Coq is a Lean Typechecker

Introduction

Presenting a custom version of Coq, extended with definitional UIP

(using the version at https://github.com/coq/coq/pull/10390 ) and

providing a brand new Lean Import command.

It is available at https://github.com/SkySkimmer/coq/tree/lean-import

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

https://github.com/SkySkimmer/coq/tree/lean-import 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: has_zero.zero
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: has_one.one
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
Done!
- 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.mk , 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

becomes

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

types.

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

max(Set+2,a,b) .

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.

Subsingletons

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 quot.mk {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 (quot.mk 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

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

  bin_tree_rect
     : 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

  bin_tree_rect
    : 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 ).

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

reduction.

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)

Options

Conversion Checking

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

variant.

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

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.

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).

subsingletons

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?