samuelgruetter / coq-tricks

Collection of Useful Features of the Coq Proof Assistant

Collection of Useful Features of the Coq Proof Assistant

Goal of this document: Let's make every Coq user's "OMG I wish I had known this before" moments happen as early as possible 😉

Pull requests are welcome!

Ltac

lazymatch

In an Ltac match, a branch is taken only if the pattern matches and the rhs tactic succeeds. If we want the decision to depend only on the pattern, but not on whether the rhs tactic succeeds, we can use lazymatch instead.

An important advantage of lazymatch compared to match is that it doesn't swallow errors, turning them into a generic "No matching clauses for match.". For instance, if pat1 and pat2 are disjoint, and x matches pat1, but tac1 fails, the following code will fail with "No matching clauses for match."

match x with
| pat1 => tac1
| pat2 => tac2
end.

However, if we replace match by lazymatch, it will fail with the error message of tac1.

control flow

tryif tac1 then tac2 else tac3

Runs tac1, if tac1 succeeds, continues with tac2, else reverts the effects of tac1 and continues with tac3.

Note: If the else branch is a sequence of several commands separated by a semicolon, they have to be wrapped in parentheses, to get the operator precedence right.

assert_fails tac1

Runs tac1, always reverts the effects of tac1, and succeeds if tac1 failed, and fails if tac1 succeeded.

assert_succeeds tac1

Runs tac1, always reverts the effects of tac1, and succeeds if tac1 succeeded, and fails if tac1 failed.

unify term1 term2
constr_eq term1 term2

unify checks that two terms are unifiable, potentially instantiating evars.

constr_eq only does a syntactic comparison (modulo alpha-conversion and casts).

However, we often want a side-effect free unification test, ie. unification which does not instantiate evars:

assert_fails (has_evar term1); assert_fails (has_evar term2); unify term1 term2.

is_var term1
is_evar term1

succeeds if term1 is a variable or evar, respectively.

More useful tactics:

• edestruct
• epose proof
• unshelve
• partial specialize: specialize H with (a := foo) (2 := H3)

And list of term classification tactics, many of which are undocumented:

• is_evar
• has_evar
• is_hyp
• is_fix
• is_cofix
• is_ind
• is_constructor
• is_proj
• is_const

Tactic redefinition with ::=

Ltac tac_name ::= ...

Overrides a previously defined tactic tac_name with a new implementation.

Useful to make libraries user-configurable, and to debug libraries without recompiling them.

Ltac Profiler

Coq has an Ltac profiler which shows for each Ltac function how much time it took.

It will be part of Coq 8.6. For earlier versions, you can get it by applying a patch to the Coq source code, as described on the LtacProf website.

Vernacular

Search

Search ident. lists all lemmas mentioning ident.

But Search is much more powerful than just that:

• Search ident1 ... identN. lists all lemmas mentioning all of ident1 ... identN.

• Search pattern. lists all lemmas in which pattern occurs. For instance, you can

  Search (length (skipn _ _)).
  

Custom reduction strategies

Declare Reduction foo := cbv [length app].

Eval foo in (length (app (cons 1 nil) (cons 2 nil))).

Tools

coqwc

coqwc is a stand-alone command line tool to print line statistics about Coq files (how many lines are spec, proof, comments).