L1 ⪤*[RN, RP, f] L2.
#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
-| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
- #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
-| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
- #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/
+| #f #I1 #I #K1 #K #HK1 #HI1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
+ #I2 #K2 #HK2 #HI2 #H destruct /4 width=6 by lexs_next/
+| #f #I1 #I #K1 #K #HK1 #HI1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
+ #I2 #K2 #HK2 #HI2 #H destruct /4 width=6 by lexs_push/
]
qed-.
(* Basic_2A1: includes: lpx_sn_conf *)
theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
∀L,f.
- (∀g,I,K,V,n. ⬇*[n] L ≡ K.ⓑ{I}V → ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K V) →
- (∀g,I,K,V,n. ⬇*[n] L ≡ K.ⓑ{I}V → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K V) →
+ (∀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 #I #V #IH #f elim (pn_split f) * #g #H destruct
+| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
#HN #HP #Y1 #H1 #Y2 #H2
- [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL1 #HV1 #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
- elim (HP … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
+ [ 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 #L1 #V1 #HL1 #HV1 #H destruct
- elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL2 #HV2 #H destruct
- elim (HN … 0 … HV1 … HV2 … HL1 … HL2) -HV1 -HV2 /2 width=2 by drops_refl/ #V #HV1 #HV2
+ | 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/
]
]
∀f2. L1 ⪤*[RN, RP, f2] L2 →
∀f. f1 ⋒ f2 ≡ f → L1 ⪤*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
-#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
∀f2. L1 ⪤*[RN, RP, f2] L2 →
∀f. f1 ⋓ f2 ≡ f → L1 ⪤*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
-#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)