Alpha Announcement: Coq is a Lean Typechecker


Presenting a custom version of Coq, extended with definitional UIP
(using the version at and
providing a brand new Lean Import command.

It is available at

This is an experimental alpha, it is useful to compare how Lean and
Coq work but probably not much beyond that.

The Lean Import command itself can be implemented as a plugin (and
will be once a few changes are upstreamed), but typechecking of Lean
terms requires UIP.

How do I install this?

Use your favorite installation procedure for Coq, using for the source.

How do I use this?

You need Lean exported files
as input.
(I used Lean version 3.4.2, but this format looks pretty stable so other versions should work)

For your convenience, I have uploaded a few examples:

  • core.out (256KB)
    This corresponds to core.lean in Lean’s stdlib.
    It contains 584 basic definitions and inductives such as equality, natural numbers, and the primitive quotient type (without the soundness axiom).
  • stdlib.out (14MB)
    This is the whole Lean stdlib, totalling 10244 definitions and inductives.
  • mathlib.7z (211MB compressed to 36MB)
    All mathlib (AFAICT): 66400 definitions and inductives (the way I counted may differ a bit from the way Lean counts).

Then start Coq and run

Require Lean.

Lean Import "/path/to/stdlib.out".

(if using the mathlib.7z I uploaded, make sure to decompress it first)

Coq will output many messages:

line 20: eq
eq is predeclared
line 21: quot
quot registered
line 38: has_add
line 59: has_add.add
line 90: add_semigroup
line 125: add_semigroup.add
line 136: add_semigroup.to_has_add
line 145: has_zero
line 159:
line 168: has_mul
line 182: has_mul.mul
line 210: semigroup
line 243: semigroup.mul
line 253: semigroup.to_has_mul
line 262: has_one
line 276:
line 285: has_le
line 305: has_le.le
line 323: list
line 331: nat
line 348: or

The Require Lean is needed for primitive quotient support (and
because quotients talk about equalities it predeclares eq). Without
it, you will get an error when trying to declare any value which
depends on quot.

Once it has finished working, Coq will output a summary:

line 586592: string.mk_iterator.equations._eqn_1
line 586615: heq.elim
- 10245 entries (24081 possible instances) (including quot).
- 274 universe expressions
- 14091 names
- 562009 expression nodes
Max universe instance length 4.
0 inductives have non syntactically arity types.

An “entry” means an axiom or constant, or an inductive type (including
its constructors and recursor), or the primitive quotient declarations
quot,, quot.lift and quot.ind.

For ease of debugging, Lean Import will succeed even when an error
occurs: this allows inspecting the state from just before the failing entry.
This may probably be changed to only happen in a debugging mode at some point.

How does it work?

The basic idea is to translate Lean Prop to Coq SProp, inductives
to inductives, etc.

We need to deal with a few issues in the translation:

Differences in name handling

Lean has . separated namespaces, so we can have foo.c depend on
bar.b which depends on foo.a. This cannot be done with Coq
modules. Instead we replace dots by underscores, and add some indexing
to deal with collisions.

For instance,

inductive nat
| zero : nat
| succ (n : nat) : nat


Inductive nat := nat_zero | nat_succ (n : nat).

then if Lean declared a nat_succ it would get renamed to nat_succ0.

Prop instantiations

Lean provides universe polymorphic values where universes may be
instantiated by Prop. For instance

inductive psum (α : Sort u) (β : Sort v)
| inl {} (val : α) : psum
| inr {} (val : β) : psum

provides discriminated sums of relevant as well as irrelevant types:
we can have psum@{0 0} true true as well as psum@{1 1} nat bool or
even psum@{0 1} true nat.

In Coq this is not possible. Instead we duplicate every value
according to which universes are instantiated to Prop. This
duplication is what the “possible instances” refers to in the end of
processing summary.

The version where no universe is Prop is considered the default and
gets the base name. The others have _instX appended to their name,
where X is the decimal representation of the number where bit n is
set if and only if universe n is instantiated by Prop. (this
naming scheme is subject to change)

By default, we produce the base instance, and the others are produced
as needed when encountered in other base instances. In other words,
when encountering the entry for psum we declare

Inductive psum (α:Type) (β:Type) := psum_inl (val:α) | psum_inr (val:β).

Then if we later encounter def bla := ... psum@{0 u} ... we will produce

Inductive psum_inst1 (α:Type) (β:Type) := psum_inl_inst1 (val:α) | psum_inr_inst1 (val:β).

and the same for psum_inst2 and psum_inst3.

