(** For compatibility, old names and old-style lemmas *)

Notation Psucc_S := Pos2Nat.inj_succ (only parsing).Notation Pplus_plus := Pos2Nat.inj_add (only parsing).Notation Pmult_mult := Pos2Nat.inj_mul (only parsing).Notation Pcompare_nat_compare := Pos2Nat.inj_compare (only parsing).Notation nat_of_P_xH := Pos2Nat.inj_1 (only parsing).Notation nat_of_P_xO := Pos2Nat.inj_xO (only parsing).Notation nat_of_P_xI := Pos2Nat.inj_xI (only parsing).Notation nat_of_P_is_S := Pos2Nat.is_succ (only parsing).Notation nat_of_P_pos := Pos2Nat.is_pos (only parsing).Notation nat_of_P_inj_iff := Pos2Nat.inj_iff (only parsing).Notation nat_of_P_inj := Pos2Nat.inj (only parsing).Notation Plt_lt := Pos2Nat.inj_lt (only parsing).Notation Pgt_gt := Pos2Nat.inj_gt (only parsing).Notation Ple_le := Pos2Nat.inj_le (only parsing).Notation Pge_ge := Pos2Nat.inj_ge (only parsing).Notation Pminus_minus := Pos2Nat.inj_sub (only parsing).Notation iter_nat_of_P := @Pos2Nat.inj_iter (only parsing).Notation nat_of_P_of_succ_nat := SuccNat2Pos.id_succ (only parsing).Notation P_of_succ_nat_of_P := Pos2SuccNat.id_succ (only parsing).Notation nat_of_P_succ_morphism := Pos2Nat.inj_succ (only parsing).Notation nat_of_P_plus_morphism := Pos2Nat.inj_add (only parsing).Notation nat_of_P_mult_morphism := Pos2Nat.inj_mul (only parsing).Notation nat_of_P_compare_morphism := Pos2Nat.inj_compare (only parsing).Notation lt_O_nat_of_P := Pos2Nat.is_pos (only parsing).Notation ZL4 := Pos2Nat.is_succ (only parsing).Notation nat_of_P_o_P_of_succ_nat_eq_succ := SuccNat2Pos.id_succ (only parsing).Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (only parsing).Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (only parsing).

Theorem Pmult_nat_plus_carry_morphism : forall p q n, Pmult_nat (Pos.add_carry p q) n = n + Pmult_nat (p + q) n.