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.
]
qed-.
-(* Basic_2A1: includes: lpx_sn_trans *)
theorem lexs_trans (RN) (RP) (f): (∀g,I,K. lexs_transitive RN RN RN RN RP g K I) →
(∀g,I,K. lexs_transitive RP RP RP RN RP g K I) →
Transitive … (lexs RN RP f).
/2 width=9 by lexs_trans_gen/ qed-.
theorem lexs_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤*[R1, cfull, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤*[R2, cfull, f] L2 → L1 ⪤*[R3, cfull, f] L2.
+ ∀L2. L ⪤*[R2, cfull, f] L2 → L1 ⪤*[R3, cfull, f] L2.
#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
[ #f #Hf #L2 #H >(lexs_inv_atom1 … H) -L2 // ]
#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
/3 width=1 by lexs_push/
qed-.
-(* 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