Unification¶
This flag makes higherorder unification used by
rewrite
rely on a set of keys to drive unification. The subterms, considered as rewriting candidates, must start with the same key as the left or righthand side of the lemma given to rewrite, and the arguments are then unified up to full reduction.
 Command Print Equivalent Keys¶
 Tactic rewrite * ><? one_term in ident? at rewrite_occs? by ltac_expr3?¶
 Tactic rewrite * ><? one_term at rewrite_occs in ident by ltac_expr3?¶
 Tactic replace ><? one_term_{from} with one_term_{to} occurrences? by ltac_expr3?¶
 Tactic replace ><? one_term_{from} occurrences?¶
The first form, when used with
<
or no arrow, replaces all free occurrences ofone_term_{from}
in the current goal withone_term_{to}
and generates an equalityone_term_{to} = one_term_{from}
as a subgoal. Note that this equality is reversed with respect to the order of the two terms. When used with>
, it generates instead an equalityone_term_{from} = one_term_{to}
. Whenby ltac_expr3
is not present, this equality is automatically solved if it occurs among the hypotheses, or if its symmetric form occurs.The second form, with
>
or no arrow, replacesone_term_{from}
withterm_{to}
using the first hypothesis whose type has the formone_term_{from} = term_{to}
. If<
is given, the tactic uses the first hypothesis with the reverse form, i.e.term_{to} = one_term_{from}
.occurrences
The
type of
andvalue of
forms are not supported. Note you mustRequire Setoid
to use theat
clause inoccurrences
.by ltac_expr3
Applies the
ltac_expr3
to solve the generated equality.
 Error Terms do not have convertible types.¶
 Tactic substitute ><? one_term_with_bindings¶
 Tactic subst ident*¶
For each
ident
, in order, for which there is a hypothesis in the formident = term
orterm = ident
, replacesident
withterm
everywhere in the hypotheses and the conclusion and clearsident
and the hypothesis from the context. If there are multiple hypotheses that match theident
, the first one is used. If noident
is given, replacement is done for all hypotheses in the appropriate form in top to bottom order.If
ident
is a local definition of the formident := term
, it is also unfolded and cleared.If
ident
is a section variable it must have no indirect occurrences in the goal, i.e. no global declarations implicitly depending on the section variable may be present in the goal.Note
If the hypothesis is itself dependent in the goal, it is replaced by the proof of reflexivity of equality.
 Flag Regular Subst Tactic¶
This flag 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
ident_{1} = ident_{2}
andident_{1} = t
, ort′ = ident_{1}
witht′
not a variable, and no other hypotheses of the formident_{2} = u
oru = ident_{2}
; without the flag, a second call to subst would be necessary to replaceident_{2}
byt
ort′
respectively.The presence of a recursive equation which without the flag would be a cause of failure of
subst
.A context with cyclic dependencies as with hypotheses
ident_{1} = f ident_{2}
andident_{2} = g ident_{1}
which without the flag would be a cause of failure ofsubst
.
Additionally, it prevents a local definition such as
ident := t
from being unfolded which otherwise would exceptionally unfold in configurations containing hypotheses of the formident = u
, oru′ = ident
withu′
not a variable. Finally, it preserves the initial order of hypotheses, which without the flag it may break.
 Error Section variable ident occurs implicitly in global declaration qualid present in hypothesis ident.¶
 Error Section variable ident occurs implicitly in global declaration qualid present in the conclusion.¶
Raised when the variable is a section variable with indirect dependencies in the goal. If
ident
is a section variable, it must not have any indirect occurrences in the goal, i.e. no global declarations implicitly depending on the section variable may be present in the goal.
 Tactic simple subst¶
 Tactic stepl one_term by ltac_expr?¶
For chaining rewriting steps. It assumes a goal in the form
R term_{1} term_{2}
whereR
is a binary relation and relies on a database of lemmas of the formforall x y z, R x y > eq x z > R z y
whereeq
is typically a setoid equality. The application ofstepl one_term
then replaces the goal byR one_term term_{2}
and adds a new goal statingeq one_term term_{1}
.If
ltac_expr
is specified, it is applied to the side condition.This tactic is especially useful for parametric setoids which are not accepted as regular setoids for
rewrite
andsetoid_replace
(see Generalized rewriting).
