-(* Basic_2A1: includes: lpx_sn_conf *)
-theorem lexs_conf: ∀RN1,RP1,RN2,RP2.
- lexs_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
- lexs_confluent RP1 RP2 RN1 RP1 RN2 RP2 →
- ∀f. confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f).
-#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L0 generalize in match f; -f
-@(f_ind … lw … L0) -L0 #x #IH *
-[ #_ #f #X1 #H1 #X2 #H2 -x
- >(lexs_inv_atom1 … H1) -X1
- >(lexs_inv_atom1 … H2) -X2 /2 width=3 by lexs_atom, ex2_intro/
-| #L0 #I #V0 #Hx #f elim (pn_split f) *
- #g #H #X1 #H1 #X2 #H2 destruct
- [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
- elim (HRP … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_push, ex2_intro/
- | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
- elim (HRN … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_next, ex2_intro/
+theorem lexs_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤*[R1, cfull, f] L → 𝐈⦃f⦄ →
+ ∀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 >(lexs_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 (lexs_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
+/3 width=1 by lexs_push/
+qed-.
+
+theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
+ ∀L,f.
+ (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
+ (∀g,I,K,n. ⬇*[n] L ≘ 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
+ /2 width=3 by lexs_atom, ex2_intro/
+| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
+ #HN #HP #Y1 #H1 #Y2 #H2
+ [ elim (lexs_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
+ elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
+ elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
+ | elim (lexs_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
+ elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
+ elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
+ elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/