(* Advanced properties ******************************************************)
lemma cpt_delta_drops (h) (n) (G):
- â\88\80L,K,V,i. â\87©[i] L â\89\98 K.â\93\93V â\86\92 â\88\80V2. â\9dªG,Kâ\9d« ⊢ V ⬆[h,n] V2 →
- â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬆[h,n] W2.
+ â\88\80L,K,V,i. â\87©[i] L â\89\98 K.â\93\93V â\86\92 â\88\80V2. â\9d¨G,Kâ\9d© ⊢ V ⬆[h,n] V2 →
+ â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9d¨G,Lâ\9d© ⊢ #i ⬆[h,n] W2.
#h #n #G #L #K #V #i #HLK #V2 *
/3 width=8 by cpg_delta_drops, ex2_intro/
qed.
lemma cpt_ell_drops (h) (n) (G):
- â\88\80L,K,V,i. â\87©[i] L â\89\98 K.â\93\9bV â\86\92 â\88\80V2. â\9dªG,Kâ\9d« ⊢ V ⬆[h,n] V2 →
- â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬆[h,↑n] W2.
+ â\88\80L,K,V,i. â\87©[i] L â\89\98 K.â\93\9bV â\86\92 â\88\80V2. â\9d¨G,Kâ\9d© ⊢ V ⬆[h,n] V2 →
+ â\88\80W2. â\87§[â\86\91i] V2 â\89\98 W2 â\86\92 â\9d¨G,Lâ\9d© ⊢ #i ⬆[h,↑n] W2.
#h #n #G #L #K #V #i #HLK #V2 *
/3 width=8 by cpg_ell_drops, ist_succ, ex2_intro/
qed.
(* Advanced inversion lemmas ************************************************)
lemma cpt_inv_atom_sn_drops (h) (n) (I) (G) (L):
- â\88\80X2. â\9dªG,Lâ\9d« ⊢ ⓪[I] ⬆[h,n] X2 →
+ â\88\80X2. â\9d¨G,Lâ\9d© ⊢ ⓪[I] ⬆[h,n] X2 →
∨∨ ∧∧ X2 = ⓪[I] & n = 0
| ∃∃s. X2 = ⋆(⫯[h]s) & I = Sort s & n = 1
- | â\88\83â\88\83K,V,V2,i. â\87©[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« ⊢ V ⬆[h,n] V2 & ⇧[↑i] V2 ≘ X2 & I = LRef i
- | â\88\83â\88\83m,K,V,V2,i. â\87©[i] L â\89\98 K.â\93\9bV & â\9dªG,Kâ\9d« ⊢ V ⬆[h,m] V2 & ⇧[↑i] V2 ≘ X2 & I = LRef i & n = ↑m.
+ | â\88\83â\88\83K,V,V2,i. â\87©[i] L â\89\98 K.â\93\93V & â\9d¨G,Kâ\9d© ⊢ V ⬆[h,n] V2 & ⇧[↑i] V2 ≘ X2 & I = LRef i
+ | â\88\83â\88\83m,K,V,V2,i. â\87©[i] L â\89\98 K.â\93\9bV & â\9d¨G,Kâ\9d© ⊢ V ⬆[h,m] V2 & ⇧[↑i] V2 ≘ X2 & I = LRef i & n = ↑m.
#h #n #I #G #L #X2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
[ #H1 #H2 destruct
/3 width=1 by or4_intro0, conj/
qed-.
lemma cpt_inv_lref_sn_drops (h) (n) (G) (L) (i):
- â\88\80X2. â\9dªG,Lâ\9d« ⊢ #i ⬆[h,n] X2 →
+ â\88\80X2. â\9d¨G,Lâ\9d© ⊢ #i ⬆[h,n] X2 →
∨∨ ∧∧ X2 = #i & n = 0
- | â\88\83â\88\83K,V,V2. â\87©[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« ⊢ V ⬆[h,n] V2 & ⇧[↑i] V2 ≘ X2
- | â\88\83â\88\83m,K,V,V2. â\87©[i] L â\89\98 K. â\93\9bV & â\9dªG,Kâ\9d« ⊢ V ⬆[h,m] V2 & ⇧[↑i] V2 ≘ X2 & n = ↑m.
+ | â\88\83â\88\83K,V,V2. â\87©[i] L â\89\98 K.â\93\93V & â\9d¨G,Kâ\9d© ⊢ V ⬆[h,n] V2 & ⇧[↑i] V2 ≘ X2
+ | â\88\83â\88\83m,K,V,V2. â\87©[i] L â\89\98 K. â\93\9bV & â\9d¨G,Kâ\9d© ⊢ V ⬆[h,m] V2 & ⇧[↑i] V2 ≘ X2 & n = ↑m.
#h #n #G #L #i #X2 * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
[ #H1 #H2 destruct
/3 width=1 by or3_intro0, conj/
(* Advanced forward lemmas **************************************************)
fact cpt_fwd_plus_aux (h) (n) (G) (L):
- â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬆[h,n] T2 → ∀n1,n2. n1+n2 = n →
- â\88\83â\88\83T. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\86[h,n1] T & â\9dªG,Lâ\9d« ⊢ T ