update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma
index 513ea525701c2ca2e2c24e219c73ec0854acf202..d54738f250b4100fea5bbdde0c20518572aad361 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma
@@ -15,9 +15,9 @@
 include "ground/arith/nat_plus.ma".
 include "ground/arith/nat_le.ma".
 
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
 
-(* Basic constructions with plus ********************************************)
+(* Constructions with nplus *************************************************)
 
 (*** monotonic_le_plus_l *)
 lemma nle_plus_bi_dx (m) (n1) (n2): n1 ≤ n2 → n1 + m ≤ n2 + m.
@@ -38,33 +38,60 @@ lemma nle_plus_dx_dx_refl (m) (n): m ≤ m + n.
 lemma nle_plus_dx_sn_refl (m) (n): m ≤ n + m.
 /2 width=1 by nle_plus_bi_sn/ qed.
 
-(*** le_plus_b *)
-lemma nle_plus_dx_dx (m) (n) (o): n + o ≤ m → n ≤ m.
-/2 width=3 by nle_trans/ qed.
-
 (*** le_plus_a *)
 lemma nle_plus_dx_sn (m) (n) (o): n ≤ m → n ≤ o + m.
 /2 width=3 by nle_trans/ qed.
 
-(* Main constructions with plus *********************************************)
+(* Main constructions with nplus ********************************************)
 
 (*** le_plus *)
 theorem nle_plus_bi (m1) (m2) (n1) (n2):
         m1 ≤ m2 → n1 ≤ n2 → m1 + n1 ≤ m2 + n2.
 /3 width=3 by nle_plus_bi_dx, nle_plus_bi_sn, nle_trans/ qed.
 
-(* Basic inversions with plus ***********************************************)
+(* Inversions with nplus ****************************************************)
+
+(*** le_plus_b *)
+lemma nle_inv_plus_sn_dx (m) (n) (o): n + o ≤ m → n ≤ m.
+/2 width=3 by nle_trans/ qed.
 
 (*** plus_le_0 *)
 lemma nle_inv_plus_zero (m) (n): m + n ≤ 𝟎 → ∧∧ 𝟎 = m & 𝟎 = n.
-/3 width=1 by nle_inv_zero_dx, eq_inv_nzero_plus/ qed-.
+/3 width=1 by nle_inv_zero_dx, eq_inv_zero_nplus/ qed-.
 
 (*** le_plus_to_le_r *)
 lemma nle_inv_plus_bi_dx (o) (m) (n): n + o ≤ m + o → n ≤ m.
-#o @(nat_ind … o) -o /3 width=1 by nle_inv_succ_bi/
+#o @(nat_ind_succ … o) -o /3 width=1 by nle_inv_succ_bi/
 qed-.
 
 (*** le_plus_to_le *)
 lemma nle_inv_plus_bi_sn (o) (m) (n): o + n ≤ o + m → n ≤ m.
 /2 width=2 by nle_inv_plus_bi_dx/ qed-.
 
+(* Destructions with nplus **************************************************)
+
+(*** plus2_le_sn_sn *)
+lemma nplus_2_des_le_sn_sn (m1) (m2) (n1) (n2):
+      m1 + n1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
+#m1 #m2 #n1 #n2 #H #Hm
+lapply (nle_plus_bi_dx n1 … Hm) -Hm >H -H
+/2 width=2 by nle_inv_plus_bi_sn/
+qed-.
+
+(*** plus2_le_sn_dx *)
+lemma nplus_2_des_le_sn_dx (m1) (m2) (n1) (n2):
+      m1 + n1 = n2 + m2 → m1 ≤ m2 → n2 ≤ n1.
+#m1 #m2 #n1 #n2 <nplus_comm in ⊢ (???%→?);
+/2 width=4 by nplus_2_des_le_sn_sn/ qed-.
+
+(*** plus2_le_dx_sn *)
+lemma nplus_2_des_le_dx_sn (m1) (m2) (n1) (n2):
+      n1 + m1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
+#m1 #m2 #n1 #n2 <nplus_comm in ⊢ (??%?→?);
+/2 width=4 by nplus_2_des_le_sn_sn/ qed-.
+
+(*** plus2_le_dx_dx *)
+lemma nplus_2_des_le_dx_dx (m1) (m2) (n1) (n2):
+      n1 + m1 = n2 + m2 → m1 ≤ m2 → n2 ≤ n1.
+#m1 #m2 #n1 #n2 <nplus_comm in ⊢ (??%?→?);
+/2 width=4 by nplus_2_des_le_sn_dx/ qed-.