Rewriting with definitional equality¶
 Tactic change one_term_{from} at occs_nums? with? one_term_{to} occurrences?¶
Replaces terms with other convertible terms. If
one_term_{from}
is not specified, thenone_term_{to}
replaces the conclusion and/or the specified hypotheses. Ifone_term_{from}
is specified, the tactic replaces occurrences ofone_term_{to}
within the conclusion and/or the specified hypotheses.one_term_{from} at occs_nums? with?
Replaces the occurrences of
one_term_{from}
specified byoccs_nums
withone_term_{to}
, provided that the twoone_term
s are convertible.one_term_{from}
may contain pattern variables such as?x
, whose value which will substituted forx
inone_term_{to}
, such as inchange (f ?x ?y) with (g (x, y))
orchange (fun x => ?f x) with f
.The
at … with …
form is deprecated in 8.14; usewith … at …
instead. Forat … with … in H 
, usewith … in H at … 
.occurrences
If
with
is not specified,occurrences
must only specify entire hypotheses and/or the goal; it must not include anyat occs_nums
clauses.
 Error Not convertible.¶
 Error Found an "at" clause without "with" clause¶
 Tactic now_show one_type¶
A synonym for
change one_type
. It can be used to make some proof steps explicit when refactoring a proof script to make it readable.
See also
 Tactic change_no_check one_term_{from} at occs_nums? with? one_term_{to} occurrences?¶
For advanced usage. Similar to
change
, but as an optimization, it skips checking thatone_term_{to}
is convertible with the goal orone_term_{from}
.Recall that the Coq kernel typechecks proofs again when they are concluded to ensure correctness. Hence, using
change
checks convertibility twice overall, whilechange_no_check
can produce illtyped terms, but checks convertibility only once. Hence,change_no_check
can be useful to speed up certain proof scripts, especially if one knows by construction that the argument is indeed convertible to the goal.In the following example,
change_no_check
replacesFalse
withTrue
, butQed
then rejects the proof, ensuring consistency.Example
 Goal False.
 1 goal ============================ False
 change_no_check True.
 1 goal ============================ True
 exact I.
 No more goals.
 Qed.
 Toplevel input, characters 04: > Qed. > ^^^^ Error: The term "I" has type "True" while it is expected to have type "False".
Example
 Goal True > False.
 1 goal ============================ True > False
 intro H.
 1 goal H : True ============================ False
 change_no_check False in H.
 1 goal H : False ============================ False
 exact H.
 No more goals.
 Qed.
 Toplevel input, characters 04: > Qed. > ^^^^ Error: The term "fun H : True => H" has type "True > True" while it is expected to have type "True > False".
Applying conversion rules¶
These tactics apply reductions and expansions, replacing convertible subterms
with others that are equal by definition in CIC.
They implement different specialized uses of the
change
tactic. Other ways to apply these reductions are through
the Eval
command, the Eval
clause in the Definition
/Example
command and the eval
tactic.
Tactics described in this section include:
lazy
andcbv
, which allow precise selection of which reduction rules to applysimpl
andcbn
, which are "clever" tactics meant to give the most readable resulthnf
andred
, which apply reduction rules only to the head of the termvm_compute
andnative_compute
, which are performanceoriented.
Conversion tactics, with two exceptions, only change the types and contexts
of existential variables
and leave the proof term unchanged. (The vm_compute
and native_compute
tactics change existential variables in a way similar to other conversions while
also adding a single explicit cast to the proof term to tell the kernel
which reduction engine to use. See Type cast.) For example:
 Goal 3 + 4 = 7.
 1 goal ============================ 3 + 4 = 7
 Show Proof.
 ?Goal
 Show Existentials.
 Existential 1 = ?Goal : [  3 + 4 = 7]
 cbv.
 1 goal ============================ 7 = 7
 Show Proof.
 (?Goal : 3 + 4 = 7)
 Show Existentials.
 Existential 1 = ?Goal : [  7 = 7]
 Abort.
