+#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2
+elim (cpye_inv_lref1_subst_ex … H … HLK) -H -HLK //
+#X2 #H0 #HXT2 lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma cpye_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdetd
+elim (cpys_inv_lift1_le … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_le … HU2 … HLK … HTU2 ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet
+elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_be … HU2 … HLK … HTU2 ? ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ yinj d + e ≤ dt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdedt
+elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_ge … HU2 … HLK … HTU2 ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] 𝐍⦃T2⦄ &
+ ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_ge_up … HU2 … HLK … HTU2 ? ? ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] 𝐍⦃W2⦄ →
+ ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
+ d ≤ yinj i → i < d + e →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ & ⇧[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #d #e * #HW12 #HW2 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_subst … HW12 … HLK … HVW1) -W1 // #V2 #HV12 #HVW2
+lapply (cny_inv_lift_subst … HLK HW2 HVW2) -L
+/3 width=3 by ex2_intro, conj/