-theorem lfxs_conf: ∀R1,R2.
- lexs_frees_confluent R1 cfull →
- lexs_frees_confluent R2 cfull →
- R_confluent2_lfxs R1 R2 R1 R2 →
- ∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
-#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
-lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
-lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01
-elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
-[ #L #HL1 #HL2
- elim (HR1 … Hf … HL01) -HL01 #f1 #Hf1 #H1
- elim (HR2 … Hf … HL02) -HL02 #f2 #Hf2 #H2
- lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
- lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
- /3 width=5 by ex2_intro/
-| #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02
- elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0
- lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01
- lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02
- elim (HR12 … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/
+theorem lfxs_bind_void: ∀R,p,I,L1,L2,V,T.
+ L1 ⪤*[R, V] L2 → L1.ⓧ ⪤*[R, T] L2.ⓧ →
+ L1 ⪤*[R, ⓑ{p,I}V.T] L2.
+#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
+lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind_void, lexs_join, isfin_tl, ex2_intro/
+qed.
+
+theorem lfxs_trans_gen: ∀R1,R2,R3.
+ c_reflexive … R1 → c_reflexive … R2 →
+ lfxs_confluent R1 R2 → lfxs_transitive R1 R2 R3 →
+ ∀L1,T,L. L1 ⪤*[R1, T] L →
+ ∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2.
+#R1 #R2 #R3 #H1R #H2R #H3R #H4R #L1 #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L1 T) -L1 -T
+#G0 #L0 #T0 #IH #G #L1 * *
+[ #s #HG #HL #HT #L #H1 #L2 #H2 destruct
+ elim (lfxs_inv_sort … H1) -H1 *
+ [ #H1 #H0 destruct
+ >(lfxs_inv_atom_sn … H2) -L2 //
+ | #I1 #I #K1 #K #HK1 #H1 #H0 destruct
+ elim (lfxs_inv_sort_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct
+ /4 width=3 by lfxs_sort, fqu_fqup/
+ ]
+| * [ | #i ] #HG #HL #HT #L #H1 #L2 #H2 destruct
+ [ elim (lfxs_inv_zero … H1) -H1 *
+ [ #H1 #H0 destruct
+ >(lfxs_inv_atom_sn … H2) -L2 //
+ | #I #K1 #K #V1 #V #HK1 #H1 #H0 #H destruct
+ elim (lfxs_inv_zero_pair_sn … H2) -H2 #K2 #V2 #HK2 #HV2 #H destruct
+ /4 width=7 by lfxs_pair, fqu_fqup, fqu_lref_O/
+ | #f1 #I #K1 #K #Hf1 #HK1 #H1 #H0 destruct
+ elim (lfxs_inv_zero_unit_sn … H2) -H2 #f2 #K2 #Hf2 #HK2 #H destruct
+ /5 width=8 by lfxs_unit, lexs_trans_id_cfull, lexs_eq_repl_back, isid_inv_eq_repl/
+ ]
+ | elim (lfxs_inv_lref … H1) -H1 *
+ [ #H1 #H0 destruct
+ >(lfxs_inv_atom_sn … H2) -L2 //
+ | #I1 #I #K1 #K #HK1 #H1 #H0 destruct
+ elim (lfxs_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct
+ /4 width=3 by lfxs_lref, fqu_fqup/
+ ]
+ ]
+| #l #HG #HL #HT #L #H1 #L2 #H2 destruct
+ elim (lfxs_inv_gref … H1) -H1 *
+ [ #H1 #H0 destruct
+ >(lfxs_inv_atom_sn … H2) -L2 //
+ | #I1 #I #K1 #K #HK1 #H1 #H0 destruct
+ elim (lfxs_inv_gref_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct
+ /4 width=3 by lfxs_gref, fqu_fqup/
+ ]
+| #p #I #V1 #T1 #HG #HL #HT #L #H1 #L2 #H2 destruct
+ elim (lfxs_inv_bind … V1 V1 … H1) -H1 // #H1V #H1T
+ elim (lfxs_inv_bind … V1 V1 … H2) -H2 // #H2V #H2T
+ /3 width=4 by lfxs_bind/
+| #I #V1 #T1 #HG #HL #HT #L #H1 #L2 #H2 destruct
+ elim (lfxs_inv_flat … H1) -H1 #H1V #H1T
+ elim (lfxs_inv_flat … H2) -H2 #H2V #H2T
+ /3 width=3 by lfxs_flat/