definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. lpx_sn (λ_.R1) L0 L1 → ∀L2. lpx_sn (λ_.R2) L0 L2 →
+ ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn (λ_.R1) L1 L2 →
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
∀T2. R2 L2 T T2 → R1 L1 T1 T2.
(* Main properties **********************************************************)
-theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn (λ_.R)).
+theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
#R #HR #L1 #L #H elim H -L1 -L //
#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
qed-.
theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
- confluent2 … (lpx_sn (λ_.R1)) (lpx_sn (λ_.R2)).
+ confluent2 … (lpx_sn R1) (lpx_sn R2).
#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
[ #_ #X1 #H1 #X2 #H2 -n
>(lpx_sn_inv_atom1 … H1) -X1