The algebra library

This is a central part of the mathematical components library.
This library register a various range of (mathematical) structures.
- types with decidable equality: eqType
- types with a finite number of elements: finType
- finite groups: finGroupType
- abelian (not possibly finite) groups: zmodType
- rings: ringType
- rings with invertible elements: unitRingType
- commutative rings: comRingType
- integral domains: idomainType
- fields: fieldType
- left modules: lmodType
- left algebra: lalgType
- algebra: algType
- finite dimensional vector spaces: vectType
- ...
Some of these structures share operators: e.g. the operator (_ + _), introduced for abelian groups (zmodType), is also available for all the structures that require it (rings, domains, fields, etc...)
All of these structures are discrete: they all have decidable equality: the operator (_ == _) is available on all of them.
Here is a picture of the begining of the hierarchy Extensive documentation in the header of:
- ssralg
- ssrnum
In addition there are structures for maps (additive morphisms, ring morphisms, etc...), and substructures (subgroup, subsemiring, subring, subfield, etc...)

Roadmap for the lesson:

introduction of the general definition pattern for algebraic structures
instantiation of a structure in the library
exploration of the theory provided by this structure and naming conventions
creation of a subalgebraic structure predicate and use

Defining a mathematical structure in Coq.

This is how mathematical structures are defined in the library. Unless you need to add a new mathematical structure to the library, you will only need to read this.

This packaging is very elementary, and the mathematical components library uses a refinement of this.

Packaging mathematical structures

We briefly explain how to do inheritance with two structures. This is another simplified version of what happens in the library. The complete process is described in Packaging Mathematical Structures (Garillot, Gonthier, Mahboubi, Rideau) and in the draft book http://math-comp.github.io/mcb/>Book.

Section AlgebraicStructuresInheritance.

Record my_mixin1 dom := My_mixin1 {
(* symbols: constants and operators *)
  my_c    : dom;
  my_op   : dom -> dom -> dom;
(* axioms: properties on the symbols *)
  my_opxc : forall x, my_op x my_c = my_c;
  my_opC : forall x y, my_op x y = my_op y x}.

Definition my_class1 := my_mixin1.

Structure my_struct1 := My_struct1 {
(* domain/carrier/sort of the structure *)
  dom1 :> Type;
  class1 : my_mixin1 dom1}.

Definition op {s : my_struct1} := my_op (class1 s).
Definition c {s : my_struct1} := my_c (class1 s).
Notation "x * y" := (op x y).

Record my_mixin2 (dom : my_struct1) := My_mixin2 {
(* symbols: constants and operators *)
  my_pred   : pred dom;
(* axioms: properties on the symbols *)
  my_predc : forall x, my_pred (op c x)}.

Record my_class2 dom := My_class2 {
  base2 :> my_class1 dom;
  mixin2 :> my_mixin2 (@My_struct1 dom base2)}.

Structure my_struct2 := My_struct2 {
  dom2  :> Type;
  class2 : my_class2 dom2}.

Definition pr {s : my_struct2} := my_pred (class2 s).

Canonical my_struct1_2 (s : my_struct2) : my_struct1 :=
  @My_struct1 (dom2 s) (class2 s : my_class1 _).

Section my_struct1_theory.

Variable s : my_struct1.

Lemma opC (x y : s) : x * y = y * x.
Proof. by case: s x y => ? []. Qed.

Lemma opxc (x : s) : x * c = c.
Proof. by case: s x => ? []. Qed.

(* Theory of the structure *)
Lemma opcx (x : s) : c * x = c.
Proof. by rewrite opC opxc. Qed.

End my_struct1_theory.

Section my_struct2_theory.

Variable s : my_struct2.

Lemma pr_xc (x : s) : pr (c * x).
Proof. by case: s x => ? [? []]. Qed.

Lemma pr_c : pr (c : s).
Proof. by rewrite -(opcx c) pr_xc. Qed.

End my_struct2_theory.

End AlgebraicStructuresInheritance.

Inhabiting the mathematical structures hierarchy.

We now show on the example of integers how to instantiate the mathematical structures that integers satisfy.
As a step to minimize the work of the user, the library provides a way to provide only the mixin. The general pattern is to build the mixin of a structure, declare the canonical structure associated with it and go one with creating the next mixin and creating the new structure. Each time we build a new structure, we provide only the mixin, as the class can be inferred from the previous canonical structures
We now show three different ways to build mixins here and an additional fourth will be shown in the exercices
- using a reference structure,
- building the required mixin from scratch,
- building a more informative mixin and using it for a weaker structure,
- (in the example) by subtyping.

First we define int

Equality, countable and choice types, by injection

We provide an injection with explicit partial inverse, grom int to nat + nat, this will be enough to provide the mixins for equality, countable and choice types.

We create the mixins for equality, countable and choice types from this injection, and gradually inhabit the hierarchy. Try to swap any of the three blocks to see what happen.

