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] â\86\92 â\86\91g = ⫱*[n] f → R_pw_transitive_sex 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 → R_pw_transitive_sex 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 = â«°*[n] f → R_pw_transitive_sex 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 = â«°*[n] f → R_pw_transitive_sex 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 â\89\98 K.â\93\98[I] â\86\92 â\86\91g = ⫱*[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] â\86\92 ⫯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] â\86\92 â\86\91g = â«°*[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] â\86\92 ⫯g = â«°*[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
qed-.
lemma sex_repl (RN) (RP) (SN) (SP) (L1) (f):
- (â\88\80g,I,K1,n. â\87©[n] L1 â\89\98 K1.â\93\98[I] â\86\92 â\86\91g = ⫱*[n] f → R_pw_replace3_sex … RN SN RN RP SN SP g K1 I) →
- (â\88\80g,I,K1,n. â\87©[n] L1 â\89\98 K1.â\93\98[I] â\86\92 ⫯g = ⫱*[n] f → R_pw_replace3_sex … RP SP RN RP SN SP g K1 I) →
+ (â\88\80g,I,K1,n. â\87©[n] L1 â\89\98 K1.â\93\98[I] â\86\92 â\86\91g = â«°*[n] f → R_pw_replace3_sex … RN SN RN RP SN SP g K1 I) →
+ (â\88\80g,I,K1,n. â\87©[n] L1 â\89\98 K1.â\93\98[I] â\86\92 ⫯g = â«°*[n] f → R_pw_replace3_sex … RP SP RN RP SN SP g K1 I) →
∀L2. L1 ⪤[RN,RP,f] L2 → ∀K1. L1 ⪤[SN,SP,f] K1 →
∀K2. L2 ⪤[SN,SP,f] K2 → K1 ⪤[RN,RP,f] K2.
#RN #RP #SN #SP #L1 elim L1 -L1