Section Generated_monoid_def.
Variable M : MONOID.
Variable A : part_set M.
Definition generated_monoid : submonoid M :=
image_monoid_hom (FM_lift (inj_part A)).
End Generated_monoid_def.
unfold included in |- *.
simpl in |- *.
intros M A H H' x H'0; try assumption.
elim H'0; intros x0; clear H'0.
generalize x; clear x.
elim x0.
intros c; try assumption.
elim c.
simpl in |- *.
intros y subtype_prf x H'0; elim H'0; intros H'1 H'2; try exact H'2;
clear H'0.
apply in_part_comp_l with y; auto with algebra.
intros f H'0 f0 H'1 x H'2; elim H'2; intros H'3 H'4; try exact H'4; clear H'2.
simpl in H'4.
apply
in_part_comp_l
with
(sgroup_law M (FM_lift_fun (inj_part A) f) (FM_lift_fun (inj_part A) f0));
auto with algebra.
simpl in |- *.
intros x H'0; elim H'0; intros H'1 H'2; try exact H'2; clear H'0.
apply in_part_comp_l with (monoid_unit M); auto with algebra.
Qed.
unfold included in |- *.
simpl in |- *.
intros M A x H'; try assumption.
exists (Var (V:=A) (Build_subtype (E:=M) (P:=A) (subtype_elt:=x) H')); split;
[ idtac | try assumption ].
auto with algebra.
simpl in |- *.
auto with algebra.
Qed.

Lemma generated_monoid_prop :
forall (M : MONOID) (A : part_set M) (y : M),
in_part y (generated_monoid A) ->
exists x : FM A, Equal y (FM_lift (inj_part A) x).