(* Advanced inversion lemmas ************************************************)
+lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
+ T2 = #0 ∨
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2.
+#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
+/4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
+qed-.
+
+lemma cpx_inv_lref1_pair: ∀h,I,G,K,V,T2,i. ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[h] T2 →
+ T2 = #(⫯i) ∨
+ ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2.
+#h #I #G #L #V #T2 #i * #c #H elim (cpg_inv_lref1_pair … H) -H *
+/4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
+qed-.
+
lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓕ{I}V2.T2
#h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex_intro/
qed-.
+
+(* Basic eliminators ********************************************************)
+
+lemma cpx_ind: ∀h. ∀R:relation4 genv lenv term term.
+ (∀I,G,L. R G L (⓪{I}) (⓪{I})) →
+ (∀G,L,s. R G L (⋆s) (⋆(next h s))) →
+ (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → R G K V1 V2 →
+ ⬆*[1] V2 ≡ W2 → R G (K.ⓑ{I}V1) (#0) W2
+ ) → (∀I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → R G K (#i) T →
+ ⬆*[1] T ≡ U → R G (K.ⓑ{I}V) (#⫯i) (U)
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ R G L V1 V2 → R G (L.ⓑ{I}V1) T1 T2 → R G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+ ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+ R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) → (∀G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T → R G (L.ⓓV) T1 T →
+ ⬆*[1] T2 ≡ T → R G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2 →
+ R G L (ⓝV.T1) T2
+ ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → R G L V1 V2 →
+ R G L (ⓝV1.T) V2
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+ R G L V1 V2 → R G L W1 W2 → R G (L.ⓛW1) T1 T2 →
+ R G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+ ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
+ R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 →
+ ⬆*[1] V ≡ V2 → R G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ ) →
+ ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2.
+#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
+* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
+qed-.