update in ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_pred_succ.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_pred_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_pred_succ.ma
index f21156455af3542b7c4a05003e72d92a83c36eb9..5472d7c1bee0a348a9233f491f820c9c7bd37030 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_pred_succ.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_pred_succ.ma
@@ -17,9 +17,45 @@ include "ground/arith/nat_pred.ma".
 
 (* PREDECESSOR FOR NON-NEGATIVE INTEGERS ************************************)
 
-(* Rewrites with nsucc ******************************************************)
+(* Constructions with npsucc ************************************************)
+
+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-.