∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-(* Basic properties ***********************************************************)
+(* Basic properties *********************************************************)
lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
/3 width=3 by lexs_atom, frees_atom, ex2_intro/
#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
qed-.
+lemma lfxs_isid: ∀R1,R2,L1,L2,T1,T2.
+ (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → 𝐈⦃f⦄) →
+ (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≡ f) →
+ L1 ⦻*[R1, T1] L2 → L1 ⦻*[R2, T2] L2.
+#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
+/4 width=7 by lexs_co_isid, ex2_intro/
+qed-.
+
(* Basic inversion lemmas ***************************************************)
-lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
-#R #I #Y2 * /2 width=4 by lexs_inv_atom1/
+lemma lfxs_inv_atom_sn: ∀R,Y2,T. ⋆ ⦻*[R, T] Y2 → Y2 = ⋆.
+#R #Y2 #T * /2 width=4 by lexs_inv_atom1/
qed-.
-lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma lfxs_inv_atom_dx: ∀R,Y1,T. Y1 ⦻*[R, T] ⋆ → Y1 = ⋆.
#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
qed-.
#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
qed-.
-(* Basic_2A1: removed theorems 24:
+lemma lfxs_fwd_dx: ∀R,I,L1,K2,T,V2. L1 ⦻*[R, T] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. L1 = K1.ⓑ{I}V1.
+#R #I #L1 #K2 #T #V2 * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
+[ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #K1 #V1 #_ #_ #H destruct
+/2 width=3 by ex1_2_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 25:
llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
llpx_sn_bind llpx_sn_flat
llpx_sn_inv_bind llpx_sn_inv_flat
llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
- llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
+ llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
+ llpx_sn_dec
*)