Set Implicit Arguments.
Unset Strict Implicit.
Require Export Group_util.
Section Free_group_def.
Variable V : SET.

Inductive FG : Type :=
  | Var : V → FG
  | Law : FG → FG → FG
  | Unit : FG
  | Inv : FG → FG.

Inductive eqFG : FG → FG → Prop :=
  | eqFG_Var : ∀ x y : V, Equal x y → (eqFG (Var x) (Var y):Prop)
  | eqFG_law :
      ∀ x x' y y' : FG,
      eqFG x x' → eqFG y y' → (eqFG (Law x y) (Law x' y'):Prop)
  | eqFG_law_assoc :
      ∀ x y z : FG, eqFG (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFG_law0r : ∀ x : FG, eqFG (Law x Unit) x:Prop
  | eqFG_inv : ∀ x y : FG, eqFG x y → eqFG (Inv x) (Inv y)
  | eqFG_invr : ∀ x : FG, eqFG (Law x (Inv x)) Unit
  | eqFG_refl : ∀ x : FG, eqFG x x:Prop
  | eqFG_sym : ∀ x y : FG, eqFG x y → (eqFG y x:Prop)
  | eqFG_trans : ∀ x y z : FG, eqFG x y → eqFG y z → (eqFG x z:Prop).
Hint Resolve eqFG_Var eqFG_law eqFG_law_assoc eqFG_law0r eqFG_invr eqFG_refl:
  algebra.
Hint Immediate eqFG_sym: algebra.

Lemma eqFG_Equiv : equivalence eqFG.
red in |- ×.
split; [ try assumption | idtac ].
exact eqFG_refl.
red in |- ×.
split; [ try assumption | idtac ].
exact eqFG_trans.
exact eqFG_sym.
Qed.

Definition FG_set := Build_Setoid eqFG_Equiv.

Definition FreeGroup : GROUP.
apply (BUILD_GROUP (E:=FG_set) (genlaw:=Law) (e:=Unit) (geninv:=Inv)).
exact eqFG_law.
exact eqFG_law_assoc.
exact eqFG_law0r.
exact eqFG_inv.
exact eqFG_invr.
Defined.
Section Universal_prop.
Variable G : GROUP.
Variable f : Hom V G.

Fixpoint FG_lift_fun (p : FreeGroup) : G :=
  match p with
  | Var v ⇒ f v
  | Law p1 p2 ⇒ sgroup_law _ (FG_lift_fun p1) (FG_lift_fun p2)
  | Unit ⇒ monoid_unit G
  | Inv p1 ⇒ group_inverse G (FG_lift_fun p1)
  end.

Definition FG_lift : Hom FreeGroup G.
apply (BUILD_HOM_GROUP (G:=FreeGroup) (G':=G) (ff:=FG_lift_fun)).
intros x y H'; try assumption.
elim H'; simpl in |- *; auto with algebra.
intros x0 y0 z H'0 H'1 H'2 H'3; try assumption.
apply Trans with (FG_lift_fun y0); auto with algebra.
simpl in |- *; auto with algebra.
simpl in |- *; auto with algebra.
Defined.

Definition FG_var : Hom V FreeGroup.
apply (Build_Map (A:=V) (B:=FreeGroup) (Ap:=Var)).
red in |- ×.
simpl in |- *; auto with algebra.
Defined.

Lemma FG_comp_prop :
 Equal f (comp_hom (FG_lift:Hom (FreeGroup:SET) G) FG_var).
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
Qed.
End Universal_prop.
End Free_group_def.
Hint Resolve FG_comp_prop: algebra.