--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/steps/rtc_plus.ma".
+include "ground_2/steps/rtc_ist.ma".
+
+(* RT-TRANSITION COUNTER ****************************************************)
+
+(* Properties with test for t-transition counter ****************************)
+
+lemma ist_plus: ān1,n2,c1,c2. šā¦n1,c1ā¦ ā šā¦n2,c2ā¦ ā šā¦n1+n2,c1+c2ā¦.
+#n1 #n2 #c1 #c2 #H1 #H2 destruct //
+qed.
+
+lemma ist_plus_O1: ān,c1,c2. šā¦0,c1ā¦ ā šā¦n,c2ā¦ ā šā¦n,c1+c2ā¦.
+/2 width=1 by ist_plus/ qed.
+
+lemma ist_plus_O2: ān,c1,c2. šā¦n,c1ā¦ ā šā¦0,c2ā¦ ā šā¦n,c1+c2ā¦.
+#n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by ist_plus/
+qed.
+
+lemma ist_succ: ān,c. šā¦n,cā¦ ā šā¦ān,c+ššā¦.
+/2 width=1 by ist_plus/ qed.
+
+(* Inversion properties with test for constrained rt-transition counter *****)
+
+lemma ist_inv_plus:
+ ān,c1,c2. šā¦n,c1 + c2ā¦ ā
+ āān1,n2. šā¦n1,c1ā¦ & šā¦n2,c2ā¦ & n1 + n2 = n.
+#n #c1 #c2 #H
+elim (plus_inv_dx ā¦ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
+elim (plus_inv_O3 ā¦ H1) -H1 #H11 #H12 destruct
+elim (plus_inv_O3 ā¦ H2) -H2 #H21 #H22 destruct
+elim (plus_inv_O3 ā¦ H3) -H3 #H31 #H32 destruct
+/3 width=5 by ex3_2_intro/
+qed-.
+
+lemma ist_inv_plus_O_dx: ān,c1,c2. šā¦n,c1 + c2ā¦ ā šā¦0,c2ā¦ ā šā¦n,c1ā¦.
+#n #c1 #c2 #H #H2
+elim (ist_inv_plus ā¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //
+qed-.
+
+lemma ist_inv_plus_SO_dx:
+ ān,c1,c2. šā¦n,c1 + c2ā¦ ā šā¦1,c2ā¦ ā
+ āām. šā¦m,c1ā¦ & n = ām.
+#n #c1 #c2 #H #H2 destruct
+elim (ist_inv_plus ā¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+/2 width=3 by ex2_intro/
+qed-.