update in ground

[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 bf889def7538d99a54e5f9f7e360ebb195260abc..61ab9b5444977d6033948a3b92e490898cb730fa 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
@@ -12,37 +12,71 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground/generated/pull_2.ma".
 include "ground/arith/nat_le_minus.ma".
 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 *)
-lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
+lemma nlt_minus_bi_dx (o) (m) (n): o ≤ m → m < n → m - o < n - o.
 #o #m #n #Hom #Hmn
-lapply (nle_minus_sn_bi … o Hmn) -Hmn
+lapply (nle_minus_bi_dx … o Hmn) -Hmn
 <(nminus_succ_sn … Hom) //
 qed.
 
+(*** monotonic_lt_minus_r *)
+lemma nlt_minus_bi_sn (o) (m) (n):
+      m < o -> m < n → o-n < o-m.
+#o #m #n #Ho #H
+lapply (nle_minus_bi_sn … o H) -H #H
+@(nle_nlt_trans … H) -n
+@nlt_i >(nminus_succ_sn … Ho) //
+qed.
+
+(* Inversions with nminus ***************************************************)
+
+lemma nlt_inv_minus_bi_dx (o) (m) (n):
+      m - o < n - o → m < n.
+#o @(nat_ind_succ … o) -o
+/3 width=1 by nlt_inv_pred_bi/
+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 elim o -o
+#o @(nat_ind_succ … o) -o
 [ #m #n <nminus_zero_dx
-  /2 width=3 by le_nlt_trans/
+  /2 width=3 by nle_nlt_trans/
 | #o #IH #m #n <nminus_succ_dx_pred_sn #H
   /3 width=2 by nlt_inv_pred_dx/
 ]
 qed-.
+
+(* Advanced eliminators for nle with nlt and nminus *************************)
+
+(*** nat_elim_le_sn *)
+lemma nle_ind_sn (Q:relation …):
+      (∀m1,m2. (∀m. m < m2-m1 → Q (m2-m) m2) → m1 ≤ m2 → Q m1 m2) →
+      ∀n1,n2. n1 ≤ n2 → Q n1 n2.
+#Q #IH #n1 #n2 #Hn
+>(nminus_minus_dx_refl_sn … Hn) -Hn
+lapply (nle_minus_sn_refl_sn n2 n1)
+let d ≝ (n2-n1)
+@(nat_ind_lt … d) -d -n1 #d
+@pull_2 #Hd
+>(nminus_minus_dx_refl_sn … Hd) in ⊢ (%→?); -Hd
+let n1 ≝ (n2-d) #IHd
+@IH -IH [| // ] #m #Hn
+/4 width=3 by nlt_des_le, nlt_nle_trans/
+qed-.