include "basic_2/relocation/ldrop_ldrop.ma".
include "basic_2/substitution/ldrops.ma".
-(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
(* Properties concerning basic local environment slicing ********************)
-lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[des] L1 ≡ L → ∀L2,i. ⇩[0, i] L ≡ L2 →
- ∃∃L0,des0,i0. ⇩[0, i0] L1 ≡ L0 & ⇩*[des0] L0 ≡ L2 &
+lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[Ⓕ, des] L1 ≡ L → ∀L2,i. ⇩[i] L ≡ L2 →
+ ∃∃L0,des0,i0. ⇩[i0] L1 ≡ L0 & ⇩*[Ⓕ, des0] L0 ≡ L2 &
@⦃i, des⦄ ≡ i0 & des ▭ i ≡ des0.
#L1 #L #des #H elim H -L1 -L -des
-[ /2 width=7/
+[ /2 width=7 by ldrops_nil, minuss_nil, at_nil, ex4_3_intro/
| #L1 #L3 #L #des3 #d #e #_ #HL3 #IHL13 #L2 #i #HL2
elim (lt_or_ge i d) #Hid
- [ elim (ldrop_trans_le … HL3 … HL2) -L /2 width=2/ #L #HL3 #HL2
- elim (IHL13 … HL3) -L3 /3 width=7/
+ [ elim (ldrop_trans_le … HL3 … HL2) -L /2 width=2 by lt_to_le/
+ #L #HL3 #HL2 elim (IHL13 … HL3) -L3 /3 width=7 by ldrops_cons, minuss_lt, at_lt, ex4_3_intro/
| lapply (ldrop_trans_ge … HL3 … HL2 ?) -L // #HL32
- elim (IHL13 … HL32) -L3 /3 width=7/
+ elim (IHL13 … HL32) -L3 /3 width=7 by minuss_ge, at_ge, ex4_3_intro/
]
]
qed-.