include "delayed_updating/syntax/path_structure.ma".
include "delayed_updating/notation/relations/black_rightarrow_ibf_3.ma".
include "ground/relocation/tr_uni.ma".
-include "ground/arith/nat_rplus.ma".
include "ground/xoa/ex_6_5.ma".
(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
definition ibfr (r): relation2 prototerm prototerm ≝
λt1,t2.
∃∃p,b,q,m,n. p●𝗔◗b●𝗟◗q = r &
- â\8a\97b ϵ ð\9d\90\81 & b ϵ ð\9d\90\82â\9d¨â\93\89,m❩ & q ϵ 𝐂❨Ⓕ,n❩ & r◖𝗱↑n ϵ t1 &
+ â\8a\97b ϵ ð\9d\90\81 & b ϵ ð\9d\90\82â\9d¨â\92»,m❩ & q ϵ 𝐂❨Ⓕ,n❩ & r◖𝗱↑n ϵ t1 &
t1[⋔r←🠡[𝐮❨↑(m+n)❩](t1⋔(p◖𝗦))] ⇔ t2
.
interpretation
"balanced focused reduction with immediate updating (prototerm)"
'BlackRightArrowIBF t1 r t2 = (ibfr r t1 t2).
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma ibfr_eq_trans (t) (t1) (t2) (r):
+ t1 ➡𝐢𝐛𝐟[r] t → t ⇔ t2 → t1 ➡𝐢𝐛𝐟[r] t2.
+#t #t1 #t2 #r
+* #p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht #Ht2
+/3 width=13 by subset_eq_trans, ex6_5_intro/
+qed-.