∀G:finite_enumerable_SemiGroup.
left_cancellable ? (op G) →
right_cancellable ? (op G) →
- ∃e:G. isMonoid (mk_PreMonoid G e).
+ ∃e:G. IsMonoid (mk_PreMonoid G e).
intros;
letin f ≝(λn.ι(G \sub O · G \sub n));
cut (∀n.n ≤ order ? (is_finite_enumerable G) → ∃m.f m = n);
clearbody GOGO;
rewrite < HH in GOGO;
rewrite < HH in GOGO:(? ? % ?);
- rewrite > (op_associative ? G) in GOGO;
+ rewrite > (op_is_associative ? G) in GOGO;
letin GaGa ≝(H ? ? ? GOGO);
clearbody GaGa;
clear GOGO;
constructor 1;
[ simplify;
- apply (semigroup_properties G)
+ apply (is_semi_group G)
| unfold is_left_unit; intro;
letin GaxGax ≝(refl_eq ? (G \sub a ·x));
clearbody GaxGax; (* demo *)
rewrite < GaGa in GaxGax:(? ? % ?);
- rewrite > (op_associative ? (semigroup_properties G)) in GaxGax;
+ rewrite > (op_is_associative ? G) in GaxGax;
apply (H ? ? ? GaxGax)
| unfold is_right_unit; intro;
letin GaxGax ≝(refl_eq ? (x·G \sub a));
clearbody GaxGax;
rewrite < GaGa in GaxGax:(? ? % ?);
- rewrite < (op_associative ? (semigroup_properties G)) in GaxGax;
+ rewrite < (op_is_associative ? G) in GaxGax;
apply (H1 ? ? ? GaxGax)
]
]