X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Farith%2Fnat_minus_plus.ma;h=bb8c911bc91b92b14eade2599003dda8ced21d7b;hb=19b0a814861157ba05f23877d5cd94059f52c2e8;hp=47e88de49c8255b43a76102bd60b7f51534553fb;hpb=5e72e41f4f86814e56d4b00959ccc56c71042a4c;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma
index 47e88de49..bb8c911bc 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma
@@ -17,33 +17,30 @@ include "ground/arith/nat_minus.ma".
 
 (* SUBTRACTION FOR NON-NEGATIVE INTEGERS ************************************)
 
-(* Rewrites with nplus ******************************************************)
+(* Constructions with nplus *************************************************)
 
 (*** minus_plus_m_m *)
 lemma nminus_plus_sn_refl_sn (m) (n): m = m + n - n.
-#m #n elim n -n //
+#m #n @(nat_ind_succ … n) -n //
 #n #IH <nplus_succ_dx <nminus_succ_bi //
 qed.
 
+(*** minus_plus_m_m *)
 lemma nminus_plus_sn_refl_dx (m) (n): m = n + m - n.
 #m #n <nplus_comm //
 qed.
 
 (*** minus_plus *)
-theorem nminus_assoc (o) (m) (n): o-m-n = o-(m+n).
-#o #m #n elim n -n //
+theorem nminus_plus_assoc (o) (m) (n): o-m-n = o-(m+n).
+#o #m #n @(nat_ind_succ … n) -n //
 #n #IH <nplus_succ_dx <nminus_succ_dx <nminus_succ_dx //
 qed-.
 
 (*** minus_plus_plus_l *)
 lemma nminus_plus_dx_bi (m) (n) (o): m - n = (m + o) - (n + o).
-#m #n #o <nminus_assoc <nminus_minus_comm //
+#m #n #o <nminus_plus_assoc <nminus_comm //
 qed.
 
-(*** plus_minus_plus_plus_l *) (**)
-lemma plus_minus_plus_plus_l: ∀z,x,y,h. z + (x + h) - (y + h) = z + x - y.
-// qed-.
-
 (* Helper constructions with nplus ******************************************)
 
 (*** plus_to_minus *)
@@ -62,17 +59,15 @@ lemma eq_inv_plus_nminus_refl_sn (m) (n) (o):
       m + o = m - n →
       ∨∨ ∧∧ 𝟎 = m & 𝟎 = o
        | ∧∧ 𝟎 = n & 𝟎 = o.
-#m elim m -m
+#m #n @(nat_ind_2_succ … m n) -m -n
 [ /3 width=1 by or_introl, conj/
-| #m #IH #n @(nat_ind … n) -n
-  [ #o #Ho
-    lapply (eq_inv_nplus_bi_sn … (𝟎) Ho) -Ho
-    /3 width=1 by or_intror, conj/
-  | #n #_ #o
-    <nminus_succ_bi >nplus_succ_shift #Ho
-    elim (IH … Ho) -IH -Ho * #_ #H
-    elim (eq_inv_nzero_succ … H)
-  ]
+| #m #o #Ho
+  lapply (eq_inv_nplus_bi_sn … (𝟎) Ho) -Ho
+  /3 width=1 by or_intror, conj/
+| #m #n #IH #o
+  <nminus_succ_bi >nplus_succ_shift #Ho
+  elim (IH … Ho) -IH -Ho * #_ #H
+  elim (eq_inv_zero_nsucc … H)
 ]
 qed-.