arithmetics for λδ

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
index 34a16e01f572fd379ba4ed35b8dece84a669e9ba..d7654a0bb43294fcfc57c6a4605e3b46959243e9 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
@@ -17,16 +17,6 @@ include "ground/arith/nat_lt_pred.ma".
 
 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
 
-(* Rewrites with nminus *****************************************************)
-
-(*** minus_pred_pred *)
-lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
-#m #n #Hm #Hn
->(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm
->(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn
-//
-qed-.
-
 (* Constructions with nminus ************************************************)
 
 (*** monotonic_lt_minus_l *)
@@ -35,3 +25,31 @@ lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
 lapply (nle_minus_sn_bi … o Hmn) -Hmn
 <(nminus_succ_sn … Hom) //
 qed.
+
+(*** monotonic_lt_minus_r *)
+lemma nlt_minus_dx_bi (o) (m) (n):
+      m < o -> m < n → o-n < o-m.
+#o #m #n #Ho #H
+lapply (nle_minus_dx_bi … o H) -H #H
+@(le_nlt_trans … H) -n
+@nlt_i >(nminus_succ_sn … Ho) //
+qed.
+
+(* Destructions with nminus *************************************************)
+
+(*** minus_pred_pred *)
+lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
+#m #n #Hm #Hn
+>(nlt_des_gen … Hm) in ⊢ (??%?); -Hm
+>(nlt_des_gen … Hn) in ⊢ (??%?); -Hn
+//
+qed-.
+
+lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n.
+#o @(nat_ind_succ … o) -o
+[ #m #n <nminus_zero_dx
+  /2 width=3 by le_nlt_trans/
+| #o #IH #m #n <nminus_succ_dx_pred_sn #H
+  /3 width=2 by nlt_inv_pred_dx/
+]
+qed-.