--- /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 "basic_2/dynamic/cnv_cpm_tdpos.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Forward lemmas with positive rt-computation for terms ********************)
+
+lemma cpms_fwd_tdpos_sn (a) (h) (o) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → 𝐏[h,o]⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∨∨ ∧∧ T1 = T2 & 0 = n
+ | ∃∃n1,n2,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & (T1 ≛[h,o] T → ⊥) & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 & n1+n2=n.
+#a #h #o #n #G #L #T1 #H1T1 #H2T1 #T2 #H
+@(cpms_ind_dx … H) -n -T2
+[ /3 width=1 by or_introl, conj/
+| #n1 #n2 #T #T2 #HT1 * *
+ [ #H1 #H2 #HT2 destruct -HT1
+ elim (cpm_fwd_tdpos_sn … H1T1 H2T1 … HT2) -H1T1 -H2T1
+ [ * #H1 #H2 destruct -HT2 /3 width=1 by or_introl, conj/
+ | #HnT2 >commutative_plus in ⊢ (??%); /4 width=7 by ex4_3_intro, or_intror/
+ ]
+ | #m1 #m2 #T0 #H1T10 #H2T10 #HT0 #H #HT2 destruct -H1T1 -H2T1 -HT1
+ >associative_plus in ⊢ (??%); /4 width=7 by cpms_step_dx, ex4_3_intro, or_intror/
+ ]
+]
+qed-.