theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
∀L1,f.
- (â\88\80g,I,K,n. â¬\87*[n] L1 â\89\98 K.â\93\98{I} â\86\92 ⫯g = ⫱*[n] f → lexs_transitive RN1 RN2 RN RN1 RP1 g K I) →
- (â\88\80g,I,K,n. â¬\87*[n] L1 â\89\98 K.â\93\98{I} â\86\92 â\86\91g = ⫱*[n] f → lexs_transitive RP1 RP2 RP RN1 RP1 g K I) →
+ (â\88\80g,I,K,n. â¬\87*[n] L1 â\89\98 K.â\93\98{I} â\86\92 â\86\91g = ⫱*[n] f → lexs_transitive RN1 RN2 RN RN1 RP1 g K I) →
+ (â\88\80g,I,K,n. â¬\87*[n] L1 â\89\98 K.â\93\98{I} â\86\92 ⫯g = ⫱*[n] f → lexs_transitive RP1 RP2 RP RN1 RP1 g K I) →
∀L0. L1 ⪤*[RN1, RP1, f] L0 →
∀L2. L0 ⪤*[RN2, RP2, f] L2 →
L1 ⪤*[RN, RP, f] L2.
(* Basic_2A1: includes: lpx_sn_conf *)
theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
∀L,f.
- (â\88\80g,I,K,n. â¬\87*[n] L â\89\98 K.â\93\98{I} â\86\92 ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
- (â\88\80g,I,K,n. â¬\87*[n] L â\89\98 K.â\93\98{I} â\86\92 â\86\91g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
+ (â\88\80g,I,K,n. â¬\87*[n] L â\89\98 K.â\93\98{I} â\86\92 â\86\91g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
+ (â\88\80g,I,K,n. â¬\87*[n] L â\89\98 K.â\93\98{I} â\86\92 ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
#RN1 #RP1 #RN2 #RP2 #L elim L -L
[ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1