propagating the arithmetics library, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_pred.ma
index 4acdf37b5f1997531e980ac9b9195b333c28c71e..d7f79d76b8084d89079406c281a86de8872545f8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_pred.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_pred.ma
@@ -18,10 +18,17 @@ include "ground/arith/nat.ma".
 
 (* PREDECESSOR FOR NON-NEGATIVE INTEGERS ************************************)
 
+definition pnpred (p): nat ≝
+           psplit … (𝟎) ninj p.
+
+interpretation
+  "positive predecessor (non-negative integers)"
+  'DownArrow p = (pnpred p).
+
 (*** pred *)
 definition npred (m): nat ≝ match m with
 [ nzero  ⇒ 𝟎
-| ninj p ⇒ psplit … (𝟎) ninj p
+| ninj p ⇒ ↓p
 ].
 
 interpretation
@@ -34,6 +41,9 @@ interpretation
 lemma npred_zero: 𝟎 = ↓𝟎.
 // qed.
 
+lemma npred_inj (p): ↓p = ↓(ninj p).
+// qed.
+
 lemma npred_one: 𝟎 = ↓𝟏.
 // qed.