Some tips, tricks, and features in Coq that are hard to discover.
If you have a trick you've found useful feel free to submit an issue or pull request!
pattern
tactic generalizes an expression over a variable. For example, pattern y
transforms a goal of P x y z
into (fun y => P x y z) y
. An especially useful way to use this is to pattern match on eval pattern y in constr:(P x y z)
to extract just the function.lazymatch
is like match
but without backtracking on failures inside the match action. If you're not using failures for backtracking (this is often the case), then lazymatch
gets better error messages because an error inside the action becomes the overall error message rather than the uninformative "no match" error message. (The semantics of match
mean Coq can't do obviously do better - it can't distinguish between a bug in the action and intentionally using the failure to prevent pattern matching.)deex
(see Deex.v) is a useful tactic for extracting the witness from an exists
hypothesis while preserving the name of the witness and hypothesis.Ltac t ::= foo.
re-defines the tactic t
to foo
. This changes the binding, so tactics that refer to t
will use the new definition. You can use this for a form of extensibility: write Ltac hook := fail
and then use
repeat match goal with
| (* initial cases *)
| _ => hook
| (* other cases *)
end
in your tactic. Now the user can insert an extra case in your tactic's core loop by overriding hook
.learn
approach - see Learn.v for a self-contained example or Clément's thesis for more detailsunshelve
tactical, especially useful with an eapply - good example use case is constructing an object by refinement where the obligations end up being your proofs with the values as evars, when you wanted to construct the values by proofunfold "+"
worksdestruct matches
tactic; see coq-tactical's SimplMatch.v
instantiate
to modify evar environment (thanks to Jonathan Leivent on coq-club)eexists ?[x]
lets one name an existential variable to be able to refer to it laterRequire Import Arith.
and use induction n as [n IHn] using lt_wf_ind.
Require Import Coq.Arith.Wf_nat.
and induction xs as [xs IHxs] using (induction_ltof1 _ (@length _)); unfold ltof in IHxs.
debug auto
, debug eauto
, and debug trivial
give traces, including failed invocations. info_auto
, info_eauto
, and info_trivial
are less verbose ways to debug which only report what the resulting proof includesconstructor
and econstructor
backtrack over the constructors over an inductive, which lets you do fun things exploring the constructors of an inductive type. See Constructors.v for some demonstrations.destruct_with_eqn x
.
You can also do destruct x eqn:H
to explicitly name the equality
hypothesis. This is similar to case_eq x; intros
; I'm not sure what the
practical differences are.rew H in t
notation to use eq_rect
for a (safe) "type cast". Need to
import EqNotations
- see RewNotation.v for a working
example.intro
-patterns can be combined in a non-trivial way: intros [=->%lemma]
-- see IntroPatterns.v.change
tactic supports patterns (?var
): e.g. repeat change (fun x => ?f x) with f
would eta-reduce all the functions (of arbitrary arity) in the goal.in *
and in H
to apply everywhere and in a specific hypotheses, but there are actually a bunch of forms, for example:
in H1, H2
(just H1
and H2
)in H1, H2 |- *
(H1
, H2
, and the goal)in * |-
(just hypotheses)in |-
(nowhere)in |- *
(just the goal, same as leaving the whole thing off)in * |- *
(everywhere, same as in *
)
These forms would be especially useful if occurrence clauses were first-class objects; that is, if tactics could take and pass occurrence clauses. Currently user-defined tactics support occurrence clauses via a set of tactic notations.Notation anyplus := (_ + _).
and then
match goal with
| |- context[anyplus] => idtac
end
I would recommend using Local Notation
so the notation isn't available outside the current file.Hint Constructors
; you can also do this locally in a proof with eauto using t
where t
is the name of the inductive.intuition
tactic has some unexpected behaviors. It takes a tactic to run on each goal, which is auto with *
by default, using hints from all hint databases. intuition idtac
or intuition eauto
are both much safer. When using these, note that intuition eauto; simpl
is parsed as intuition (eauto; simpl)
, which is unlikely to be what you want; you'll need to instead write (intuition eauto); simpl
.Coq.Program.Tactics
library has a number of useful tactics and tactic helpers. Some gems that I like: add_hypothesis
is like pose proof
but fails if the fact is already in the context (a lightweight version of the learn
approach); destruct_one_ex
implements the tricky code to eliminate an exists
while retaining names (it's a better version of our deex
); on_application
matches any application of f
by simply handling a large number of arities.-
, +
, *
so that Proof General indents them correctly. If you need more bullets you can keep going with --
, ++
, **
(although you should rarely need more then three levels of bullets in one proof).set
tactic to create shorthand names for expressions. These are special let
-bound variables and show up in the hypotheses as v := def
. To "unfold" these definitions you can do subst v
(note the explicit name is required, subst
will not do this by default). This is a good way to make large goals readable, perhaps while figuring out what lemma to extract. It can also be useful if you need to refer these expressions.Defined
instead of Qed
. The difference is that Qed
makes the proof term opaque and prevents reduction, while Defined
will simplify correctly. If you mix computational parts and proof parts (eg, functions which produce sigma types) then you may want to separate the proof into a lemma so that it doesn't get unfolded into a large proof term.unshelve (instantiate (1:=_))
. The way this work is to instantiate the evar with a fresh evar (created due to the _
) and then unshelve that evar, making it an explicit goal. See UnshelveInstantiate.v for a working example.enough
tactic behaves like assert
but puts the goal for the stated fact after the current goal rather than before.context E [x]
to bind a context variable, and then let e := eval context E [y] in ...