 Tactic lazy reductions? simple_occurrences¶
 Tactic cbv reductions? simple_occurrences¶
 reductions
::=
reduction+
head? delta_reductionsreduction
::=
head
beta
delta delta_reductions?
match
fix
cofix
iota
zetadelta_reductions
::=
? [ reference+ ]Normalize the goal as specified by
reductions
. If no reductions are specified by name, all reductions are applied. If any reductions are specified by name, then only the named reductions are applied. The reductions include:head
Do only head reduction, without going under binders. Supported by
simpl
,cbv
,cbn
andlazy
. If this is the only specified reduction, all other reductions are applied.beta
betareduction of functional application
delta delta_reductions?
deltareduction: unfolding of transparent constants, see Controlling reduction strategies and the conversion algorithm. The form in
reductions
without the keyworddelta
includesbeta
,iota
andzeta
reductions in addition todelta
using the givendelta_reductions
.? [ reference+ ]
without the

, limits delta unfolding to the listed constants. If the
is present, unfolding is applied to all constants that are not listed. Notice that thedelta
doesn't apply to variables bound by a letin construction inside the term itself (usezeta
to inline these). Opaque constants are never unfolded except byvm_compute
andnative_compute
(see #4476 and Controlling reduction strategies and the conversion algorithm).
iota
iotareduction of pattern matching (
match
) over a constructed term and reduction offix
andcofix
expressions. Shorthand formatch fix cofix
.zeta
zetareduction: reduction of letin definitions
Normalization is done by first evaluating the head of the expression into weakhead normal form, i.e. until the evaluation is blocked by a variable, an opaque constant, an axiom, such as in
x u_{1} … u_{n}
,match x with … end
,(fix f x {struct x} := …) x
, a constructed form (a \(\lambda\)expression, constructor, cofixpoint, inductive type, product type or sort) or a redex for which flags prevent reduction of the redex. Once a weakhead normal form is obtained, subterms are recursively reduced using the same strategy.There are two strategies for reduction to weakhead normal form: lazy (the
lazy
tactic), or callbyvalue (thecbv
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 propositionexists x. P(x)
reduce to a pair of a witnesst
and a proof thatt
satisfies the predicateP
. Most of the time,t
may be computed without computing the proof ofP(t)
, thanks to the lazy strategy. Flag Kernel Term Sharing¶
Turning this flag off disables the sharing of computations in
lazy
, making it a callbyname reduction. This also affects the reduction procedure used by the kernel when typechecking. By default sharing is activated.
The callbyvalue strategy is the one used in ML languages: the arguments of a function call are systematically weakly evaluated first. The lazy strategy is similar to how Haskell reduces terms. Although the lazy strategy always does fewer reductions than the callbyvalue strategy, the latter is generally more efficient for evaluating purely computational expressions (i.e. with little dead code).
 Tactic compute delta_reductions? simple_occurrences¶
A variant form of
cbv
.
Setting
Debug
"Cbv"
makescbv
(and its derivativecompute
) print information about the constants it encounters and the unfolding decisions it makes.
 Tactic simpl head? delta_reductions? reference_occspattern_occs? simple_occurrences¶
 reference_occs
::=
reference at occs_nums?pattern_occs
::=
one_term at occs_nums?Reduces a term to something still readable instead of fully normalizing it. It performs a sort of strong normalization with two key differences:
It unfolds constants only if they lead to an ιreduction, i.e. reducing a match or unfolding a fixpoint.
When reducing a constant unfolding to (co)fixpoints, the tactic uses the name of the constant the (co)fixpoint comes from instead of the (co)fixpoint definition in recursive calls.
occs_nums
Selects which occurrences of
one_term
to process (counting from left to right on the expression printed using thePrinting All
flag)simple_occurrences
Permits selecting whether to reduce the conclusion and/or one or more hypotheses. While the
at
option ofoccurrences
is not allowed here,reference_occs
andpattern_occs
have a somewhat less flexibleat
option for selecting specific occurrences.
simpl
can unfold transparent constants whose name can be reused in recursive calls as well as those designated byArguments
reference … /
commands. For instance, a constantplus' := plus
may be unfolded and reused in recursive calls, but a constant such assucc := plus (S O)
is not unfolded unless it was specifically designated in anArguments
command such asArguments succ /.
.reference_occspattern_occs
can limit the application ofsimpl
to:
 Tactic cbn reductions? simple_occurrences¶
cbn
was intended to be a more principled, faster and more predictable replacement forsimpl
. The main difference is thatcbn
may unfold constants even when they cannot be reused in recursive calls: in the previous example,succ t
is reduced toS t
. Modifiers such assimpl never
are also not treated the same, see Effects of Arguments on unfolding.Setting
Debug
"RAKAM"
makescbn
print various debugging information.RAKAM
is the Refolding Algebraic Krivine Abstract Machine.Example
Here are typical examples comparing
cbn
andsimpl
: Definition add1 (n:nat) := n + 1.
 add1 is defined
 Eval simpl in add1 0.
