+
+(* Properties on relocation *************************************************)
+
+lemma frees_lift_ge: ∀K,T,d,i. K ⊢ i ϵ𝐅*[d]⦃T⦄ →
+ ∀L,s,d0,e0. ⇩[s, d0, e0] L ≡ K →
+ ∀U. ⇧[d0, e0] T ≡ U → d0 ≤ i →
+ L ⊢ i+e0 ϵ 𝐅*[d]⦃U⦄.
+#K #T #d #i #H elim H -K -T -d -i
+[ #K #T #d #i #HnT #L #s #d0 #e0 #_ #U #HTU #Hd0i -K -s
+ @frees_eq #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
+| #I #K #K0 #T #V #d #i #j #Hdj #Hji #HnT #HK0 #HV #IHV #L #s #d0 #e0 #HLK #U #HTU #Hd0i
+ elim (lt_or_ge j d0) #H1
+ [ elim (ldrop_trans_lt … HLK … HK0) // -K #L0 #W #HL0 #HLK0 #HVW
+ @(frees_be … HL0) -HL0 -HV
+ /3 width=3 by lt_plus_to_minus_r, lt_to_le_to_lt/
+ [ #X #HXU >(plus_minus_m_m d0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
+ elim (lift_div_le … HXU … HTU ?) -U /2 width=2 by monotonic_pred/
+ | >minus_plus <plus_minus // <minus_plus
+ /3 width=5 by monotonic_le_minus_l2/
+ ]
+ | lapply (ldrop_trans_ge … HLK … HK0 ?) // -K #HLK0
+ lapply (ldrop_inv_gen … HLK0) >commutative_plus -HLK0 #HLK0
+ @(frees_be … HLK0) -HLK0 -IHV
+ /2 width=1 by yle_plus_dx1_trans, lt_minus_to_plus/
+ #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
+ ]
+]
+qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma frees_inv_lift_be: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+ ∀K,s,d0,e0. ⇩[s, d0, e0+1] L ≡ K →
+ ∀T. ⇧[d0, e0+1] T ≡ U → d0 ≤ i → i ≤ d0 + e0 → ⊥.
+#L #U #d #i #H elim H -L -U -d -i
+[ #L #U #d #i #HnU #K #s #d0 #e0 #_ #T #HTU #Hd0i #Hide0
+ elim (lift_split … HTU i e0) -HTU /2 width=2 by/
+| #I #L #K0 #U #W #d #i #j #Hdi #Hij #HnU #HLK0 #_ #IHW #K #s #d0 #e0 #HLK #T #HTU #Hd0i #Hide0
+ elim (lt_or_ge j d0) #H1
+ [ elim (ldrop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
+ @(IHW … HKL0 … HVW)
+ [ /2 width=1 by monotonic_le_minus_l2/
+ | >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
+ ]
+ | elim (lift_split … HTU j e0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
+ ]
+]
+qed-.
+
+lemma frees_inv_lift_ge: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+ ∀K,s,d0,e0. ⇩[s, d0, e0] L ≡ K →
+ ∀T. ⇧[d0, e0] T ≡ U → d0 + e0 ≤ i →
+ K ⊢ i-e0 ϵ𝐅*[d-yinj e0]⦃T⦄.
+#L #U #d #i #H elim H -L -U -d -i
+[ #L #U #d #i #HnU #K #s #d0 #e0 #HLK #T #HTU #Hde0i -L -s
+ elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i @frees_eq #X #HXT -K
+ elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
+| #I #L #K0 #U #W #d #i #j #Hdi #Hij #HnU #HLK0 #_ #IHW #K #s #d0 #e0 #HLK #T #HTU #Hde0i
+ elim (lt_or_ge j d0) #H1
+ [ elim (ldrop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
+ elim (le_inv_plus_l … Hde0i) #H0 #He0i
+ @(frees_be … H) -H
+ [ /3 width=1 by yle_plus_dx1_trans, monotonic_yle_minus_dx/
+ | /2 width=3 by lt_to_le_to_lt/
+ | #X #HXT elim (lift_trans_ge … HXT … HTU) -T /2 width=2 by/
+ | lapply (IHW … HKL0 … HVW ?) // -I -K -K0 -L0 -V -W -T -U -s
+ >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
+ ]
+ | elim (lt_or_ge j (d0+e0)) [ >commutative_plus |] #H2
+ [ -L -I -W lapply (lt_plus_to_minus ???? H2) // -H2 #H2
+ elim (lift_split … HTU j (e0-1)) -HTU //
+ [ >minus_minus_associative /2 width=2 by ltn_to_ltO/ <minus_n_n
+ #X #_ #H elim (HnU … H)
+ | >commutative_plus /3 width=1 by le_minus_to_plus, monotonic_pred/
+ ]
+ | lapply (ldrop_conf_ge … HLK … HLK0 ?) // -L #HK0
+ elim (le_inv_plus_l … H2) -H2 #H2 #He0j
+ @(frees_be … HK0)
+ [ /2 width=1 by monotonic_yle_minus_dx/
+ | /2 width=1 by monotonic_lt_minus_l/
+ | #X #HXT elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
+ | >arith_b1 /2 width=5 by/
+ ]
+ ]
+ ]
+]
+qed-.