+
+(* Basic eliminators ********************************************************)
+
+lemma cpm_ind (h) (Q:relation5 …):
+ (∀I,G,L. Q 0 G L (⓪[I]) (⓪[I])) →
+ (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
+ (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → Q n G K V1 V2 →
+ ⇧[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
+ ) → (∀n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → Q n G K V1 V2 →
+ ⇧[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
+ ) → (∀n,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h,n] T → Q n G K (#i) T →
+ ⇧[1] T ≘ U → Q n G (K.ⓘ[I]) (#↑i) (U)
+ ) → (∀n,p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 →
+ Q 0 G L V1 V2 → Q n G (L.ⓑ[I]V1) T1 T2 → Q n G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
+ ) → (∀n,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
+ ) → (∀n,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,n] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
+ ) → (∀n,G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → ❪G,L❫ ⊢ T ➡[h,n] T2 →
+ Q n G L T T2 → Q n G L (+ⓓV.T1) T2
+ ) → (∀n,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
+ ) → (∀n,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] V2 →
+ Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
+ ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 →
+ Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
+ Q n G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+ ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 →
+ Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
+ ⇧[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
+ ) →
+ ∀n,G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → Q n G L T1 T2.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
+* #c #HC #H generalize in match HC; -HC generalize in match n; -n
+elim H -c -G -L -T1 -T2
+[ #I #G #L #n #H <(isrt_inv_00 … H) -H //
+| #G #L #s1 #s2 #HRs #n #H <(isrt_inv_01 … H) -H destruct //
+| /3 width=4 by ex2_intro/
+| #c #G #L #V1 #V2 #W2 #HV12 #HVW2 #IH #x #H
+ elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
+ /3 width=4 by ex2_intro/
+| /3 width=4 by ex2_intro/
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
+ elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
+ /3 width=3 by ex2_intro/
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
+ elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
+ /3 width=3 by ex2_intro/
+| #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #n #H
+ elim (isrt_inv_max_eq_t … H) -H // #HcV #HcT
+ /3 width=3 by ex2_intro/
+| #c #G #L #V #T1 #T #T2 #HT1 #HT2 #IH #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
+ /3 width=4 by ex2_intro/
+| #c #G #L #U #T1 #T2 #HT12 #IH #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
+ /3 width=3 by ex2_intro/
+| #c #G #L #U1 #U2 #T #HU12 #IH #x #H
+ elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
+ /3 width=3 by ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #IHV #IHW #IHT #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
+ elim (isrt_inv_max_shift_sn … H) -H #H #HcT
+ elim (isrt_O_inv_max … H) -H #HcV #HcW
+ /3 width=3 by ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #IHV #IHW #IHT #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
+ elim (isrt_inv_max_shift_sn … H) -H #H #HcT
+ elim (isrt_O_inv_max … H) -H #HcV #HcW
+ /3 width=4 by ex2_intro/
+]
+qed-.