-lemma lpxs_inv_pair_sn: ∀h,G,I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢⬈*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
-#h #G #I #L2 #K1 #V1 #H
-elim (lpxs_inv_bind_sn … H) -H #Y #K2 #HK12 #H0 #H destruct
-elim (ext2_inv_pair_sn … H0) -H0 #V2 #HV12 #H destruct
-/2 width=5 by ex3_2_intro/
-qed-.
+(* Basic_2A1: was: lpxs_inv_pair1 *)
+lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ❪G,K1.ⓑ[I]V1❫ ⊢ ⬈*[h] L2 →
+ ∃∃K2,V2. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ[I]V2.
+/2 width=1 by lex_inv_pair_sn/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_atom2 *)
+lemma lpxs_inv_atom_dx (h) (G): ∀L1. ❪G,L1❫ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
+/2 width=2 by lex_inv_atom_dx/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_pair2 *)
+lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ❪G,L1❫ ⊢ ⬈*[h] K2.ⓑ[I]V2 →
+ ∃∃K1,V1. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ[I]V1.
+/2 width=1 by lex_inv_pair_dx/ qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_2A1: was: lpxs_ind_alt *)
+lemma lpxs_ind (h) (G): ∀Q:relation lenv.
+ Q (⋆) (⋆) → (
+ ∀I,K1,K2.
+ ❪G,K1❫ ⊢ ⬈*[h] K2 →
+ Q K1 K2 → Q (K1.ⓘ[I]) (K2.ⓘ[I])
+ ) → (
+ ∀I,K1,K2,V1,V2.
+ ❪G,K1❫ ⊢ ⬈*[h] K2 → ❪G,K1❫ ⊢ V1 ⬈*[h] V2 →
+ Q K1 K2 → Q (K1.ⓑ[I]V1) (K2.ⓑ[I]V2)
+ ) →
+ ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 → Q L1 L2.
+/3 width=4 by lex_ind/ qed-.