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diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
index f0b3f5f31..67c22257b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
@@ -12,23 +12,26 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/xoa/ex_3_1.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_equivalence.ma".
+include "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/substitution/lift.ma".
 include "delayed_updating/syntax/path_structure.ma".
 include "delayed_updating/syntax/path_balanced.ma".
-include "delayed_updating/substitution/fsubst.ma".
-(*
-include "delayed_updating/notation/functions/pitchforkleftarrow_3.ma".
-*)
+include "delayed_updating/syntax/path_depth.ma".
+include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
+include "ground/xoa/ex_1_2.ma".
+include "ground/xoa/and_4.ma".
 
-(* DELAYED FOCALIZED REDUCTION **********************************************)
+(* DELAYED FOCUSED REDUCTION ************************************************)
 
-inductive dfr (p) (q) (t): predicate preterm ≝
-| dfr_beta (b) (n):
-  let r ≝ p;;(𝗔;b;;(𝗟;q,𝗱❨n❩)) in
-  r ϵ t → ⊓⊗b → dfr p q t (t[⋔r←t⋔p,𝗦])
+definition dfr (p) (q): relation2 prototerm prototerm ≝
+           λt1,t2. ∃∃b,n.
+           let r ≝ p●𝗔◗b●𝗟◗q in
+           ∧∧ ⊗b ϵ 𝐁 & ❘q❘ = (↑[q]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
+              t1[⋔r←𝛗n.(t1⋔(p◖𝗦))] ⇔ t2
 .
-(*
+
 interpretation
-  "focalized substitution (preterm)"
-  'PitchforkLeftArrow t p u = (fsubst p u t).
-*)
+  "focused balanced reduction with delayed updating (prototerm)"
+  'BlackRightArrowDF t1 p q t2 = (dfr p q t1 t2).