to substitute back into the context. See
Context.v for an example.@?z x
, which bind z
to a function) and you want to apply the function, there's a trick involving a seemingly useless match. See LtacGallinaApplication.v for an example.auto with foo nocore
with the pseudo-database nocore
disables the default core
hint databases and only uses hints from foo
(and the context).apply thm in H as [x H]
, for example, might be used then thm
produces an existential for a variable named x
.H: a = b
and need f a = f b
, you can use apply (f_equal f) in H
. (Strictly speaking this is just using the f_equal
theorem in the standard library, but it's also very much like the inverse direction for the f_equal
tactic.)constr
, you can do so by wrapping the side effect in let _ := match goal with _ => side_effect_tactic end in ...
. See https://stackoverflow.com/questions/45949064/check-for-evars-in-a-tactic-that-returns-a-value/46178884#46178884 for Jason Gross's much more thorough explanation.lia
to help with non-linear arithmetic involving division or modulo (or the similar quot
and rem
), you can do that for simple cases with Ltac Zify.zify_post_hook ::= Z.div_mod_to_equations.
See DivMod.v for an example and the micromega documentation the full details.admit
will force you to use Admitted
. If you want to use Qed, you can instead use Axiom falso : False. Ltac admit := destruct falso.
This can be useful for debugging Qed errors (say, due to universes) or slow Qeds.tactics in terms, eg ltac:(eauto)
can provide a proof argument
maximally inserted implicit arguments are implicit even when for identifier alone (eg, nil
is defined to include the implicit list element type)
maximally inserted arguments can be defined differently for different numbers of arguments - undocumented but eq_refl
provides an example
r.(Field)
syntax: same as Field r
, but convenient when Field
is a projection function for the (record) type of r
. If you use these, you might also want Set Printing Projections
so Coq re-prints calls to projections with the same syntax.
Function
vernacular provides a more advanced way to define recursive functions, which removes the restriction of having a structurally decreasing argument; you just need to specify a well-founded relation or a decreasing measure maps to a nat, then prove all necessary obligations to show this function can terminate. See manual and examples in Function.v
for more details.
Two alternatives are considerable as drop-in replacements for Function
.
Program Fixpoint
may be useful when defining a nested recursive function. See manual and this StackOverflow post.Fix
combinator.One can pattern-match on tuples under lambdas: Definition fst {A B} : (A * B) -> A := fun '(x,_) => x.
Records fields can be defined with :>
, which make that field accessor a coercion. There are three ways to use this (since there are three types of coercion classes). See Coercions.v for some concrete examples.
Type
), then the record can be used as a type.When a Class field (as opposed to a record) is defined with :>
, it becomes a hint for typeclass resolution. This is useful when a class includes a "super-class" requirement as a field. For example, Equivalence
has fields for reflexivity, symmetry, and transitivity. The reflexivity field can be used to generically take an Equivalence
instance and get a reflexivity instance for free.
The type classes in RelationClasses are useful but can be repetitive to prove. RelationInstances.v goes through a few ways of making these more convenient, and why you would want to do so (basically you can make reflexivity
, transitivity
, and symmetry
more powerful).
The types of inductives can be definitions, as long as they expand to an "arity" (a function type ending in Prop
, Set
, or Type
). See ArityDefinition.v.
Record fields that are functions can be written in definition-style syntax with the parameters bound after the record name, eg {| func x y := x + y; |}
(see RecordFunction.v for a complete example).
If you have a coercion get_function : MyRecord >-> Funclass
you can use Add Printing Coercion get_function
and then add a notation for get_function
so your coercion can be parsed as function application but printed using some other syntax (and maybe you want that syntax to be printing only
).
