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Free_group.v
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(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
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 : forall x y : V, Equal x y -> (eqFG (Var x) (Var y):Prop)
| eqFG_law :
forall x x' y y' : FG,
eqFG x x' -> eqFG y y' -> (eqFG (Law x y) (Law x' y'):Prop)
| eqFG_law_assoc :
forall x y z : FG, eqFG (Law (Law x y) z) (Law x (Law y z)):Prop
| eqFG_law0r : forall x : FG, eqFG (Law x Unit) x:Prop
| eqFG_inv : forall x y : FG, eqFG x y -> eqFG (Inv x) (Inv y)
| eqFG_invr : forall x : FG, eqFG (Law x (Inv x)) Unit
| eqFG_refl : forall x : FG, eqFG x x:Prop
| eqFG_sym : forall x y : FG, eqFG x y -> (eqFG y x:Prop)
| eqFG_trans : forall 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.