This library contains an automatic translation in Coq of (for the moment) some small part the HOL-Light library using hol2dk and lambdapi.
As HOL-Light is based on higher-order logic, it assumes in Coq classical logic, Hilbert's ε operator, functional and propositional extensionnality (see coq.v for more details):
Axiom classic : forall P:Prop, P \/ ~ P.
Axiom constructive_indefinite_description : forall (A : Type) (P : A->Prop), (exists x, P x) -> { x : A | P x }.
Axiom fun_ext : forall {A B : Type} {f g : A -> B}, (forall x, (f x) = (g x)) -> f = g.
Axiom prop_ext : forall {P Q : Prop}, (P -> Q) -> (Q -> P) -> P = Q.
Axiom proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.
For the moment, it only contains the HOL-Light base library up to lists.ml, that is, basic theorems on first-order logic, on natural numbers arithmetic on the Coq functions pred, add, mul, pow, div, modulo, max, min, and the Coq predicates le, lt, ge, gt, Even, Odd, and on basic functions on lists.
We can translate to Coq much bigger parts of the HOL-Light library (at least all the base library hol.ml). However, to get a library that is directly usable in Coq, we need to align HOL-Light functions and predicates to the ones defined in Coq standard library. This requires to manually prove that the HOL-Light definitions are equal to the corresponding Coq definitions. This has been done for the above functions and predicates but remains to be done for other types like real numbers.
The translated theorems are provided as axioms in order to have fast Require's in Coq because the proofs currently extracted from HOL-Light are very big and not very informative for they are low level (the translation is done at the kernel level, not at the source level). If you are skeptical, you can however generate and check them again by using hol2dk.
The current library contains 667 lemmas. The corresponding Coq proof files have a size of 86 Mo and take about 4m to check. The whole HOL-Light base library hol.ml has 2834 theorems. The corresponding Coq proof files have a size of 1.1 Go and take about 31m to check (with 32 processors and 64 Go RAM). The full HOL-Light library has 36471 theorems.
Installation using opam
opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-hol-light
Usage in a Coq file
Require Import HOLLight.hol_light.
Check thm_DIV_DIV.
Reproducibility
First, install HOL-Light and hol2dk as described in README.md. Then, do run reproduce.sh. If every thing works well, the proofs will be in the directory reproduce/output.