Abelian group structure, from scratch

We now create the abelian group structure of integers (here called Z-module), from scratch, introducing the operators and proving exactly the required properties.

Remark: we may develop here a piece of abelian group theory which is specific to the theory of integers.

Ring and Commutative ring structure, the stronger the better

This time, we will build directly a rich commutative ring mixin first and use it to instanciate both the ring structure and the commutative ring struture at the same time. This is not only a structural economy of space, but a mathematical economy of proofs, since the commutativity property reduces the number of ring axioms to prove.

Other structures

Extensions of rings

read the documentation of ssralg and ssrnum (algebraic structures with order)

Structures for morphisms

Substructures

Naming conventions.

The two most important things to get your way out of a situation:

Knowing your math
Searching the library for what you think you know

For that you have the ssreflect Search command. To use its full power, one should combine seach by identifier (Coq definition), pattern (partial terms) and name (a string contained in the name of the theorem).

The two first methods are straightforward to use (except if you instanciate your patterns more than necessary), but searching by name requires to be aware of naming conventions.

Names in the library are usually obeying one of following the convention

```
(condition_)?mainSymbol_suffixes
```
```
mainSymbol_suffixes(_condition)?
```

Or in the presence of a property denoted by a nary or unary predicate:

```
naryPredicate_mainSymbol+
```
```
mainSymbol_unaryPredicate
```

Where :

mainSymbol is the most meaningful part of the lemma. It generally is the head symbol of the right-hand side of an equation or the head symbol of a theorem. It can also simply be the main object of study, head symbol or not. It is usually either
- one of the main symbols of the theory at hand. For example, it will be opp, add, mul, etc...
- or a special canonical operation, such as a ring morphism or a subtype predicate. e.g. linear, raddf, rmorph, rpred, etc ...
condition is used when the lemma applies under some hypothesis.
suffixes are there to refine what shape and/or what other symbols the lemma has. It can either be the name of a symbol (add, mul, etc...), or the (short) name of a predicate (inj for injectivity, id for identity, ...) or an abbreviation.

Abbreviations are in the header of the file which introduce them. We list here the main abbreviations.

A -- associativity, as in andbA : associative andb.
AC -- right commutativity.
ACA -- self-interchange (inner commutativity), e.g., orbACA : (a || b) || (c || d) = (a || c) || (b || d).
b -- a boolean argument, as in andbb : idempotent andb.
C -- commutativity, as in andbC : commutative andb, or predicate complement, as in predC.
CA -- left commutativity.
D -- predicate difference, as in predD.
E -- elimination, as in negbFE : ~~ b = false -> b.
F or f -- boolean false, as in andbF : b && false = false.
I -- left/right injectivity, as in addbI : right_injective addb or predicate intersection, as in predI.
l -- a left-hand operation, as andb_orl : left_distributive andb orb.
N or n -- boolean negation, as in andbN : a && (~~ a) = false.
n -- alternatively, it is a natural number argument
P -- a characteristic property, often a reflection lemma, as in andP : reflect (a /\ b) (a && b).
r -- a right-hand operation, as orb_andr : right_distributive orb andb.
- - alternatively, it is a ring argument
T or t -- boolean truth, as in andbT: right_id true andb.
U -- predicate union, as in predU.
W -- weakening, as in in1W : {in D, forall x, P} -> forall x, P.
0 -- ring 0, as in addr0 : x + 0 = x.
1 -- ring 1, as in mulr1 : x * 1 = x.
D -- ring addition, as in linearD : f (u + v) = f u + f v.
B -- ring subtraction, as in opprB : - (x - y) = y - x.
M -- ring multiplication, as in invfM : (x * y)^-1 = x^-1 * y^-1.
Mn -- ring by nat multiplication, as in raddfMn : f (x *+ n) = f x *+ n.
mx -- a matrix argument
N -- ring opposite, as in mulNr : (- x) * y = - (x * y).
V -- ring inverse, as in mulVr : x^-1 * x = 1.
X -- ring exponentiation, as in rmorphX : f (x ^+ n) = f x ^+ n.
Z -- (left) module scaling, as in linearZ : f (a *: v) = s *: f v.
z -- a int operation

My most used search pattern

Search _ "prefix" "suffix"* (symbol|pattern)* in library.

Examples

A short parenthesis on subtyping.

In Coq, sT := {x : T | P x} is a way to form a sigma-type. We say it is a subtype if there is only one element it sT for each element in T. This happens when P is a boolean predicate. Another way to form a subtype is to create a record : Record sT := ST {x : T; px : P x}.
In mathcomp, to deal with subtypes independently from how they are form, we have a ganonical structure.

The most important operators to know on subtypes are val : sT -> T, insub : T -> option sT and insubd : sT -> T -> sT.
And the most important theorems to know are val_inj, val_eqE, val_insubd, insubdK and insub

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