include "lambda-delta/language/lenv.ma".
include "lambda-delta/substitution/lift.ma".
-(* SUBSTITUTION *************************************************************)
+(* TELESCOPIC SUBSTITUTION **************************************************)
inductive subst: lenv → term → nat → nat → term → Prop ≝
| subst_sort : ∀L,k,d,e. subst L (⋆k) d e (⋆k)
| subst_lref_lt: ∀L,i,d,e. i < d → subst L (#i) d e (#i)
| subst_lref_O : ∀L,V,e. 0 < e → subst (L. ♭Abbr V) #0 0 e V
| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
- d ≤ i → i < d + e → subst L #i d e T1 → [1,d]↑ T1 ≡ T2 →
+ d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T1 ≡ T2 →
subst (L. ♭I V) #(i + 1) (d + 1) e T2
| subst_lref_ge: ∀L,i,d,e. d + e ≤ i → subst L (#i) d e (#(i - e))
| subst_con2 : ∀L,I,V1,V2,T1,T2,d,e.
interpretation "telescopic substritution" 'RSubst L T1 d e T2 = (subst L T1 d e T2).
-lemma subst_lift_inv: ∀d,e,T1,T2. [d,e]↑ T1 ≡ T2 → ∀L. [d,e]← L / T2 ≡ T1.
+lemma subst_lift_inv: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀L. L ⊢ ↓[d,e] T2 ≡ T1.
#d #e #T1 #T2 #H elim H -H d e T1 T2 /2/
-#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/
+#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/ (**) (* use \ldots *)
qed.
-