 = add1 0 : nat
 Eval cbn in add1 0.
 = 1 : nat
 Definition pred_add n m := pred (n + m).
 pred_add is defined
 Eval simpl in pred_add 0 0.
 = pred_add 0 0 : nat
 Eval cbn in pred_add 0 0.
 = 0 : nat
 Parameter n : nat.
 n is declared
 Eval simpl in pred_add 0 n.
 = pred_add 0 n : nat
 Eval cbn in pred_add 0 n.
 = pred_add 0 n : nat
 Tactic hnf simple_occurrences¶
Replaces the current goal with its weakhead normal form according to the βδιζreduction rules, i.e. it reduces the head of the goal until it becomes a product or an irreducible term. All inner βιredexes are also reduced. While
hnf
behaves similarly tosimpl
andcbn
, unlike them, it does not recurse into subterms. The behavior ofhnf
can be tuned using theArguments
command.Example: The term
fun n : nat => S n + S n
is not reduced byhnf
.Note
The δ rule only applies to transparent constants (see Controlling reduction strategies and the conversion algorithm on transparency and opacity).
 Tactic red simple_occurrences¶
βιζreduces the head constant of
T
, if possible, in the selected hypotheses and/or the goal which have the form:forall open_binders ,? T
(where
T
does not begin with aforall
) toc t_{1} … t_{n}
wherec
is a constant. Ifc
is transparent then it replacesc
with its definition and reduces again until no further reduction is possible.In the term
forall open_binders ,? t_{1} ... t_{n}
, wheret_{1}
is not aterm_application
,t_{1}
is the head of the term. In a term with the formforall open_binders ,? c t_{1} ... t_{n}
, wherec
is a constant,c
is the head constant. Error No head constant to reduce.¶
 Tactic unfold reference_occs+, occurrences?¶
Applies deltareduction to the constants specified by each
reference_occs
. The selected hypotheses and/or goals are then reduced to βιζnormal form. Use the general reduction tactics if you want to only apply the δ rule, for examplecbv
delta [ reference ]
.reference_occs
If
reference
is aqualid
, it must be a defined transparent constant or local definition (see Toplevel definitions and Controlling reduction strategies and the conversion algorithm).If
reference
is astring scope_key?
, thestring
is the discriminating symbol of a notation (e.g. "+") or an expression defining a notation (e.g."_ + _"
) and the notation is an application whose head symbol is an unfoldable constant, then the tactic unfolds it.occurrences
If
occurrences
is specified, the specified occurrences will be replaced in the selected hypotheses and/or goal. Otherwise every occurrence of the constants in the goal is replaced. If multiplereference_occs
are given, anyat
clauses must be in thereference_occs
rather than inoccurrences
.
 Error Cannot turn inductiveconstructor into an evaluable reference.¶
Occurs when trying to unfold something that is defined as an inductive type (or constructor) and not as a definition.
Example
 Goal 0 <= 1.
 1 goal ============================ 0 <= 1
 unfold le.
 Toplevel input, characters 79: > unfold le. > ^^ Error: Cannot coerce le to an evaluable reference.
 Tactic fold one_term+ simple_occurrences¶
First, this tactic reduces each
one_term
using thered
tactic. Then, every occurrence of the resulting terms in the selected hypotheses and/or goal will be replaced by its associatedone_term
. This tactic is particularly useful for reversing undesired unfoldings, which may make the goal very hard to read. The undesired unfoldings may be due to the limited capabilities of other reduction tactics. On the other hand, when an unfolded function applied to its argument has been reduced, thefold
tactic doesn't do anything.fold
one_term_{1} one_term_{2}
is equivalent tofold one_term_{1}; fold one_term_{2}
.Example:
fold
doesn't always undounfold
 Goal ~0=0.
 1 goal ============================ 0 <> 0
 unfold not.
 1 goal ============================ 0 = 0 > False
This
fold
doesn't undo the preceedingunfold
(it makes no change): fold not.
 1 goal ============================ 0 = 0 > False
However, this
pattern
followed byfold
does: pattern (0 = 0).
