(* Relocation properties ****************************************************)
lemma cpds_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 →
- ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •*➡*[g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- ⦃h, L⦄ ⊢ U1 •*➡*[g] U2.
+ ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •*➡*[h, g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 •*➡*[h, g] U2.
#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 * #T
elim (lift_total T d e) /3 width=11/
qed.
lemma cpds_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •*➡*[g] U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •*➡*[g] T2.
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃G, L⦄ ⊢ U1 •*➡*[h, g] U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •*➡*[h, g] T2.
#L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 * #U #HU1 #HU2
elim (sstas_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HT1 #HTU
elim (cprs_inv_lift1 … HU2 … HLK … HTU) -U -L /3 width=5/