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.
lapply (sex_inv_atom1 … H1) -H1 #H destruct
lapply (sex_inv_atom1 … H2) -H2 #H destruct
/2 width=1 by sex_atom/
-| #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct
+| #K1 #I1 #IH #f elim (pr_map_split_tl f) * #g #H destruct
#HN #HP #L0 #H1 #L2 #H2
[ elim (sex_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
elim (sex_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
/2 width=9 by sex_trans_gen/ qed-.
theorem sex_trans_id_cfull (R1) (R2) (R3):
- â\88\80L1,L,f. L1 ⪤[R1,cfull,f] L â\86\92 ð\9d\90\88â\9dªfâ\9d« →
+ â\88\80L1,L,f. L1 ⪤[R1,cfull,f] L â\86\92 ð\9d\90\88â\9d¨fâ\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 // ]
#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
-[ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
+[ elim (pr_isi_inv_next … Hf) | lapply (pr_isi_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
elim (sex_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
/3 width=1 by sex_push/
qed-.
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
/2 width=3 by sex_atom, ex2_intro/
-| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
+| #L #I0 #IH #f elim (pr_map_split_tl f) * #g #H destruct
#HN #HP #Y1 #H1 #Y2 #H2
[ elim (sex_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
elim (sex_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
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
lapply (sex_inv_atom1 … HY) -HY #H destruct
lapply (sex_inv_atom1 … HY1) -HY1 #H destruct
lapply (sex_inv_atom1 … HY2) -HY2 #H destruct //
-| #L1 #I1 #IH #f elim (pn_split f) * #g #H destruct
+| #L1 #I1 #IH #f elim (pr_map_split_tl f) * #g #H destruct
#HN #HP #Y #HY #Y1 #HY1 #Y2 #HY2
[ elim (sex_inv_push1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct
elim (sex_inv_push1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct
∀f. f1 ⋒ f2 ≘ 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
+elim (pr_map_split_tl f2) * #g2 #H2 destruct
try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
-[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
-| elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
+[ elim (pr_sand_inv_next_push … Hf) | elim (pr_sand_inv_next_bi … Hf)
+| elim (pr_sand_inv_push_bi … Hf) | elim (pr_sand_inv_push_next … Hf)
] -Hf /3 width=5 by sex_next, sex_push/
qed-.
∀f. f1 ⋓ f2 ≘ 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
+elim (pr_map_split_tl f2) * #g2 #H2 destruct
try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
-[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
-| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
+[ elim (pr_sor_inv_next_push … Hf) | elim (pr_sor_inv_next_bi … Hf)
+| elim (pr_sor_inv_push_bi … Hf) | elim (pr_sor_inv_push_next … Hf)
] -Hf /3 width=5 by sex_next, sex_push/
qed-.