#[deprecated(since="8.19",note="Use Rplus_diag.")]
Notation double := (fun r1 => eq_sym (Rplus_diag r1)) (only parsing).
#[deprecated(since="8.19",note="Use Rplus_half_diag.")]
Notation double_var := (fun r1 => eq_sym (Rplus_half_diag r1)) (only parsing).
#[deprecated(since="8.19",note="Use eq_IZR_contrapositive.")]
Notation IZR_neq := (fun z1 z2 => (eq_IZR_contrapositive z1 z2)) (only parsing).

(* Specifications: *)
Lemma S_O_plus_INR_depr : forall n, INR (1 + n) = INR 1 + INR n.
#[deprecated(since="8.19")]
Notation S_O_plus_INR := S_O_plus_INR_depr (only parsing).
Lemma single_z_r_R1_depr :
forall r (n m:Z),
r < IZR n -> IZR n <= r + 1 -> r < IZR m -> IZR m <= r + 1 -> n = m.