(* 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 Cfield_facts.
Section Def.
Variable R : CFIELD.

Record Ctype : Type :=  {real :> R; imag : R}.

Definition Cadd (z z' : Ctype) : Ctype :=
  Build_Ctype (sgroup_law R (real z) (real z'))
    (sgroup_law R (imag z) (imag z')).

Definition Cmult (z z' : Ctype) : Ctype :=
  Build_Ctype
    (sgroup_law R (ring_mult (real z) (real z'))
       (group_inverse R (ring_mult (imag z) (imag z'))))
    (sgroup_law R (ring_mult (real z) (imag z'))
       (ring_mult (imag z) (real z'))).

Definition Copp (z : Ctype) : Ctype :=
  Build_Ctype (group_inverse R (real z)) (group_inverse R (imag z)).

Definition Cone : Ctype := Build_Ctype (ring_unit R) (monoid_unit R).

Definition Czero : Ctype := Build_Ctype (monoid_unit R) (monoid_unit R).

Definition Ceq (z z' : Ctype) :=
  Equal (real z) (real z') /\ Equal (imag z) (imag z').

Definition Cset : Setoid.
(* Goal: Setoid *)
apply (Build_Setoid (Carrier:=Ctype) (Equal:=Ceq)).
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))) *)
red in |- *.
(* Goal: and (@transitive Ctype Ceq) (@symmetric Ctype Ceq) *)
split; [ red in |- * | red in |- * ].
(* Goal: forall x : Ctype, @app_rel Ctype Ceq x x *)
(* Goal: and (@transitive Ctype Ceq) (@symmetric Ctype Ceq) *)
intros x; red in |- *; red in |- *; auto with algebra.
(* Goal: and (@transitive Ctype Ceq) (@symmetric Ctype Ceq) *)
split; [ red in |- * | red in |- * ].
(* Goal: forall (x y : Ctype) (_ : @app_rel Ctype Ceq x y), @app_rel Ctype Ceq y x *)
unfold app_rel, Ceq in |- *.
(* Goal: forall (x y z : Ctype) (_ : and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real x) (real y)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag y))) (_ : and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real y) (real z)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag y) (imag z))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real x) (real z)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag z)) *)
(* Goal: forall (x y : Ctype) (_ : @app_rel Ctype Ceq x y), @app_rel Ctype Ceq y x *)
intros x y z H' H'0; split; [ try assumption | idtac ].
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real x) (real z) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag z) *)
(* Goal: forall (x y : Ctype) (_ : @app_rel Ctype Ceq x y), @app_rel Ctype Ceq y x *)
apply Trans with (real y); intuition.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag z) *)
(* Goal: forall (x y : Ctype) (_ : @app_rel Ctype Ceq x y), @app_rel Ctype Ceq y x *)
apply Trans with (imag y); intuition.
(* Goal: forall (x y : Ctype) (_ : @app_rel Ctype Ceq x y), @app_rel Ctype Ceq y x *)
unfold app_rel, Ceq in |- *.
(* Goal: forall (x y : Ctype) (_ : and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real x) (real y)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag y))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real y) (real x)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag y) (imag x)) *)
intuition.
Defined.
Require Export Ring_util.

Lemma Build_Ctype_comp :
 forall x x' y y' : R,
 Equal x x' ->
 Equal y y' -> Equal (s:=Cset) (Build_Ctype x y) (Build_Ctype x' y').
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (imag y)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (imag y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (real y)))) (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (real y')) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (imag y')))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (imag y')) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (real y')))) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
intros x x' y y' H' H'0; try assumption.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: Ceq (Build_Ctype x y) (Build_Ctype x' y') *)
red in |- *; auto with algebra.
Qed.
Hint Resolve Build_Ctype_comp: algebra.

Lemma real_comp : forall x x' : Cset, Equal x x' -> Equal (real x) (real x').
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: forall (x x' : Ctype) (_ : and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real x) (real x')) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag x'))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (real x') *)
intros x x' H'; elim H'; intros H'0 H'1; try exact H'0; clear H'.
Qed.
Hint Resolve real_comp: algebra.

