+(* Forward lemmas with strongly rt-normalizing terms ************************)
+
+fact rsx_fwd_lref_pair_csx_aux (h) (G):
+ ∀L. G ⊢ ⬈*𝐒[h,#0] L →
+ ∀I,K,V. L = K.ⓑ[I]V → ❪G,K❫ ⊢ ⬈*𝐒[h] V.
+#h #G #L #H
+@(rsx_ind … H) -L #L #_ #IH #I #K #V1 #H destruct
+@csx_intro #V2 #HV12 #HnV12
+@(IH … I) -IH [1,4: // | -HnV12 | -G #H ]
+[ /2 width=1 by lpx_pair/
+| elim (reqx_inv_zero_pair_sn … H) -H #Y #X #_ #H1 #H2 destruct -I
+ /2 width=1 by/
+]
+qed-.
+
+lemma rsx_fwd_lref_pair_csx (h) (G):
+ ∀I,K,V. G ⊢ ⬈*𝐒[h,#0] K.ⓑ[I]V → ❪G,K❫ ⊢ ⬈*𝐒[h] V.
+/2 width=4 by rsx_fwd_lref_pair_csx_aux/ qed-.
+
+lemma rsx_fwd_lref_pair_csx_drops (h) (G):
+ ∀I,K,V,i,L. ⇩[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*𝐒[h,#i] L → ❪G,K❫ ⊢ ⬈*𝐒[h] V.
+#h #G #I #K #V #i elim i -i
+[ #L #H >(drops_fwd_isid … H) -H
+ /2 width=2 by rsx_fwd_lref_pair_csx/
+| #i #IH #L #H1 #H2
+ elim (drops_inv_bind2_isuni_next … H1) -H1 // #J #Y #HY #H destruct
+ lapply (rsx_inv_lifts … H2 … (𝐔❨1❩) ?????) -H2
+ /3 width=6 by drops_refl, drops_drop/
+]
+qed-.
+
+(* Inversion lemmas with strongly rt-normalizing terms **********************)
+
+lemma rsx_inv_lref_pair (h) (G):
+ ∀I,K,V. G ⊢ ⬈*𝐒[h,#0] K.ⓑ[I]V →
+ ∧∧ ❪G,K❫ ⊢ ⬈*𝐒[h] V & G ⊢ ⬈*𝐒[h,V] K.
+/3 width=2 by rsx_fwd_lref_pair_csx, rsx_fwd_pair, conj/ qed-.
+
+lemma rsx_inv_lref_pair_drops (h) (G):
+ ∀I,K,V,i,L. ⇩[i] L ≘ K.ⓑ[I]V → G ⊢ ⬈*𝐒[h,#i] L →
+ ∧∧ ❪G,K❫ ⊢ ⬈*𝐒[h] V & G ⊢ ⬈*𝐒[h,V] K.
+/3 width=5 by rsx_fwd_lref_pair_csx_drops, rsx_fwd_lref_pair_drops, conj/ qed-.
+
+lemma rsx_inv_lref_drops (h) (G):
+ ∀L,i. G ⊢ ⬈*𝐒[h,#i] L →
+ ∨∨ ⇩*[Ⓕ,𝐔❨i❩] L ≘ ⋆
+ | ∃∃I,K. ⇩[i] L ≘ K.ⓤ[I]
+ | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*𝐒[h] V & G ⊢ ⬈*𝐒[h,V] K.
+#h #G #L #i #H elim (drops_F_uni L i)
+[ /2 width=1 by or3_intro0/
+| * * /4 width=10 by rsx_fwd_lref_pair_csx_drops, rsx_fwd_lref_pair_drops, ex3_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
+]
+qed-.
+
+(* Properties with strongly rt-normalizing terms ****************************)