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https://github.com/math-comp/hierarchy-builder

High level commands to declare a hierarchy based on packed classes
https://github.com/math-comp/hierarchy-builder

coq elpi mathcomp

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High level commands to declare a hierarchy based on packed classes

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# Hierarchy Builder



Hierarchy Builder (HB) provides high level commands to declare a hierarchy of algebraic structure

(or interfaces if you prefer the glossary of computer science) for the Coq system.



Given a structure one can develop its theory, and that theory becomes automatically applicable to

all the examples of the structure. One can also declare alternative interfaces, for convenience

or backward compatibility, and provide glue code linking these interfaces to the structures part of

the hierarchy.



HB commands compile down to Coq modules, sections, records, coercions, canonical structure instances

and notations following the *packed classes* discipline which is at the core of the [Mathematical

Components](https://github.com/math-comp/math-comp) library. All that complexity is hidden behind

a few concepts and a few declarative Coq commands.



## Example



```coq

From HB Require Import structures.

From Coq Require Import ssreflect ZArith.



HB.mixin Record IsAddComoid A := {

  zero : A;

  add : A -> A -> A;

  addrA : forall x y z, add x (add y z) = add (add x y) z;

  addrC : forall x y, add x y = add y x;

  add0r : forall x, add zero x = x;

}.



HB.structure Definition AddComoid := { A of IsAddComoid A }.



Notation "0" := zero.

Infix "+" := add.



Check forall (M : AddComoid.type) (x : M), x + x = 0.

```



This is all we need to do in order to declare the `AddComoid` structure

and write statements in its signature.



We proceed by declaring how to obtain an Abelian group out of the

additive, commutative, monoid.



```coq

HB.mixin Record IsAbelianGrp A of IsAddComoid A := {

  opp : A -> A;

  addNr : forall x, opp x + x = 0;

}.



HB.structure Definition AbelianGrp := { A of IsAbelianGrp A & IsAddComoid A }.



Notation "- x" := (opp x).

```



Abelian groups feature the operations and properties given by the

`IsAbelianGrp` mixin (and its dependency `IsAddComoid`).



```coq

Lemma example (G : AbelianGrp.type) (x : G) : x + (- x) = - 0.

Proof. by rewrite addrC addNr -[LHS](addNr zero) addrC add0r. Qed.

```



We proceed by showing that `Z` is an example of both structures, and use

the lemma just proved on a statement about `Z`.



```coq

HB.instance Definition Z_CoMoid :=

  IsAddComoid.Build Z 0%Z Z.add Z.add_assoc Z.add_comm Z.add_0_l.

 

HB.instance Definition Z_AbGrp :=

  IsAbelianGrp.Build Z Z.opp Z.add_opp_diag_l.



Lemma example2 (x : Z) : x + (- x) = - 0.

Proof. by rewrite example. Qed.

```



## Documentation



This [paper](https://hal.inria.fr/hal-02478907) describes the language

in details, and the corresponding talk [is available on youtube](https://www.youtube.com/watch?v=F6iRaTlQrlo).

The [wiki](https://github.com/math-comp/hierarchy-builder/wiki) gathers some

tricks and FAQs. If you want to work on the implementation of HB, this

[recorded hacking session](https://www.youtube.com/watch?v=gmaJjCbzqO0) may be relevant to you.



### Installation & availability



(click to expand)



- You can install HB via OPAM



```shell

opam repo add coq-released https://coq.inria.fr/opam/released

opam install coq-hierarchy-builder

```



- You can use it in nix with the attribute `coqPackages_8_XX.hierarchy-builder` e.g.

  via `nix-shell -p coq_8_13 -p coqPackages_8_13.hierarchy-builder`

 




### Key concepts



(click to expand)



- a *mixin* is a bare bone building block of the hierarchy, it packs operations

  and axioms.



- a *factory* is a package of operations and properties that is elaborated by

  HB to one or more mixin. A mixin is hence a trivial factory.



- a *structure* is declared by attaching zero or more factories to a type.



- a *builder* is a user provided piece of code capable of

  building one or more mixins from a factory.



- an *instance* is an example of a structure: it provides all operation and

  fulfills all axioms.






### The commands of HB



(click to expand)



- HB core commands:

  - `HB.mixin` declares a mixin,

  - `HB.structure` declares a structure,

  - `HB.factory` declares a factory,

  - `HB.builders` and `HB.end` declare a set of builders,

  - `HB.instance` declares a structure instance,

  - `HB.declare` declares a context with parameters, key and mixins.

  - `HB.saturate` reconsiders all mixin instances to see if some newly declared

    structure can be inhabited



- HB core tactic-in-term:

  - `HB.pack` to synthesize a structure instance in the middle of a term.



- HB utility commands:

  - `HB.export` exports a module and schedules it for re-export

  - `HB.reexport` exports all modules, instances and constants scheduled for

    re-export

  - `HB.lock` locks a definition behind an opaque symbol and an unfolding

    equation using Coq module system



- HB queries:

  - `HB.about` is similar to `About` but prints more info on HB structures, like

    the known instances and where they are declared

  - `HB.locate` is similar to `Locate`, prints file name and line of any global

    constant synthesized by HB

  - `HB.graph` prints the structure hierarchy to a dot file

  - `HB.howto` prints sequences of factories to equip a type with a given structure



- HB debug commands:

  - `HB.status` dumps the contents of the hierarchy (debug purposes)

  - `HB.check` is similar to `Check` (testing purposes)



The documentation of all commands can be found in the comments of

[structures.v](structures.v), search for `Elpi Command` and you will

find them. All commands can be prefixed with the attribute `#[verbose]`

to get an idea of what they are doing.



For debugging and teaching purposes, passing the attributes

`#[log]` or `#[log(raw)]` to a HB command prints Coq commands which are

*almost* equivalent to its effect. Hence, copy-pasting the displayed commands into

your source file is not expected to work, and we strongly recommend

against it.






### Demos



(click to expand)



- [demo1](examples/demo1/) and [demo3](examples/demo3/) declare and evolve a hierarchy up to

  rings with various clients that are tested not to break when the hierarchy

  evolves

- [demo2](examples/demo2/) describes the subtle triangular interaction between groups,

  topological space and uniform spaces. Indeed, 1. all uniform spaces induce a

  topology, which makes them topological spaces, but 2. all topological groups

  (groups that are topological spaces such that the addition and opposite are

  continuous) induce a uniformity, which makes them uniform spaces. We solve

  this seamingly mutual dependency using HB.