Lemma imag_comp : forall x x' : Cset, Equal x x' -> Equal (imag x) (imag x').
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: forall (x x' : Ctype) (_ : and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real x) (real x')) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag x) (imag x'))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (imag x') *)
intros x x' H'; elim H'; intros H'0 H'1; try exact H'1; clear H'.
Qed.
Hint Resolve imag_comp: algebra.

Lemma Build_Ctype_comp2 :
 forall x x' y y' : R,
 Equal x x' -> Equal y y' -> Ceq (Build_Ctype x y) (Build_Ctype x' y').
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (imag y)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (imag y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (real y)))) (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (real y')) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (imag y')))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (imag y')) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (real y')))) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
intros x x' y y' H' H'0; try assumption.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: Ceq (Build_Ctype x y) (Build_Ctype x' y') *)
red in |- *; auto with algebra.
Qed.
Hint Resolve Build_Ctype_comp2: algebra.

Definition Cring : RING.
apply
 (BUILD_RING (E:=Cset) (ringplus:=Cadd) (ringmult:=Cmult) (zero:=Czero)
    (un:=Cone) (ringopp:=Copp)).
(* Goal: forall x y : Carrier Cset, @Equal Cset (Cadd x y) (Cadd y x) *)
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Cmult x y) (Cmult x' y') *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cadd in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x y : Carrier Cset, @Equal Cset (Cadd x y) (Cadd y x) *)
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Cmult x y) (Cmult x' y') *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cadd in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x : Carrier Cset, @Equal Cset (Cadd x Czero) x *)
(* Goal: forall (x y : Carrier Cset) (_ : @Equal Cset x y), @Equal Cset (Copp x) (Copp y) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cadd x (Copp x)) Czero *)
(* Goal: forall x y : Carrier Cset, @Equal Cset (Cadd x y) (Cadd y x) *)
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Cmult x y) (Cmult x' y') *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cadd, Czero in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))) *)
intros x; red in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall (x y : Carrier Cset) (_ : @Equal Cset x y), @Equal Cset (Copp x) (Copp y) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cadd x (Copp x)) Czero *)
(* Goal: forall x y : Carrier Cset, @Equal Cset (Cadd x y) (Cadd y x) *)
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Cmult x y) (Cmult x' y') *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Copp in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x : Carrier Cset, @Equal Cset (Cadd x (Copp x)) Czero *)
(* Goal: forall x y : Carrier Cset, @Equal Cset (Cadd x y) (Cadd y x) *)
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Cmult x y) (Cmult x' y') *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cadd, Czero, Copp in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x y : Carrier Cset, @Equal Cset (Cadd x y) (Cadd y x) *)
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Cmult x y) (Cmult x' y') *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cadd in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cmult in |- *.
(* Goal: forall (x x' y y' : Carrier Cset) (_ : @Equal Cset x x') (_ : @Equal Cset y y'), @Equal Cset (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (imag y)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (imag y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (real y)))) (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (real y')) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (imag y')))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (imag y')) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (real y')))) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
intros x x' y y' H' H'0; try assumption.
(* Goal: @Equal Cset (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (imag y)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (imag y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x) (real y)))) (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (real y')) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (imag y')))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x') (imag y')) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag x') (real y')))) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
apply Build_Ctype_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cmult x y) z) (Cmult x (Cmult y z)) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cmult in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall x y z : Ctype, Ceq (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag y)))) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag y)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y))) (imag z)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag y)))) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag y)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y))) (real z)))) (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag z))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real z)))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real z)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag z))))))) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
intros x y z; try assumption.
(* Goal: Ceq (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag y)))) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag y)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y))) (imag z)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag y)))) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag y)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y))) (real z)))) (Build_Ctype (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag z))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real z)))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real z)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag z))))))) *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult x Cone) x *)
