-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_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 elim L0 -L0
+[ #f #L1 #HL01 #L2 #HL02 -HRN -HRP
+ lapply (lexs_inv_atom1 … HL01) -HL01 #H destruct
+ lapply (lexs_inv_atom1 … HL02) -HL02 #H destruct
+ /2 width=3 by ex2_intro/
+| #K0 #I #V0 #IH #f #L1 #HL01 #L2 #HL02
+ elim (pn_split f) * #g #H destruct
+ [ elim (lexs_inv_push1 … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct
+ elim (lexs_inv_push1 … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct
+ elim (IH … HK01 … HK02) -IH #K #HK1 #HK2
+ elim (HRP … HV01 … HV02 … HK01 … HK02) -HRP -HRN -K0 -V0
+ /3 width=5 by lexs_push, ex2_intro/
+ | elim (lexs_inv_next1 … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct
+ elim (lexs_inv_next1 … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct
+ elim (IH … HK01 … HK02) -IH #K #HK1 #HK2
+ elim (HRN … HV01 … HV02 … HK01 … HK02) -HRN -HRP -K0 -V0
+ /3 width=5 by lexs_next, ex2_intro/