Alpha Announcement: Coq is a Lean Typechecker

Warning

Currently in the process of converting to a plugin, use tag lean-import-alpha1 at https://github.com/SkySkimmer/coq instead.

Introduction

Presenting a custom version of Coq, extended with definitional UIP (using the version at coq/coq#10390) and providing a brand new Lean Import command.

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

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

  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).

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 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

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.