(* Inversion lemmas about local env. full refinement for substitution *******)
(* Note: ldrop_ldrop not needed *)
-lemma sfr_inv_ldrop: â\88\80I,L,K,V,i. â\87©[0, i] L â\89¡ K. â\93\91{I}V â\86\92 â\88\80d,e. â\89¼ [d, e] L →
+lemma sfr_inv_ldrop: â\88\80I,L,K,V,i. â\87©[0, i] L â\89¡ K. â\93\91{I}V â\86\92 â\88\80d,e. â\89½ [d, e] L →
d ≤ i → i < d + e → I = Abbr.
#I #L elim L -L
[ #K #V #i #H
[ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
- elim (lsubs_inv_abbr1 … H ?) -H // -Hide #K #_ #H destruct //
+ elim (lsubs_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
| #Hi #HLK #d @(nat_ind_plus … d) -d
[ #e #H #_ #Hide
elim (sfr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
(* Note: ldrop_ldrop not needed *)
lemma sfr_ldrop: ∀L,d,e.
(∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
- â\89¼ [d, e] L.
+ â\89½ [d, e] L.
#L elim L -L //
#L #I #V #IHL #d @(nat_ind_plus … d) -d
[ #e @(nat_ind_plus … e) -e //
]
qed.
-lemma sfr_ldrop_trans_le: â\88\80L1,L2,d,e. â\87©[d, e] L1 â\89¡ L2 â\86\92 â\88\80dd,ee. â\89¼ [dd, ee] L1 →
- dd + ee â\89¤ d â\86\92 â\89¼ [dd, ee] L2.
+lemma sfr_ldrop_trans_le: â\88\80L1,L2,d,e. â\87©[d, e] L1 â\89¡ L2 â\86\92 â\88\80dd,ee. â\89½ [dd, ee] L1 →
+ dd + ee â\89¤ d â\86\92 â\89½ [dd, ee] L2.
#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
qed.
lemma sfr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
- â\88\80dd,ee. â\89¼ [dd, ee] L1 →
+ â\88\80dd,ee. â\89½ [dd, ee] L1 →
dd ≤ d + e → d + e ≤ dd + ee →
- â\89¼ [d, dd + ee - d - e] L2.
+ â\89½ [d, dd + ee - d - e] L2.
#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
@sfr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
qed.
-lemma sfr_ldrop_trans_ge: â\88\80L1,L2,d,e. â\87©[d, e] L1 â\89¡ L2 â\86\92 â\88\80dd,ee. â\89¼ [dd, ee] L1 →
- d + e â\89¤ dd â\86\92 â\89¼ [dd - e, ee] L2.
+lemma sfr_ldrop_trans_ge: â\88\80L1,L2,d,e. â\87©[d, e] L1 â\89¡ L2 â\86\92 â\88\80dd,ee. â\89½ [dd, ee] L1 →
+ d + e â\89¤ dd â\86\92 â\89½ [dd - e, ee] L2.
#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd