Functional Scheme Qpositive_plus_ind := Induction for Qpositive_plus Sort Prop.

(* Specifications: *)
Lemma what_nR_does : forall p : Qpositive, nR p = Qpositive_plus p One.
Functional Scheme Qpositive_mult_ind := Induction for Qpositive_mult Sort Prop.
Lemma what_dL_does :
forall p : Qpositive,
dL p = Qpositive_mult p (Qpositive_inv (Qpositive_plus p One)).
Lemma Qpos_nR : forall p : Qpositive, Qpos (nR p) = Qplus (Qpos p) Qone.
Lemma Qmult_Z_nR :
forall (a : Z) (p : Qpositive),
Qmult (Z_to_Q a) (Qpos (nR p)) =
Qplus (Qmult (Z_to_Q a) (Qpos p)) (Z_to_Q a).
Lemma Qpos_dL :
forall p : Qpositive,
Qpos (dL p) = Qmult (Qpos p) (Qinv (Qplus (Qpos p) Qone)).
Lemma Qmult_Z_dL :
forall (a : Z) (p : Qpositive),
Qmult (Z_to_Q a) (Qpos (dL p)) =
Qmult (Qmult (Z_to_Q a) (Qpos p)) (Qinv (Qplus (Qpos p) Qone)).