Inductive FMd : Type :=
  | Var : V -> FMd
  | Law : FMd -> FMd -> FMd
  | Unit : FMd
  | Inv : FMd -> FMd
  | Op : R -> FMd -> FMd.Inductive FMd : Type :=
  | Var : V -> FMd
  | Law : FMd -> FMd -> FMd
  | Unit : FMd
  | Inv : FMd -> FMd
  | Op : R -> FMd -> FMd.Inductive FMd : Type :=
  | Var : V -> FMd
  | Law : FMd -> FMd -> FMd
  | Unit : FMd
  | Inv : FMd -> FMd
  | Op : R -> FMd -> FMd.Inductive FMd : Type :=
  | Var : V -> FMd
  | Law : FMd -> FMd -> FMd
  | Unit : FMd
  | Inv : FMd -> FMd
  | Op : R -> FMd -> FMd.Inductive FMd : Type :=
  | Var : V -> FMd
  | Law : FMd -> FMd -> FMd
  | Unit : FMd
  | Inv : FMd -> FMd
  | Op : R -> FMd -> FMd.Inductive FMd : Type :=
  | Var : V -> FMd
  | Law : FMd -> FMd -> FMd
  | Unit : FMd
  | Inv : FMd -> FMd
  | Op : R -> FMd -> FMd.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.Inductive eqFMd : FMd -> FMd -> Prop :=
  | eqFMd_Var : forall x y : V, Equal x y -> (eqFMd (Var x) (Var y):Prop)
  | eqFMd_law :
      forall x x' y y' : FMd,
      eqFMd x x' -> eqFMd y y' -> (eqFMd (Law x y) (Law x' y'):Prop)
  | eqFMd_law_assoc :
      forall x y z : FMd, eqFMd (Law (Law x y) z) (Law x (Law y z)):Prop
  | eqFMd_law0r : forall x : FMd, eqFMd (Law x Unit) x:Prop
  | eqFMd_inv : forall x y : FMd, eqFMd x y -> eqFMd (Inv x) (Inv y)
  | eqFMd_invr : forall x : FMd, eqFMd (Law x (Inv x)) Unit
  | eqFMd_refl : forall x : FMd, eqFMd x x:Prop
  | eqFMd_sym : forall x y : FMd, eqFMd x y -> (eqFMd y x:Prop)
  | eqFMd_trans :
      forall x y z : FMd, eqFMd x y -> eqFMd y z -> (eqFMd x z:Prop)
  | eqFMd_com : forall x y : FMd, eqFMd (Law x y) (Law y x)
  | eqFMd_op_comp :
      forall (a b : R) (x y : FMd),
      Equal a b -> eqFMd x y -> (eqFMd (Op a x) (Op b y):Prop)
  | eqFMd_oplin_l :
      forall (a b : R) (x : FMd),
      eqFMd (Op (sgroup_law R a b) x) (Law (Op a x) (Op b x))
  | eqFMd_oplin_r :
      forall (a : R) (x y : FMd),
      eqFMd (Op a (Law x y)) (Law (Op a x) (Op a y))
  | eqFMd_op_assoc :
      forall (a b : R) (x : FMd),
      eqFMd (Op a (Op b x)) (Op (ring_mult a b) x)
  | eqFMd_op_unit : forall x : FMd, eqFMd (Op (ring_unit R) x) x.

Lemma eqFMd_Equiv : equivalence eqFMd.