propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / rtmap_basic_after.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_basic_after.ma b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_basic_after.ma
index 3bcde30924d58b3c7b563a61d1aad15681754aa4..792fa995d660583de99c41043499c052495000a2 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_basic_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_basic_after.ma
@@ -12,28 +12,26 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/relocation/rtmap_after.ma".
+include "ground/arith/nat_le_pred.ma".
+include "ground/relocation/rtmap_after_uni.ma".
 include "ground/relocation/rtmap_basic.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
 (* Properties with composition **********************************************)
 
-lemma after_basic_rc (m2,m1,n2,n1):
-                     m1 ≤ m2 → m2 ≤ m1+n1 → 𝐁❨m2,n2❩ ⊚ 𝐁❨m1,n1❩ ≘ 𝐁❨m1,n2+n1❩.
-#m2 elim m2 -m2
-[ #m1 #n2 #n1 #Hm21 #_
-  <(le_n_O_to_eq … Hm21) -m1 //
-| #m2 #IH *
-  [ #n2 #n1 #_ < plus_O_n #H
-    elim (le_inv_S1 … H) -H #x #Hx #H destruct
-    <plus_n_Sm
-    @after_push [4:|*: // ]
-    @(IH 0 … Hx) -IH -n2 -x //
-  | #m1 #n2 #n1 #H1 #H2
-    lapply (le_S_S_to_le … H1) -H1 #H1
-    lapply (le_S_S_to_le … H2) -H2 #H2
-    /3 width=7 by after_refl/
-  ]
+lemma after_basic_rc (m2,m1):
+      m1 ≤ m2 → ∀n2,n1.m2 ≤ n1+m1 → 𝐁❨m2,n2❩ ⊚ 𝐁❨m1,n1❩ ≘ 𝐁❨m1,n1+n2❩.
+#m2 #m1 @(nat_ind_2_succ … m2 m1) -m2 -m1
+[ #m1 #H #n2 #n1 #_
+  <(nle_inv_zero_dx … H) -m1 //
+| #m2 #IH #_ #n2 #n1 <nplus_zero_dx #H
+  elim (nle_inv_succ_sn … H) -H #Hm2 #Hn1
+  >Hn1 -Hn1 <nplus_succ_sn
+  /3 width=7 by after_push/
+| #m2 #m1 #IH #H1 #n2 #n1 <nplus_succ_dx #H2
+  lapply (nle_inv_succ_bi … H1) -H1 #H1
+  lapply (nle_inv_succ_bi … H2) -H2 #H2
+  /3 width=7 by after_refl/
 ]
 qed.