+
+lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄.
+/2 width=1 by isrt_plus/ qed.
+
+lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄.
+#n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by isrt_plus/
+qed.
+
+lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n,c⦄ → 𝐑𝐓⦃↑n,c+𝟘𝟙⦄.
+/2 width=1 by isrt_plus/ qed.
+
+(* Inversion properties with test for constrained rt-transition counter *****)
+
+lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ →
+ ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & n1 + n2 = n.
+#n #c1 #c2 * #ri #rs #H
+elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
+elim (plus_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
+qed-.
+
+lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄.
+#n #c1 #c2 #H #H2
+elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
+qed-.
+
+lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃1,c2⦄ →
+ ∃∃m. 𝐑𝐓⦃m,c1⦄ & n = ↑m.
+#n #c1 #c2 #H #H2
+elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+lapply (isrt_inj … Hn2 H2) -c2 #H destruct
+/2 width=3 by ex2_intro/
+qed-.