theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
∀L1,f.
- (â\88\80g,I,K,n. â¬\87*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) →
- (â\88\80g,I,K,n. â¬\87*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → sex_transitive RP1 RP2 RP RN1 RP1 g K I) →
+ (â\88\80g,I,K,n. â\87©*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) →
+ (â\88\80g,I,K,n. â\87©*[n] L1 ≘ K.ⓘ{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.
theorem sex_conf (RN1) (RP1) (RN2) (RP2):
∀L,f.
- (â\88\80g,I,K,n. â¬\87*[n] L ≘ K.ⓘ{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 ≘ K.ⓘ{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 ≘ K.ⓘ{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 ≘ K.ⓘ{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