theorem sex_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} → ↑g = ⫱*[n] f → sex_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} → ⫯g = ⫱*[n] f → sex_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] → ↑g = ⫱*[n] f → sex_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] → ⫯g = ⫱*[n] f → sex_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.
Transitive … (sex RN RP f).
/2 width=9 by sex_trans_gen/ qed-.
-theorem sex_trans_id_cfull: â\88\80R1,R2,R3,L1,L,f. L1 ⪤[R1,cfull,f] L â\86\92 ð\9d\90\88â¦\83fâ¦\84 →
+theorem sex_trans_id_cfull: â\88\80R1,R2,R3,L1,L,f. L1 ⪤[R1,cfull,f] L â\86\92 ð\9d\90\88â\9dªfâ\9d« →
∀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 >(sex_inv_atom1 … H) -L2 // ]
theorem sex_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} → ↑g = ⫱*[n] f → R_pw_confluent2_sex 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} → ⫯g = ⫱*[n] f → R_pw_confluent2_sex 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] → ↑g = ⫱*[n] f → R_pw_confluent2_sex 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] → ⫯g = ⫱*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L.
#RN1 #RP1 #RN2 #RP2 #L elim L -L
[ #f #_ #_ #L1 #H1 #L2 #H2 >(sex_inv_atom1 … H1) >(sex_inv_atom1 … H2) -H2 -H1