This lazyness has an exception: each instance of an inductive type
with large elimination has 2 instances of the recursor, depending on
whether we’re eliminating to a Prop motive. These 2 instances are
always declared, so we don’t wait until psum_rec_inst1 is needed to
declare it.

Instances may be eagerly declared by using Set Lean Upfront Instantiation.

Algebraic universes

Lean uses non cumulative universes, such that Π (x:A), B lives
exactly in the impredicative maximum of the domain and codomain
universes: imax(uA,uB). We also get max in the level of inductive

Thanks to the previous section, every universe name can be determined
to be either SProp or strictly greater than Set, so we can reduce
universe expressions to Coq algebraic universes. However this leaves
us with 2 issues:

  • Coq expects universes in terms to be atomic, except in the codomain
    of the type of a global declaration. This is mostly required for the
    elaborator, so we could ignore it.

  • Coq universe polymorphic values may only be instantiated by atomic
    universes (and, as we mentioned, ones which are not SProp).
    However Lean can (and must, due to lack of cumulativity) instantiate
    polymorphic universes with arbitrary expressions.

So we need to replace algebraic universes in universe instances by
some atomic name. In order to preserve conversions, we also need to
replace algebraic universes in terms (so for instance if we have def univ@{u} := Sort u, the translation of univ@{max(u,v)} must be
convertible to Sort (max (u,v))).

The full process of translating a universe expression is then:

  • first, produce a Coq algebraic universe:
    • Prop is translated to SProp
    • a+1 is translated to the Coq successor of the translation of a
      (note that the successor of SProp is Set + 1, not SProp + 1
      which is invalid)
    • a Lean universe parameter is translated to SProp or a Coq named
      universe (depending on which instance we are currently producing)
    • max(a,b) is translated to the max of the respective translations
    • imax(a,b) is translated to SProp if b is translated to
      SProp, otherwise to the max of the respective translations.

We also need to make sure that every universe parameter not
instantiated by Prop is considered strictly greater than Set. This
is because Lean recognizes that imax(Prop+1,l)+1 <= max(Prop+1,l+1):
either l=Prop, in which case the problem reduces to Prop+1 <= Prop+1, or l=l'+1, in which case it reduces to l'+1 <= l'+2.
However in our translation we reduced to max(Set+1,l)+1 <= max(Set+1,l+1) which is only true when Set < l.

To ensure this we keep constraints Set < l for every universe
parameter l, and we also apply a simplification step to the
translated universes which removes any Set+n subexpression when it
is together with a l+k with n <= k + 1.

We then associate a unique surrogate name for each simplified
algebraic universe.

At the end, we will produce a top-level universe polymorphic value
with the original parameters appended with the surrogates. It will
have constraints such that Set < l for each original parameter l,
and each pair of parameter (surrogate or original) is related by any
constraint relating its corresponding algebraic universes. For
instance, if AB is the surrogate for max(a,b) and ABC is the
surrogate for max(a,b,c) we must have all of a <= AB, b <= AB, a <= ABC, b <= ABC, c <= ABC and AB <= ABC.

Since we have added universe parameters, we must adapt instances in
terms accordingly: if a definition foo@{u} is translated to foo@{u, U1} where U1 is the surrogate for max(Set+2,u), its use as
foo@{max(a,b)} must be translated to foo@{AB AB2} where AB is the
surrogate for max(a,b) and AB2 is the surrogate for

By default, surrogate names are based on their corresponding universe.
For instance the surrogate for max(u,v) is Lean.max__u_v.0. If a
strangely-crafted input uses this to cause collisions, you can Unset Lean Fancy Universes to get guaranteed unique names bla.XXX where
bla is the current file and XXX some unique index.

Note that once the kernel has accepted a declaration the universe
names are used only for printing.


Even with Coq accepting UIP, the rules for which inductives enjoy
unrestricted eliminations are different between Coq and Lean.
Typically, the accessibility predicate Acc is unrestricted in Lean
but is not accepted in SProp by Coq.

This is because it leads to an undecidable theory (of course we now
know that UIP combined with impredicativity is enough for that).

The workaround is simple: we detect when Coq is stricter than Lean and
in that case disable universe checking while declaring the inductive.

Sadly this is not enough to make the translation work in Coq without
UIP, because such a Coq also lacks the special reduction of the
eliminator of equality.

Note that this translation breaks extraction: for instance the recursor
of the translated acc cannot be extracted. However a more careful
translation could take advantage of Coq’s non recursively uniform
parameter feature to fix extraction.

We may note that Lean is sometimes stricter than Coq. Specifically, if
an inductive has a Prop and a non-Prop instantiation, it may
happen that Coq only squashes the Prop instantiation.

Primitive quotients

Lean’s quotient primitives are

constant quot {α : Sort u} (r : α → α → Prop) : Sort u

constant {α : Sort u} (r : α → α → Prop) (a : α) : quot r

constant quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) :
  (∀ a b : α, r a b → eq (f a) (f b)) → quot r → β

constant quot.ind {α : Sort u} {r : α → α → Prop} {β : quot r → Prop} :
  (∀ a : α, β ( r a)) → ∀ q : quot r, β q

with the appropriate reductions.

Coq can emulate this using “Private inductive types”. This emulation
has been done for you in the Coq module named Coq.Lean: simply
Require it before running Lean Import.

Because the lift mentions equality, Coq.Lean also predeclares the
equality type (we can’t use the one from Coq’s standard library since
it’s not polymorphic). Declaring the eliminators for equality is still
done by Lean Import.

Additional note on recursors

Coq autogenerates recursors for inductives types called (for an
inductive foo) foo_sind, foo_ind, foo_rec and foo_rect
(respectively for SProp, Prop, Set and Type motives).
These names are automatically detected by tactics like induction.

When the inductive is universe polymorphic (which is always the case
for our translations) the recursors are also universe polymorphic, and
notably the motive of the _rect version is a universe parameter.

However we cannot directly reuse the generated _sind and _rect
recursors as the 2 instantiations of the translated Lean recursor:

  • In Lean the motive universe is the first parameter, in Coq it is the
    last. This is could be handled during the translation though.

  • Coq generates non dependent eliminators for SProp inductives, but
    if the original Lean inductive has a non-Prop instantiation Lean
    expects a dependent eliminator.

  • Each recursive argument of each constructor corresponds to an
    inductive hypothesis in the function for the branch of that
    constructor (the first P n in nat_rect : forall P : nat -> Type, P 0 -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n). In
    Coq each inductive hypothesis comes immediately after the recursive
    argument, but in Lean the inductive hypotheses come after all the
    constructor argument.

    This produces different types when a recursive argument is not the
    last constructor argument, for instance with

    Inductive bin_tree := Leaf | Node (a b : bin_tree).

    Coq generates

       : forall P : bin_tree -> Type,
         P Leaf ->
         (forall a : bin_tree, P a -> forall b : bin_tree, P b -> P (Node a b)) ->
         forall b : bin_tree, P b

    but Lean expects

      : forall P : bin_tree -> Type,
        P Leaf ->
        (forall a b : bin_tree, P a -> P b -> P (Node a b)) ->
        forall b : bin_tree, P b

To avoid these issues, we explicitly ask Coq for a term implementing
the recursor with the expected dependency, then post-process it to fix
universe and argument order. Since the result may not be compatible
with induction's expectation, we use our own suffixes _indl and
_recl (l for Lean).

Since we use _indl for the Prop-motive recursor, any _inst
suffix corresponds to the instantiation of the inductive we eliminate.
For instance psum_inst3_indl is instance 5 (all universes Prop) of
psum.rec, its principal argument is of type psum_inst3.

Experimental results

All times are on my laptop, which may have caused variance through
thermal throttling or whatever.

The export for just core.lean passes without issue in about 2s.

The whole stdlib cannot be checked as some conversion problems are
pathological. two_pos seems a typical example (0 < 2 in an ordered
field). It’s interesting to note that on this specific example,
changing the default conversion procedure to use Coq’s VM makes it
succeeds in about 1 second (tested by importing with Unset Conversion Checking (see next section), then Require Import the resulting
.vo and do Definition foo := Eval cbv [two_pos] in two_pos.).
Sadly using the VM makes other declarations take too long, and anyway
it hasn’t been updated for proof irrelevance and for UIP’s special

As a superset of the stdlib, mathlib also cannot be fully checked.
Worse, even with Unset Conversion Checking it tries to use more RAM
and takes longer than I was willing to try.

Some stats:

  • stdlib:
    lean --export in 46s, about 450MB RAM
    leanchecker in 8s, 80MB RAM

    Lean Import with Unset Conversion Checking: 43s, 723MB RAM
    resulting vo size 53MB

    Lean Import with 10s line timeout: 451s, 720MB RAM
    resulting vo size 50MB
    89 skipped entries out of 10244 (32 timeout, rest due to missing value from previous errors)

  • mathlib:
    lean --export: didn’t measure, took long enough and enough RAM that I don’t want to retry
    (at least 1h / 10GB RAM I guess)
    leanchecker: 6min, 1GB RAM

    Lean Import with Just Parsing: 347s, 745MB RAM

    Lean Import with Unset Conversion Checking: killed at 4GB RAM on filter_mem_inf_sets_of_right

    Lean Import with Unset Conversion Checking and 10s timeout: 1h13min, 10GB RAM
    resulting vo size 1.4GB
    11867 skipped entries out of 66400 (first one is real.linear_order._proof_5)


Conversion Checking

On by default. This is a new kernel option, only available in this Coq

With it off, the kernel will skip conversion checks. This is useful to
get past declarations where Coq’s heuristics don’t work without
breaking their dependents.

Lean Fancy Universes

See explanation of surrogate universe names.

Lean Upfront Instantiation

See explanation of universe polymorphism and Prop.

Lean Skip Missing Quotient

On by default, this means that encountering the primitive quotient
entry when the primitive quotient has not been predeclared is not an
error (i.e. when Coq.Lean is not loaded).

This means you will instead get a quot was not instantiated error
when a declaration refers to it.

Lean Just Parsing

Off by default, if on Lean Import will not actually translate
anything. Useful to get the summary of how many entries are
encountered quickly.

Lean Print Squash Info

Off by default, this may be useful for debugging if Lean Import
misdetects whether Lean would allow unrestricted elimination for some
inductive type.

Lean Skip Errors

Off by default. With it on, when an error is encountered, skip the
failed line and keep going.

Useful to tell how much the current system can handle.

Note that timeouts and interrupts are also absorbed by this option. If
you turn it on and start loading mathlib, then change your mind and
decide to stop, you will need to kill the Coq process.

Lean Line Timeout

An integer option, off by default. Use Set Lean Line Timeout 10. to
cause a failure whenever some entry takes more than 10s. Combined with
Lean Skip Errors, this allows processing all the entries which do
not depend on something that takes more than 10s.


Very cool. This really helps understanding the differences.

Why is this disallowed in Coq? What would break if we added it?

How often is this feature used in lean? As you point out it has quite negative consequences.

Any thoughts on what happens here with native_compute or certicoq?

Prop instantiations

In Coq instantiating with either Prop or SProp is forbidden (for the translation we only care about SProp and never touch Prop)
AFAICT instantiating with Prop is mostly a matter of changing stuff so that we do typechecking without ambient Set <= u constraints by default. It wouldn’t be complete (ie Definition product would land in the max universe since we don’t have imax, and imax seems harder to do when you have to generate cumulativity constraints and refresh universes) but I guess that’s OK.
For SProp it’s harder, because if the definition requires a constraint u < v we don’t know if the implicit u <= v was used (in which case we can’t instantiate u by SProp).


How often is this feature used in lean? As you point out it has quite negative consequences.

Not sure tbh

Any thoughts on what happens here with native_compute or certicoq?

What do you mean?

Updated stdlib stats after fixing the issue examplified by two_pos (we need to check for proof irrelevance before comparing structurally):

Lean Import with 10s timeout, set upfront instances and set conv check:
320s, 1GB RAM
vo size 111MB
11 skips (3 timeout, 8 not instantiated)

The timeouts are bla.equations._eqn_1, where bla in

  • tactic.delta_config.max_steps._default
  • char.of_nat
  • simp.default_max_steps
    These all involve moderately sized integers (of_nat through char.is_valid)

I actually already did a hack for std.prec.max.equations._eqn_1, setting prec.max to the Expand reduction strategy. That may be correct for the max_steps but doesn’t seem right for of_nat.

The stdlib now fully passes after copying Lean’s definitional height unfolding heuristic. Current stats:

  • 284s runtime, 1.3GB RAM usage
  • resulting vo takes 116MB
  • coqchk takes 540s and 1.1GB RAM
    If we skip structure validation coqchk takes 51s and 720MB RAM
    If we use OCaml’s Marshal instead of our custom Analyze coqchk takes 45s and 380MB RAM
1 Like

You said things improved with vm_compute, so I was wondering about other evaluation machinery, such as native_compute or certicoq.