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
apply Build_Ctype_comp2.
apply
 Trans
  with
    (sgroup_law R
       (sgroup_law R (ring_mult (ring_mult x y) z)
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) z))
       (group_inverse R
          (sgroup_law R (ring_mult (ring_mult x (imag y)) (imag z))
             (ring_mult (ring_mult (imag x) y) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (sgroup_law R (ring_mult x (ring_mult y z))
          (ring_mult x (group_inverse R (ring_mult (imag y) (imag z)))))
       (group_inverse R
          (sgroup_law R (ring_mult (imag x) (ring_mult y (imag z)))
             (ring_mult (imag x) (ring_mult (imag y) z))))).
apply
 Trans
  with
    (sgroup_law R (ring_mult (ring_mult x y) z)
       (sgroup_law R
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) z)
          (group_inverse R
             (sgroup_law R (ring_mult (ring_mult x (imag y)) (imag z))
                (ring_mult (ring_mult (imag x) y) (imag z)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x (ring_mult y z))
       (sgroup_law R
          (ring_mult x (group_inverse R (ring_mult (imag y) (imag z))))
          (group_inverse R
             (sgroup_law R (ring_mult (imag x) (ring_mult y (imag z)))
                (ring_mult (imag x) (ring_mult (imag y) z)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) z)
       (sgroup_law R
          (group_inverse R (ring_mult (ring_mult (imag x) y) (imag z)))
          (group_inverse R (ring_mult (ring_mult x (imag y)) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (ring_mult x (group_inverse R (ring_mult (imag y) (imag z))))
       (sgroup_law R
          (group_inverse R (ring_mult (imag x) (ring_mult (imag y) z)))
          (group_inverse R (ring_mult (imag x) (ring_mult y (imag z)))))).
apply
 Trans
  with
    (sgroup_law R
       (group_inverse R (ring_mult (ring_mult x (imag y)) (imag z)))
       (sgroup_law R
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) z)
          (group_inverse R (ring_mult (ring_mult (imag x) y) (imag z))))).
apply
 Trans
  with
    (sgroup_law R
       (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) z)
       (sgroup_law R
          (group_inverse R (ring_mult (ring_mult x (imag y)) (imag z)))
          (group_inverse R (ring_mult (ring_mult (imag x) y) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (group_inverse R (ring_mult (ring_mult x (imag y)) (imag z)))
       (sgroup_law R
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) z)
          (group_inverse R (ring_mult (ring_mult (imag x) y) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
apply
 Trans with (ring_mult (group_inverse R (ring_mult x (imag y))) (imag z)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans with (ring_mult (ring_mult x (group_inverse R (imag y))) (imag z)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans with (ring_mult x (ring_mult (group_inverse R (imag y)) (imag z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
apply
 Trans with (group_inverse R (ring_mult (ring_mult (imag x) (imag y)) z)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (sgroup_law R (ring_mult (ring_mult x y) (imag z))
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) (imag z)))
       (sgroup_law R (ring_mult (ring_mult x (imag y)) z)
          (ring_mult (ring_mult (imag x) y) z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (sgroup_law R (ring_mult x (ring_mult y (imag z)))
          (ring_mult x (ring_mult (imag y) z)))
       (sgroup_law R (ring_mult (imag x) (ring_mult y z))
          (ring_mult (imag x) (group_inverse R (ring_mult (imag y) (imag z)))))).
apply
 Trans
  with
    (sgroup_law R (ring_mult (ring_mult x y) (imag z))
       (sgroup_law R
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) (imag z))
          (sgroup_law R (ring_mult (ring_mult x (imag y)) z)
             (ring_mult (ring_mult (imag x) y) z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x (ring_mult y (imag z)))
       (sgroup_law R (ring_mult x (ring_mult (imag y) z))
          (sgroup_law R (ring_mult (imag x) (ring_mult y z))
             (ring_mult (imag x)
                (group_inverse R (ring_mult (imag y) (imag z))))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult (ring_mult x (imag y)) z)
       (sgroup_law R
          (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) (imag z))
          (ring_mult (ring_mult (imag x) y) z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult (ring_mult (imag x) y) z)
       (ring_mult (group_inverse R (ring_mult (imag x) (imag y))) (imag z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with (group_inverse R (ring_mult (ring_mult (imag x) (imag y)) (imag z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with (group_inverse R (ring_mult (imag x) (ring_mult (imag y) (imag z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cmult, Cone in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall (x : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
intros x; try assumption.
(* Goal: forall _ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
elim x.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall real imag : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (ring_unit (cring_ring (cfield_ring R))) real) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) imag))) real) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (ring_unit (cring_ring (cfield_ring R))) imag) (@ring_mult (cring_ring (cfield_ring R)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) real)) imag) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
intros real0 imag0; split; [ try assumption | idtac ].
apply Trans with (sgroup_law R real0 (group_inverse R (monoid_unit R)));
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
 auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply Trans with (sgroup_law R real0 (monoid_unit R)); auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply Trans with (sgroup_law R (monoid_unit R) imag0); auto with algebra.
(* Goal: forall x : Carrier Cset, @Equal Cset (Cmult Cone x) x *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cmult, Cone in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall (x : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
intros x; try assumption.
(* Goal: forall _ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
elim x.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall real imag : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (ring_unit (cring_ring (cfield_ring R))) real) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) imag))) real) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (ring_unit (cring_ring (cfield_ring R))) imag) (@ring_mult (cring_ring (cfield_ring R)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) real)) imag) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult x (Cadd y z)) (Cadd (Cmult x y) (Cmult x z)) *)
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
intros real0 imag0; split; [ try assumption | idtac ].
apply Trans with (sgroup_law R real0 (group_inverse R (monoid_unit R)));
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
 auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply Trans with (sgroup_law R real0 (monoid_unit R)); auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply Trans with (sgroup_law R imag0 (monoid_unit R)); auto with algebra.
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cmult, Cadd in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall x y z : Ctype, and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (real x) (real y)) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (imag x) (imag y)) (imag z)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag z)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag z)))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (real x) (real y)) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (imag x) (imag y)) (real z))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real z))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real z))))) *)
intros x y z; split; [ try assumption | idtac ].
apply
 Trans
  with
    (sgroup_law R (sgroup_law R (ring_mult x y) (ring_mult x z))
       (group_inverse R (ring_mult (imag x) (sgroup_law R (imag y) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x y)
       (sgroup_law R (ring_mult x z)
          (group_inverse R
             (ring_mult (imag x) (sgroup_law R (imag y) (imag z)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x y)
       (sgroup_law R (group_inverse R (ring_mult (imag x) (imag y)))
          (sgroup_law R (ring_mult x z)
             (group_inverse R (ring_mult (imag x) (imag z)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x z)
       (ring_mult (group_inverse R (imag x)) (sgroup_law R (imag y) (imag z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x z)
       (sgroup_law R (ring_mult (group_inverse R (imag x)) (imag y))
          (ring_mult (group_inverse R (imag x)) (imag z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult (group_inverse R (imag x)) (imag y))
       (sgroup_law R (ring_mult x z)
          (ring_mult (group_inverse R (imag x)) (imag z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (sgroup_law R (ring_mult x (imag y)) (ring_mult x (imag z)))
       (sgroup_law R (ring_mult (imag x) y) (ring_mult (imag x) z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x (imag y))
       (sgroup_law R (ring_mult x (imag z))
          (sgroup_law R (ring_mult (imag x) y) (ring_mult (imag x) z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x (imag y))
       (sgroup_law R (ring_mult (imag x) y)
          (sgroup_law R (ring_mult x (imag z)) (ring_mult (imag x) z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall x y z : Carrier Cset, @Equal Cset (Cmult (Cadd x y) z) (Cadd (Cmult x z) (Cmult y z)) *)
unfold Cmult, Cadd in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: forall x y z : Ctype, and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (real x) (real y)) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (imag x) (imag y)) (imag z)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag z)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real z)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag z)))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (real x) (real y)) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (imag x) (imag y)) (real z))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real z))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag z)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real z))))) *)
intros x y z; split; [ try assumption | idtac ].
apply
 Trans
  with
    (sgroup_law R (sgroup_law R (ring_mult x z) (ring_mult y z))
       (group_inverse R
          (sgroup_law R (ring_mult (imag x) (imag z))
             (ring_mult (imag y) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x z)
       (sgroup_law R (ring_mult y z)
          (group_inverse R
             (sgroup_law R (ring_mult (imag x) (imag z))
                (ring_mult (imag y) (imag z)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x z)
       (sgroup_law R (group_inverse R (ring_mult (imag x) (imag z)))
          (sgroup_law R (ring_mult y z)
             (group_inverse R (ring_mult (imag y) (imag z)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult y z)
       (sgroup_law R (group_inverse R (ring_mult (imag y) (imag z)))
          (group_inverse R (ring_mult (imag x) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult y z)
       (sgroup_law R (group_inverse R (ring_mult (imag x) (imag z)))
          (group_inverse R (ring_mult (imag y) (imag z))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (sgroup_law R (ring_mult x (imag z)) (ring_mult y (imag z)))
       (sgroup_law R (ring_mult (imag x) z) (ring_mult (imag y) z))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x (imag z))
       (sgroup_law R (ring_mult y (imag z))
          (sgroup_law R (ring_mult (imag x) z) (ring_mult (imag y) z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R (ring_mult x (imag z))
       (sgroup_law R (ring_mult (imag x) z)
          (sgroup_law R (ring_mult y (imag z)) (ring_mult (imag y) z)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
Defined.

