1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
|
Require Import ZArith ssreflect ssrfun.
From HB Require Import structures.
HB.mixin Record is_semigroup (S : Type) := {
add : S -> S -> S;
addrA : associative add;
}.
HB.structure Definition SemiGroup :=
{ S & is_semigroup S }.
HB.mixin Record semigroup_is_monoid M of is_semigroup M := {
zero : M;
add0r : forall x, add zero x = x;
addr0 : forall x, add x zero = x;
}.
HB.factory Record is_monoid (M : Type) := {
zero : M;
add : M -> M -> M;
addrA : associative add;
add0r : forall x, add zero x = x;
addr0 : forall x, add x zero = x;
}.
HB.builders Context (M : Type) (f : is_monoid M).
HB.instance Definition _ : is_semigroup M :=
is_semigroup.Build M add addrA.
HB.instance Definition _ : semigroup_is_monoid M :=
semigroup_is_monoid.Build M zero add0r addr0.
HB.end.
HB.structure Definition Monoid :=
{ M & is_monoid M }.
HB.instance Definition Z_is_monoid : is_monoid Z :=
is_monoid.Build Z 0%Z Z.add
Z.add_assoc Z.add_0_l Z.add_0_r.
|
