+
+(* Basic eliminators ********************************************************)
+
+lemma cpx_ind (Q:relation4 …):
+ (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
+ (∀G,L,s1,s2. Q G L (⋆s1) (⋆s2)) →
+ (∀I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈ 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 ⬈ 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 ⬈ V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ 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 ⬈ V2 → ❪G,L❫ ⊢ T1 ⬈ 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 ⬈ T2 → Q G L T T2 →
+ Q G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
+ ) → (∀G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈ V2 → Q G L V1 V2 →
+ Q G L (ⓝV1.T) V2
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈ 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 ⬈ V → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈ 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 ⬈ T2 → Q G L T1 T2.
+#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-.