 1 goal ============================ (fun P : Prop => P > False) (0 = 0)
 fold not.
 1 goal ============================ 0 <> 0
Example: Use
fold
to reverse unfolding offold_right
 Require Import Coq.Lists.List.
 Local Open Scope list_scope.
 Goal forall x xs, fold_right and True (x::xs).
 1 goal ============================ forall (x : Prop) (xs : list Prop), fold_right and True (x :: xs)
 red.
 1 goal ============================ forall (x : Prop) (xs : list Prop), x /\ (fix fold_right (l : list Prop) : Prop := match l with  nil => True  b :: t => b /\ fold_right t end) xs
 fold (fold_right and True).
 1 goal ============================ forall (x : Prop) (xs : list Prop), x /\ fold_right and True xs
 Tactic pattern pattern_occs+, occurrences?¶
Performs betaexpansion (the inverse of betareduction) for the selected hypotheses and/or goals. The
one_term
s inpattern_occs
must be free subterms in the selected items. The expansion is done for each selected itemT
for a set ofone_term
s in thepattern_occs
by:replacing all selected occurrences of the
one_term
s inT
with fresh variablesabstracting these variables
applying the abstracted goal to the
one_term
s
For instance, if the current goal
T
is expressible asφ(t_{1} … t_{n})
where the notation captures all the instances of thet_{i}
in φ, thenpattern
t_{1}, …, t_{n}
generates the equivalent goal(fun (x_{1}:A_{1} … (x_{n}:A_{n}) => φ(x_{1} … x_{n})) t_{1} … t_{n}
. Ift_{i}
occurs in one of the generated typesA_{j}
(forj > i
), occurrences will also be considered and possibly abstracted.This tactic can be used, for instance, when the tactic
apply
fails on matching or to better control the behavior ofrewrite
.See the example here.
Fast reduction tactics: vm_compute and native_compute¶
vm_compute
is a bruteforce but efficient tactic that
first normalizes the terms before comparing them. It is based on a
bytecode representation of terms similar to the bytecode
representation used in the ZINC virtual machine [Ler90]. It is
especially useful for intensive computation of algebraic values, such
as numbers, and for reflectionbased tactics.
native_compute
is based on on converting the Coq code to OCaml.
Note that both these tactics ignore Opaque
markings
(see issue #4776), nor do they
apply unfolding strategies such as from Strategy
.
native_compute
is typically two to five
times faster than vm_compute
at applying conversion rules
when Coq is running native code, but native_compute
requires
considerably more overhead. We recommend using native_compute
when all of the following are true (otherwise use vm_compute
):
the running time in
vm_compute
at least 510 secondsthe size of the input term is small (e.g. handgenerated code rather than automaticallygenerated code that may have nested destructs on inductives with dozens or hundreds of constructors)
the output is small (e.g. you're returning a boolean, a natural number or an integer rather than a large abstract syntax tree)
These tactics change existential variables in a way similar to other conversions while also adding a single explicit cast (see Type cast) to the proof term to tell the kernel which reduction engine to use.
 Tactic vm_compute reference_occspattern_occs? occurrences?¶
Evaluates the goal using the optimized callbyvalue evaluation bytecodebased virtual machine described in [GregoireL02]. This algorithm is dramatically more efficient than the algorithm used for the
cbv
tactic, but it cannot be finetuned. It is especially useful for full evaluation of algebraic objects. This includes the case of reflectionbased tactics.
 Tactic native_compute reference_occspattern_occs? occurrences?¶
Evaluates the goal by compilation to OCaml as described in [BDenesGregoire11]. Depending on the configuration, this tactic can either default to
vm_compute
, recompile dependencies or fail due to some missing precompiled dependencies, see the nativecompiler option for details. Flag NativeCompute Timing¶
This flag causes all calls to the native compiler to print timing information for the conversion to native code, compilation, execution, and reification phases of native compilation. Timing is printed in units of seconds of wallclock time.
 Flag NativeCompute Profiling¶
On Linux, if you have the
perf
profiler installed, this flag makes it possible to profilenative_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 ofnative_compute
in a script. From the Linux command line, runperf report
on the profile file to see the results. Consult theperf
documentation for more details.
Computing in a term: eval and Eval¶
Evaluation of a term can be performed with:
 Tactic eval red_expr in term¶
 red_expr
::=
lazy reductions?
cbv reductions?
compute delta_reductions?
vm_compute reference_occspattern_occs?
native_compute reference_occspattern_occs?
red
hnf
simpl head? delta_reductions? reference_occspattern_occs?
cbn reductions?
unfold reference_occs+,
fold one_term+
pattern pattern_occs+,
identeval
is avalue_tactic
. It returns the result of applying the conversion rules specified byred_expr
. It does not change the proof state.The
red_expr
alternatives that begin with a keyword correspond to the tactic with the same name, though in several cases with simpler syntax than the tactic.ident
is a named reduction expression created withDeclare Reduction
.See also
Section Applying conversion rules.
 Command Eval red_expr in term¶
Performs the specified reduction on
term
and displays the resulting term with its type. If a proof is open,term
may reference hypotheses of the selected goal.Eval
is aquery_command
, so it may be prefixed with a goal selector.
 Command Declare Reduction ident := red_expr¶
Declares a short name for the reduction expression
red_expr
, for instancelazy beta delta [foo bar]
. This short name can then be used inEval ident in
oreval
constructs. This command accepts thelocal
attribute, which indicates that the reduction will be discarded at the end of the file or module. The name is not qualified. In particular declaring the same name in several modules or in several functor applications will be rejected if these declarations are not local. The nameident
cannot be used directly as an Ltac tactic, but nothing prevents the user from also performing aLtac ident := red_expr
.
Controlling reduction strategies and the conversion algorithm¶
The commands to finetune the reduction strategies and the lazy conversion algorithm are described in this section. Also see Effects of Arguments on unfolding, which supports additional finetuning.
 Command Opaque !? reference+¶
Marks the specified constants as opaque so tactics won't unfold them with deltareduction. "Constants" are items defined by commands such as
Definition
,Let
(with an explicit body),Fixpoint
,CoFixpoint
andFunction
.This command accepts the
global
attribute. By default, the scope ofOpaque
is limited to the current section or module.Opaque
also affects Coq's conversion algorithm, causing it to delay unfolding the specified constants as much as possible when it has to check that two distinct applied constants are convertible. See Section Conversion rules.In the particular case where the constants refer to primitive projections, a
can be used to make the compatibility constants opaque, while by default the projection themselves are made opaque and the compatibility constants always remain transparent. This mechanism is only intended for debugging purposes.
 Command Transparent !? reference+¶
The opposite of
Opaque
, it marks the specified constants as transparent so that tactics may unfold them. SeeOpaque
above.This command accepts the
global
attribute. By default, the scope ofTransparent
is limited to the current section or module.Note that constants defined by proofs ending with
Qed
are irreversibly opaque;Transparent
will not make them transparent. This is consistent with the usual mathematical practice of proof irrelevance: what matters in a mathematical development is the sequence of lemma statements, not their actual proofs. This distinguishes lemmas from the usual defined constants, whose actual values are of course relevant in general.In the particular case where the constants refer to primitive projections, a
can be used to make the compatibility constants transparent (see
Opaque
for more details).
See also
 Command Strategy strategy_level [ reference+ ]+¶
 strategy_level
::=
opaque
integer
expand
transparentGeneralizes the behavior of the
Opaque
andTransparent
commands. It is used to finetune the strategy for unfolding constants, both at the tactic level and at the kernel level. This command associates astrategy_level
with the qualified names in thereference
sequence. Whenever two expressions with two distinct head constants are compared (for example, typecheckingf x
wheref : A > B
andx : C
will result in convertingA
andC
), the one with lower level is expanded first. In case of a tie, the second one (appearing in the cast type) is expanded.This command accepts the
local
attribute, which limits its effect to the current section or module, in which case the section and module behavior is the same asOpaque
andTransparent
(withoutglobal
).Levels can be one of the following (higher to lower):
opaque
: level of opaque constants. They cannot be expanded by tactics (behaves like +∞, see next item).integer
: levels indexed by an integer. Level 0 corresponds to the default behavior, which corresponds to transparent constants. This level can also be referred to astransparent
. Negative levels correspond to constants to be expanded before normal transparent constants, while positive levels correspond to constants to be expanded after normal transparent constants.expand
: level of constants that should be expanded first (behaves like −∞)transparent
: Equivalent to level 0
 Command Print Strategy reference¶
This command prints the strategy currently associated with
reference
. It fails ifreference
is not an unfoldable reference, that is, neither a variable nor a constant. Error The reference is not unfoldable.¶
 Command Print Strategies¶
Print all the currently nontransparent strategies.
 Tactic with_strategy strategy_level_or_var [ reference+ ] ltac_expr3¶
 strategy_level_or_var
::=
strategy_level
identExecutes
ltac_expr3
, applying the alternate unfolding behavior that theStrategy
command controls, but only forltac_expr3
. This can be useful for guarding calls to reduction in tactic automation to ensure that certain constants are never unfolded by tactics likesimpl
andcbn
or to ensure that unfolding does not fail.Example
 Opaque id.
 Goal id 10 = 10.
 1 goal ============================ id 10 = 10
 Fail unfold id.
 The command has indeed failed with message: id is opaque.
 with_strategy transparent [id] unfold id.
 1 goal ============================ 10 = 10
Warning
Use this tactic with care, as effects do not persist past the end of the proof script. Notably, this finetuning of the conversion strategy is not in effect during
Qed
norDefined
, so this tactic is most useful either in combination withabstract
, which will check the proof early while the finetuning is still in effect, or to guard calls to conversion in tactic automation to ensure that, e.g.,unfold
does not fail just because the user made a constantOpaque
.This can be illustrated with the following example involving the factorial function.
 Fixpoint fact (n : nat) : nat := match n with  0 => 1  S n' => n * fact n' end.
 fact is defined fact is recursively defined (guarded on 1st argument)
Suppose now that, for whatever reason, we want in general to unfold the
id
function very late during conversion: Strategy 1000 [id].
If we try to prove
id (fact n) = fact n
byreflexivity
, it will now take time proportional to \(n!\), because Coq will keep unfoldingfact
and*
and+
before it unfoldsid
, resulting in a full computation offact n
(in unary, because we are usingnat
), which takes time \(n!\). We can see this cross the relevant threshold at around \(n = 9\): Goal True.
 1 goal ============================ True
 Time assert (id (fact 8) = fact 8) by reflexivity.
 Finished transaction in 0.195 secs (0.167u,0.024s) (successful) 1 goal H : id (fact 8) = fact 8 ============================ True
 Time assert (id (fact 9) = fact 9) by reflexivity.
 Finished transaction in 1.328 secs (1.328u,0.s) (successful) 1 goal H : id (fact 8) = fact 8 H0 : id (fact 9) = fact 9 ============================ True
Note that behavior will be the same if you mark
id
asOpaque
because while most reduction tactics refuse to unfoldOpaque
constants, conversion treatsOpaque
as merely a hint to unfold this constant last.We can get around this issue by using
with_strategy
: Goal True.
 1 goal ============================ True
 Fail Timeout 1 assert (id (fact 100) = fact 100) by reflexivity.
 The command has indeed failed with message: Timeout!
 Time assert (id (fact 100) = fact 100) by with_strategy 1 [id] reflexivity.
 Finished transaction in 0.002 secs (0.002u,0.s) (successful) 1 goal H : id (fact 100) = fact 100 ============================ True
However, when we go to close the proof, we will run into trouble, because the reduction strategy changes are local to the tactic passed to
with_strategy
. exact I.
 No more goals.
 Timeout 1 Defined.
 Toplevel input, characters 018: > Timeout 1 Defined. > ^^^^^^^^^^^^^^^^^^ Error: Timeout!
We can fix this issue by using
abstract
: Goal True.
 1 goal ============================ True
 Time assert (id (fact 100) = fact 100) by with_strategy 1 [id] abstract reflexivity.
 Finished transaction in 0.006 secs (0.006u,0.s) (successful) 1 goal H : id (fact 100) = fact 100 ============================ True
 exact I.
 No more goals.
 Time Defined.
 Finished transaction in 0.001 secs (0.001u,0.s) (successful)
On small examples this sort of behavior doesn't matter, but because Coq is a superlinear performance domain in so many places, unless great care is taken, tactic automation using
with_strategy
may not be robustly performant when scaling the size of the input.Warning
In much the same way this tactic does not play well with
Qed
andDefined
without usingabstract
as an intermediary, this tactic does not play well withcoqchk
, even when used withabstract
, due to the inability of tactics to persist information about conversion hints in the proof term. See #12200 for more details.