You can pass implicit arguments explicitly in a keyword-argument-like style, eg nil (A:=nat)
. Use About
to figure out argument names.
If you do nasty dependent pattern matches or use inversion
on a goal and it produces equalities of existT
's, you may benefit from small inversions, described in this blog post. While the small inversion tactic is still not available anywhere I can find, some support is built in to Coq's match return type inference; see SmallInversions.v for examples of how to use that.
You can use tactics-in-terms with notations to write function-like definitions that are written in Ltac. For example, you can use this facility to write macros that inspect and transform Gallina terms, producing theorem statements and optionally their proofs automatically. A simple example is given in DefEquality.v of writing a function that produces an equality for unfolding a definition.
Notations can be dangerous since they by default have global scope and are imported by Import
, with no way to selectively import. A pattern I now use by default to make notations controllable is to define every notation in a module with a scope; see NotationModule.v.
This pattern has several advantages:
Import
and Local Open Scope
, restoring the convenience of a global notationCoq has a module system, modeled after ML (eg, the one used in OCaml). You can see some simple examples of using it in Modules.v. In user code, I've found modules to be more trouble than their worth 90% of the time - the biggest issue is that once something is in a module type, the only way to extend it is with a new module that wraps an existing module, and the only way to use the extension is to instantiate it. At the same time, you can mostly simulate module types with records.
Coq type class resolution is extremely flexible. There's a hint database called typeclass_instances
and typeclass resolution is essentially eauto with typeclass_instances
. Normally you add to this database with commands like Instance
, but you can add whatever you want to it, including Hint Extern
s. See coq-record-update for a practical example.
Classes are a bit special compared to any other type. First of all, in (_ : T x1 x2)
Coq will only trigger type class resolution to fill the hole when T
is a class. Second, classes get special implicit generalization behavior; specifically, you can write {T}
and Coq will automatically generalize the arguments to T, which you don't even have to write down. See the manual on implicit generalization for more details. Note that you don't have to use Class
at declaration time to make something a class; you can do it after the fact with Existing Class T
.
Search
vernacular variants; see Search.v for examples.
Search s -Learnt
for a search of local hypotheses excluding Learnt
Locate
can search for notation, including partial searches.
Optimize Heap
(undocumented) runs GC (specifically Gc.compact
)
Optimize Proof
(undocumented) runs several simplifications on the current proof term (see Proofview.compact
)
(in Coq 8.12 and earlier) Generalizable Variable A
enables implicit generalization; Definition id `(x:A) := x
will implicitly add a parameter A
before x
. Generalizable All Variables
enables implicit generalization for any identifier. Note that this surprisingly allows generalization without a backtick in Instances. Issue #6030 generously requests this behavior be documented, but it should probably require enabling some option. This has been fixed in Coq 8.13; the old behavior requires Set Instance Generalized Output
. In Coq 8.14 the option has been removed.
Check
supports partial terms, printing a type along with a context of evars. A cool example is Check (id _ _)
, where the first underscore must be a function (along with other constraints on the types involved).
The above also works with named existentials. For example, Check ?[x] + ?[y]
works.
Unset Intuition Negation Unfolding
will cause intuition
to stop unfolding not
.
Definitions can be normalized (simplified/computed) easily with Definition bar := Eval compute in foo.
Set Uniform Inductive Parameters
(in Coq v8.9+beta onwards) allows you to omit the uniform parameters to an inductive in the constructors.
Lemma
and Theorem
are synonymous, except that coqdoc
will not show lemmas. Also synonymous: Corollary
, Remark
, and Fact
. Definition
is nearly synonymous, except that Theorem x := def
is not supported (you need to use Definition
).
Sections are a powerful way to write a collection of definitions and lemmas that all take the same generic arguments. Here are some tricks for working with sections, which are illustrated in Sections.v:
Context
, which is strictly more powerful than Variable
- you can declare multiple dependent parameters and get type inference, and can write {A}
to make sure a parameter is implicit and maximally inserted.t
that are specific to the automation of a file, and callers don't see it. Similarly with adding hints to core
with abandon.Context (A:Type). Notation list := (List.list A). Implicit Types (l:list).
and then in the whole section you basically never need to write type annotations. The notation and implicit type disappears at the end of the section so no worries about leaking it. Furthermore, don't write Theorem foo : forall l,
but instead write Theorem foo l :
; you can often also avoid using intros
with this trick (though be careful about doing induction and ending up with a weak induction hypothesis).t
that solves most goals in a section, it gets annoying to write Proof. t. Qed.
every time. Instead, define Notation magic := ltac:(t) (only parsing).
and write Definition foo l : l = l ++ [] = magic.
