+
+(* Basic eliminators ********************************************************)
+
+lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
+ (∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
+ (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
+ (∀I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
+ ⇧*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
+ ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
+ ⇧*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q 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 →
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) → (∀G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 →
+ Q G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
+ ) → (∀G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
+ Q 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 →
+ Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
+ Q 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 →
+ Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
+ ⇧*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ ) →
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
+#h #Q #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-.