include "ground/arith/nat_succ.ma".
include "ground/arith/nat_pred.ma".
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* PREDECESSOR FOR NON-NEGATIVE INTEGERS ************************************)
-(* Basic rewrites with succ *************************************************)
+(* Constructions with npsucc ************************************************)
-(*** pred_Sn *)
+lemma pnpred_succ (n): n = pnpred (npsucc n).
+* //
+qed.
+
+lemma npsucc_pred (p): p = npsucc (pnpred p).
+* //
+qed.
+
+(* Constructions with nsucc and psucc ***************************************)
+
+lemma pnpred_psucc (p): pnpred (psucc p) = nsucc (pnpred p).
+* // qed.
+
+(* Constructions with nsucc *************************************************)
+
+lemma nsucc_pnpred (p):
+ ninj p = ↑(pnpred p).
+// qed.
+
+(*** pred_Sn pred_S *)
lemma npred_succ (n): n = ↓↑n.
* //
qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_pnpred_bi:
+ injective … pnpred.
+#p1 #p2 #Hp
+>(npsucc_pred p1) >(npsucc_pred p2)
+<Hp -Hp @refl
+qed-.
+
+(* Inversions with nsucc ****************************************************)
+
+(*** nat_split *)
+lemma nat_split_zero_pos (n): ∨∨ 𝟎 = n | n = ↑↓n.
+#n @(nat_ind_succ … n) -n
+/2 width=1 by or_introl, or_intror/
+qed-.