. You do unfortunately have to write Definition
; Lemma
and Theorem
do not support :=
definitions. You don't have to call it magic
but of course it's more fun that way. Note that this isn't the best plan because you end up with transparent proofs, which isn't great; ideally Coq would just support Theorem foo :=
syntax for opaque proofs.Haskell has an operator f $ x
, which is the same as f x
except that its parsed differently: f $ 1 + 1
means f (1 + 1)
, avoiding parentheses. You can simulate this in Coq with a notation: Notation "f $ x" := (f x) (at level 60, right associativity, only parsing).
(from jwiegley/coq-haskell).
A useful convention for notations is to have them start with a word and an exclamation mark. This is borrowed from @andres-erbsen, who borrowed it from the Rust macro syntax. An example of using this convention is in Macros.v. There are three big advantages to this approach: first, using it consistently alerts readers that a macro is being used, and second, using names makes it much easier to create many macros compared to inventing ASCII syntax, and third, starting every macro with a keyword makes them much easier to get parsing correctly.
To declare an axiomatic instance of a typeclass, use Declare Instance foo : TypeClass
. This better than the pattern of Axiom
+ Existing Instance
.
To make Ltac scripts more readable, you can use Set Default Goal Selector "!".
, which will enforce that every Ltac command (sentence) be applied to exactly one focused goal. You achieve that by using a combination of bullets and braces. As a result, when reading a script you can always see the flow of where multiple goals are generated and solved.
Arguments foo _ & _
(in Coq 8.11) adds a bidirectionality hint saying that an application of foo
should infer a type from its arguments after typing the first argument. See src/Bidirectional.v for an example and the latest Coq documentation.
Coq 8.11 introduced compiled interfaces, aka vos
files (as far as I can tell there are a more principled replacement for vio
files). Suppose you make a change deep down to Lib.v
and want to start working on Proof.v
which imports Lib.v
through many dependencies. With vos
files, you can recompile all the signatures that Proof.v
depends on, skippinng proofs, and keep working. The basic way to use them is to compile Proof.required_vos
, a special dependency coqdep
generates that will build everything needed to work on Proof.v
. Coq natively looks for vos
files in interactive mode, and uses empty vos
files to indicate that the file is fully compiled in a vo
file.
Note that Coq also has vok
files; it's possible to check the missing proofs in a vos
file, but this does not produce a vo
and so all Coq can do is record that the proofs have been checked. They can also be compiled in parallel within a single file, although I don't know how to do that. Compiling vok
s lets you fairly confidently check proofs, but to really check everything (particularly universe constraints) you need to build vo
files from scratch.
Signature files have one big caveat: if Coq cannot determine the type of a theorem or the proof ends with Defined
(and thus might be relevant to later type-checking), it has to run the proof. It does so silently, potentially eliminating any performance benefit. The main reason this happens is due to proofs in a section that don't annotate which section variables are used with Proof using
. Generally this can be fixed with Set Default Proof Using "Type"
, though only on Coq master and not on Coq 8.11.0.
Coq 8.12+alpha has a new feature Set Printing Parentheses
that prints parentheses as if no notations had an associativity. For example, this will print (1,2,3)
as ((1,2),3)
. This is much more readable than entirely disabling notations.
You can use Export Set
to set options in a way that affects any file directly importing the file (but not transitively importing, the way Global Set
works). This allows a project to locally set up defaults with an options.v
file with all of its options, which every file imports. You can use this for basic sanity settings, like Set Default Proof Using "Type".
and Set Default Goal Selector "!"
without forcing them on all projects that import your project.
You can use all: fail "goals remaining".
to assert that a proof is complete. This is useful when you'll use Admitted.
but want to document (and check in CI) that the proof is complete other than the admit.
tactics used.
You can also use Fail idtac.
to assert that a proof is complete, which is shorter than the above but more arcane.
You can use Fail Fail Qed.
to really assert that a proof is complete, including doing universe checks, but then still be able to Restart
it. I think this is only useful for illustrating small examples but it's amusing that it works. (The new Succeed
vernacular in Coq 8.15 is a better replacement, because it preserves error messages on failure.)
Hints can now be set to global, export, or local. Global (the current default) means the hint applies to any module that transitivity imports this one; export makes the hint visible only if the caller imports this module directly. The behavior will eventually change for hints at the top-level so that they become export instead of global (see https://github.com/coq/coq/pull/13384), so this might be worth understanding now. HintLocality.v walks through what the three levels do.
-noinit
to coqc
or coqtop
to avoid loading the standard library.Declare ML Module "ltac_plugin".
(see NoInit.v).Require Import Coq.Init.Datatypes
.