Definition Ccring : CRING.
(* Goal: Ob CRING *)
apply (Build_cring (cring_ring:=Cring)).
(* Goal: cring_on Cring *)
apply (Build_cring_on (R:=Cring)).
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))) *)
red in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
unfold Cmult, Ceq in |- *; simpl in |- *.
(* Goal: forall x y : Ctype, and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (imag y)))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (imag x))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag y)) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (imag x)) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (real x)))) *)
intros x y; split; [ try assumption | idtac ].
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply Trans with (sgroup_law R (ring_mult (imag y) x) (ring_mult y (imag x)));
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
 auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply SGROUP_comp; auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (imag y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (imag y) (real x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real y) (imag x)) *)
exact (CRING_com (R1:=R) x (imag y)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (real y)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real y) (imag x)) *)
exact (CRING_com (R1:=R) (imag x) y).
Defined.

Definition conjugate (z : Ctype) : Ctype :=
  Build_Ctype (real z) (group_inverse R (imag z)).

Definition CdivR (z : Ctype) (r : R) : Ctype :=
  Build_Ctype (field_div (real z) r) (field_div (imag z) r).

Definition Cinv (z : Ctype) : Ctype :=
  CdivR (conjugate z) (Cmult z (conjugate z)).

Definition Cinv_map : MAP Ccring Ccring.
(* Goal: Carrier (MAP (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply (Build_Map (A:=Ccring) (B:=Ccring) (Ap:=Cinv)).
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))) *)
red in |- *.
(* Goal: forall (x y : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x y), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (Cinv x) (Cinv y) *)
unfold Cinv in |- *.
(* Goal: forall (x y : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x y), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (CdivR (conjugate x) (real (Cmult x (conjugate x)))) (CdivR (conjugate y) (real (Cmult y (conjugate y)))) *)
unfold CdivR in |- *.
(* Goal: forall (x y : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x y), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (Build_Ctype (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real (conjugate x)) (real (Cmult x (conjugate x)))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (imag (conjugate x)) (real (Cmult x (conjugate x))))) (Build_Ctype (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real (conjugate y)) (real (Cmult y (conjugate y)))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (imag (conjugate y)) (real (Cmult y (conjugate y))))) *)
unfold conjugate in |- *.
(* Goal: forall (x y : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x y), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (Build_Ctype (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real (Build_Ctype (real x) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag x)))) (real (Cmult x (Build_Ctype (real x) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (imag (Build_Ctype (real x) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag x)))) (real (Cmult x (Build_Ctype (real x) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag x))))))) (Build_Ctype (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real (Build_Ctype (real y) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag y)))) (real (Cmult y (Build_Ctype (real y) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag y)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (imag (Build_Ctype (real y) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag y)))) (real (Cmult y (Build_Ctype (real y) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (imag y))))))) *)
intros x y H'; try assumption.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
unfold Ceq in |- *; simpl in |- *.
(* Goal: and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real y) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) *)
simpl in H'.
(* Goal: and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real y) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) *)
red in H'.
(* Goal: False *)
elim H'; intros H'0 H'1; try exact H'0; clear H'.
(* Goal: and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real y) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) *)
split; [ try assumption | idtac ].
(* Goal: forall _ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
unfold field_div in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i r)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply RING_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y)))))) *)
apply FIELD_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: forall _ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
unfold field_div in |- *.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i r)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply RING_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y)))))) *)
apply FIELD_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
Defined.
Hypothesis
  sum_of_square :
    forall x y : R,
    Equal (sgroup_law R (ring_mult x x) (ring_mult y y)) (monoid_unit R) ->
    Equal x (monoid_unit R) /\ Equal y (monoid_unit R).

Lemma C_inv_r :
 forall x : Ccring,
 ~ Equal x (monoid_unit Ccring) ->
 Equal (ring_mult x (Ap Cinv_map x)) (ring_unit Ccring).
(* Goal: forall (x : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
intros x; try assumption.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
unfold Ceq in |- *; simpl in |- *.
(* Goal: forall _ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
unfold field_div in |- *.
(* Goal: forall _ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag x) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real x) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x))))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
elim x.
(* Goal: forall (real0 imag0 : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))))) (_ : not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (real (Build_Ctype real0 imag0)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (imag (Build_Ctype real0 imag0)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))))), and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real (Build_Ctype real0 imag0)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real (Build_Ctype real0 imag0)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real (Build_Ctype real0 imag0)) (real (Build_Ctype real0 imag0))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag (Build_Ctype real0 imag0)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag (Build_Ctype real0 imag0))))))))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag (Build_Ctype real0 imag0)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag (Build_Ctype real0 imag0))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real (Build_Ctype real0 imag0)) (real (Build_Ctype real0 imag0))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag (Build_Ctype real0 imag0)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag (Build_Ctype real0 imag0))))))))))) (ring_unit (cring_ring (cfield_ring R)))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real (Build_Ctype real0 imag0)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag (Build_Ctype real0 imag0))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real (Build_Ctype real0 imag0)) (real (Build_Ctype real0 imag0))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag (Build_Ctype real0 imag0)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag (Build_Ctype real0 imag0))))))))) (@ring_mult (cring_ring (cfield_ring R)) (imag (Build_Ctype real0 imag0)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (real (Build_Ctype real0 imag0)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real (Build_Ctype real0 imag0)) (real (Build_Ctype real0 imag0))) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag (Build_Ctype real0 imag0)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag (Build_Ctype real0 imag0)))))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) *)
intros r i; try assumption.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real x) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (real y) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real x) (real x)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag x) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag x)))))) (@field_div (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) (real y) (real y)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) (imag y) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (imag y))))))) *)
intros H'; split; [ try assumption | idtac ].
apply
 Trans
  with
    (sgroup_law R
       (ring_mult (ring_mult r r)
          (field_inverse
             (sgroup_law R (ring_mult r r)
                (group_inverse R (ring_mult i (group_inverse R i))))))
       (group_inverse R
          (ring_mult (ring_mult i (group_inverse R i))
             (field_inverse
                (sgroup_law R (ring_mult r r)
                   (group_inverse R (ring_mult i (group_inverse R i)))))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law (Build_field R)
       (ring_mult (ring_mult r r)
          (field_inverse
             (sgroup_law (Build_field R) (ring_mult r r)
                (group_inverse (Build_field R)
                   (ring_mult i (group_inverse (Build_field R) i))))))
       (ring_mult
          (group_inverse (Build_field R)
             (ring_mult i (group_inverse (Build_field R) i)))
          (field_inverse
             (sgroup_law (Build_field R) (ring_mult r r)
                (group_inverse (Build_field R)
                   (ring_mult i (group_inverse (Build_field R) i))))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (ring_mult
       (sgroup_law (Build_field R) (ring_mult r r)
          (group_inverse (Build_field R)
             (ring_mult i (group_inverse (Build_field R) i))))
       (field_inverse
          (sgroup_law (Build_field R) (ring_mult r r)
             (group_inverse (Build_field R)
                (ring_mult i (group_inverse (Build_field R) i)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (ring_unit (cring_ring (cfield_ring R))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply (FIELD_inverse_r (K:=R)).
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))) *)
red in |- *.
(* Goal: forall _ : @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))), False *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
intros H'0; try assumption.
cut
 (Equal (sgroup_law (Build_field R) (ring_mult r r) (ring_mult i i))
    (monoid_unit (Build_field R))).
(* Goal: forall _ : @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))), False *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
intros H'1; try assumption.
absurd (Equal r (monoid_unit R) /\ Equal i (monoid_unit R));
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
 auto with algebra.
apply
 Trans
  with
    (sgroup_law (Build_field R) (ring_mult r r)
       (group_inverse (Build_field R)
          (ring_mult i (group_inverse (Build_field R) i)))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i i)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r (@ring_mult (cring_ring (cfield_ring R)) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i))))))) (@ring_mult (cring_ring (cfield_ring R)) i (@ring_mult (cring_ring (cfield_ring R)) r (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (@ring_mult (cring_ring (cfield_ring R)) r r) (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) (@ring_mult (cring_ring (cfield_ring R)) i (group_inverse (abelian_group_group (ring_group (cring_ring (cfield_ring R)))) i)))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply SGROUP_comp.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (group_inverse (Build_field R)
       (group_inverse (Build_field R) (ring_mult i i))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law R
       (ring_mult (ring_mult r (group_inverse R i))
          (field_inverse
             (sgroup_law R (ring_mult r r)
                (group_inverse R (ring_mult i (group_inverse R i))))))
       (ring_mult (ring_mult i r)
          (field_inverse
             (sgroup_law R (ring_mult r r)
                (group_inverse R (ring_mult i (group_inverse R i))))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (ring_mult
       (sgroup_law (Build_field R)
          (ring_mult r (group_inverse (Build_field R) i)) 
          (ring_mult i r))
       (field_inverse
          (sgroup_law R (ring_mult r r)
             (group_inverse R (ring_mult i (group_inverse R i)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (ring_mult (monoid_unit R)
       (field_inverse
          (sgroup_law R (ring_mult r r)
             (group_inverse R (ring_mult i (group_inverse R i)))))).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i r)) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (@field_inverse (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)) (sgroup_law (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) r r) (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) (@ring_mult (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))) i (group_inverse (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R))))) i)))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) *)
apply RING_comp.
apply
 Trans
  with
    (sgroup_law (Build_field R)
       (group_inverse (Build_field R) (ring_mult r i)) 
       (ring_mult i r)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
apply
 Trans
  with
    (sgroup_law (Build_field R)
       (group_inverse (Build_field R) (ring_mult i r)) 
       (ring_mult i r)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
Qed.

