theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
∀L1,f.
- (â\88\80g,I,K,n. â¬\87*[n] L1 â\89¡ K.ⓘ{I} → ⫯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¡ K.ⓘ{I} → ↑g = ⫱*[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.ⓘ{I} → ⫯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.ⓘ{I} → ↑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¡ K.ⓘ{I} → ⫯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¡ K.ⓘ{I} → ↑g = ⫱*[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.ⓘ{I} → ⫯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.ⓘ{I} → ↑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
lemma lexs_meet: ∀RN,RP,L1,L2.
∀f1. L1 ⪤*[RN, RP, f1] L2 →
∀f2. L1 ⪤*[RN, RP, f2] L2 →
- â\88\80f. f1 â\8b\92 f2 â\89¡ f → L1 ⪤*[RN, RP, f] L2.
+ â\88\80f. f1 â\8b\92 f2 â\89\98 f → L1 ⪤*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
lemma lexs_join: ∀RN,RP,L1,L2.
∀f1. L1 ⪤*[RN, RP, f1] L2 →
∀f2. L1 ⪤*[RN, RP, f2] L2 →
- â\88\80f. f1 â\8b\93 f2 â\89¡ f → L1 ⪤*[RN, RP, f] L2.
+ â\88\80f. f1 â\8b\93 f2 â\89\98 f → L1 ⪤*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct