(* *)
(**************************************************************************)
+include "ground_2/insert_eq/insert_eq_0.ma".
include "ground_2/lib/functions.ma".
(* LABELLED TRANSITIVE CLOSURE **********************************************)
(∀a,b1. R a b1 b2 → Q a b1) →
(∀a1,a2,b1,b. R a1 b1 b → ltc … f … R a2 b b2 → Q a2 b → Q (f a1 a2) b1) →
∀a,b1. ltc … f … R a b1 b2 → Q a b1.
-#A #f #B #R #Q #b2 #Hf #IH1 #IH2 #a #b1 @(insert_eq … b2)
+#A #f #B #R #Q #b2 #Hf #IH1 #IH2 #a #b1 @(insert_eq_0 … b2)
#b0 #H elim H -a -b1 -b0 /2 width=2 by/
#a1 #a2 #b1 #b #b0 #H #Hb2 #_
generalize in match Hb2; generalize in match a2; -Hb2 -a2
(∀a,b2. R a b1 b2 → Q a b2) →
(∀a1,a2,b,b2. ltc … f … R a1 b1 b → Q a1 b → R a2 b b2 → Q (f a1 a2) b2) →
∀a,b2. ltc … f … R a b1 b2 → Q a b2.
-#A #f #B #R #Q #b1 #Hf #IH1 #IH2 #a #b2 @(insert_eq … b1)
+#A #f #B #R #Q #b1 #Hf #IH1 #IH2 #a #b2 @(insert_eq_0 … b1)
#b0 #H elim H -a -b0 -b2 /2 width=2 by/
#a1 #a2 #b0 #b #b2 #Hb0 #H #IHb0 #_
generalize in match IHb0; generalize in match Hb0; generalize in match a1; -IHb0 -Hb0 -a1
@(ltc_ind_dx … R … H1f … IH2 … H) -a -b2 -H1f #a #b2 #Hb12
>(H2f a) -H2f /3 width=4 by ltc_rc/
qed-.
+
+(* Properties with lsub *****************************************************)
+
+lemma ltc_lsub_trans: ∀A,f. associative … f →
+ ∀B,C,R,S. (∀n. lsub_trans B C (λL. R L n) S) →
+ ∀n. lsub_trans B C (λL. ltc A f … (R L) n) S.
+#A #f #Hf #B #C #R #S #HRS #n #L2 #T1 #T2 #H
+@(ltc_ind_dx … Hf ???? H) -n -T2
+/3 width=5 by ltc_dx, ltc_rc/
+qed-.