Definition C_field_on : field_on Ccring.
(* Goal: field_on (cring_ring Ccring) *)
apply (Build_field_on (R:=Ccring) (field_inverse_map:=Cinv_map)).
(* Goal: forall (x : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: forall (x : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
exact C_inv_r.
(* Goal: forall (x : Carrier (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) (_ : not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) x (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))))), @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
intros x H'; try assumption.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply Trans with (ring_mult x (Ap Cinv_map x)).
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
auto with algebra.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
apply C_inv_r; auto with algebra.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (Ceq (ring_unit Cring) Czero) *)
unfold Ceq in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (real (ring_unit Cring)) (real Czero)) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (field_ring (@Build_field (cring_ring (cfield_ring R)) (cfield_on_def R)))))))) (imag (ring_unit Cring)) (imag Czero))) *)
simpl in |- *.
(* Goal: not (and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))))) *)
red in |- *.
(* Goal: forall _ : and (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (ring_unit (cring_ring (cfield_ring R))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))) (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R))))))) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring (cfield_ring R)))))))), False *)
intros H'; try assumption.
(* Goal: False *)
elim H'; intros H'0 H'1; try exact H'0; clear H'.
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x) x) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) *)
(* Goal: @Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (@ring_mult (cring_ring Ccring) x (@Ap (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) Cinv_map x)) (ring_unit (cring_ring Ccring)) *)
(* Goal: not (@Equal (sgroup_set (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring)))))) (ring_unit (cring_ring Ccring)) (@monoid_unit (monoid_sgroup (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))) (monoid_on_def (group_monoid (abelian_group_group (ring_group (cring_ring Ccring))))))) *)
absurd (Equal (ring_unit R) (monoid_unit R)); auto with algebra.
Defined.

Definition CC : CFIELD := Build_cfield C_field_on.
End Def.