]
qed-.
-lemma frees_S: â\88\80L,U,d,i. L â\8a¢ i ϵ ð\9d\90\85*[yinj d]â¦\83Uâ¦\84 â\86\92 â\88\80I,K,W. â\87©[d] L ≡ K.ⓑ{I}W →
+lemma frees_S: â\88\80L,U,d,i. L â\8a¢ i ϵ ð\9d\90\85*[yinj d]â¦\83Uâ¦\84 â\86\92 â\88\80I,K,W. â¬\87[d] L ≡ K.ⓑ{I}W →
(K ⊢ i-d-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯d]⦃U⦄.
#L #U #d #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
* #I #K #W #j #Hdj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
(* Properties on relocation *************************************************)
lemma frees_lift_ge: ∀K,T,d,i. K ⊢ i ϵ𝐅*[d]⦃T⦄ →
- â\88\80L,s,d0,e0. â\87©[s, d0, e0] L ≡ K →
- â\88\80U. â\87§[d0, e0] T ≡ U → d0 ≤ i →
+ â\88\80L,s,d0,e0. â¬\87[s, d0, e0] L ≡ K →
+ â\88\80U. â¬\86[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
(* Inversion lemmas on relocation *******************************************)
lemma frees_inv_lift_be: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- â\88\80K,s,d0,e0. â\87©[s, d0, e0+1] L ≡ K →
- â\88\80T. â\87§[d0, e0+1] T ≡ U → d0 ≤ i → i ≤ d0 + e0 → ⊥.
+ â\88\80K,s,d0,e0. â¬\87[s, d0, e0+1] L ≡ K →
+ â\88\80T. â¬\86[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/
qed-.
lemma frees_inv_lift_ge: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- â\88\80K,s,d0,e0. â\87©[s, d0, e0] L ≡ K →
- â\88\80T. â\87§[d0, e0] T ≡ U → d0 + e0 ≤ i →
+ â\88\80K,s,d0,e0. â¬\87[s, d0, e0] L ≡ K →
+ â\88\80T. â¬\86[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