milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma
index 2b3502259f796845194b82ee7d94f64ecca9112a..e05ff9473f0c36213c42ded225f6108a4defff77 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma
@@ -12,20 +12,50 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/arith/nat_pred_succ.ma".
+include "ground/arith/nat_le_pred.ma".
 include "ground/arith/nat_lt.ma".
 
 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
 
-(* Constructions with npred *************************************************)
+(* Destructions with npred **************************************************)
 
-lemma nlt_zero_sn (m): m = ↑↓m → 𝟎 < m.
-// qed.
+(*** S_pred lt_succ_pred lt_inv_O1 *)
+lemma nlt_des_gen (m) (n): m < n → n = ↑↓n.
+#m #n @(nat_ind_succ … n) -n //
+#H elim (nlt_inv_zero_dx … H)
+qed-.
 
 (* Inversions with npred ****************************************************)
 
-(*** S_pred *)
-lemma nlt_inv_zero_sn (m): 𝟎 < m → m = ↑↓m.
-#m @(nat_ind … m) -m //
-#H elim (nlt_inv_refl … H)
+(*** lt_inv_gen *)
+lemma nlt_inv_gen (m) (n): m < n → ∧∧ m ≤ ↓n & n = ↑↓n.
+/2 width=1 by nle_inv_succ_sn/ qed-.
+
+(*** lt_inv_S1 *)
+lemma nlt_inv_succ_sn (m) (n): ↑m < n → ∧∧ m < ↓n & n = ↑↓n.
+/2 width=1 by nle_inv_succ_sn/ qed-.
+
+lemma nlt_inv_pred_dx (m) (n): m < ↓n → ↑m < n.
+#m #n #H >(nlt_des_gen (𝟎) n)
+[ /2 width=1 by nlt_succ_bi/
+| /3 width=3 by nle_nlt_trans, nlt_nle_trans/
+]
+qed-.
+
+lemma nlt_inv_pred_bi (m) (n):
+      ↓m < ↓n → m < n.
+/3 width=3 by nlt_inv_pred_dx, nle_nlt_trans/
 qed-.
+
+(* Constructions with npred *************************************************)
+
+lemma nlt_zero_sn (n): n = ↑↓n → 𝟎 < n.
+// qed.
+
+(*** monotonic_lt_pred *)
+lemma nlt_pred_bi (m) (n): 𝟎 < m → m < n → ↓m < ↓n.
+#m #n #Hm #Hmn
+@nle_inv_succ_bi
+<(nlt_des_gen … Hm) -Hm
+<(nlt_des_gen … Hmn) //
+qed.