Module SuccNat2Pos.
(** Composition of [Pos.of_succ_nat] and [Pos.to_nat] is successor on [nat] *)
(** [Pos.of_succ_nat] is hence injective *)
(** Another formulation *)
(** Successor and comparison are morphisms with respect to [Pos.of_succ_nat] *)
(** Other operations, for instance [Pos.add] and [plus] aren't directly related this way (we would need to compensate for the successor hidden in [Pos.of_succ_nat] *)
End SuccNat2Pos.
(** 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).
Proof (fun H => Pos2Nat.inj_sub p q (Pos.gt_lt _ _ H)).

Lemma nat_of_P_lt_Lt_compare_morphism p q :
Pos.compare_cont Eq p q = Lt -> Pos.to_nat p < Pos.to_nat q.