X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Farith%2Fnat_le_pred.ma;h=1666a554894cf8986dd8d3519dca338c176b06c4;hb=41a54a797af98d2867d4bf979d424283fb44a1fc;hp=da1de97f5574a6b0d0741850097451eace80609c;hpb=5e72e41f4f86814e56d4b00959ccc56c71042a4c;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma
index da1de97f5..1666a5548 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma
@@ -17,11 +17,39 @@ include "ground/arith/nat_le.ma".
 
 (* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
 
+(* Inversions with npred ****************************************************)
+
+lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
+#m #n @(nat_ind_succ … m) -m
+/2 width=1 by nle_succ_bi/
+qed-.
+
+(*** le_inv_S1 *)
+lemma nle_inv_succ_sn (m) (n):
+      ↑m ≤ n → ∧∧ m ≤ ↓n & n = ↑↓n.
+#m #n * -n
+[ /2 width=3 by nle_refl, conj/
+| #n #Hn /3 width=1 by nle_des_succ_sn, conj/
+]
+qed-.
+
+lemma nle_inv_succ_dx (m) (n):
+      m ≤ ↑n → ∨∨ 𝟎 = m | ∧∧ ↓m ≤ n & m = ↑↓m.
+#m #n @(nat_ind_succ … m) -m
+[ /2 width=1 by or_introl/
+| #m #_ #H0
+  /4 width=1 by nle_inv_succ_bi, or_intror, conj/
+]
+qed-.
+
 (* Constructions with npred *************************************************)
 
+lemma nle_succ_pred_dx_refl (m): m ≤ ↑↓m.
+#m @nle_inv_pred_sn // qed.
+
 (*** le_pred_n *)
 lemma nle_pred_sn_refl (m): ↓m ≤ m.
-#m elim m -m //
+#m @(nat_ind_succ … m) -m //
 qed.
 
 (*** monotonic_pred *)
@@ -31,13 +59,6 @@ lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n.
 qed.
 
 lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n.
-#m #n elim m -m //
+#m #n @(nat_ind_succ … m) -m //
 /2 width=1 by nle_pred_bi/
-qed-.
-
-(* Destructions with npred **************************************************)
-
-lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
-#m #n elim m -m
-/2 width=1 by nle_succ_bi/
-qed-.
+qed.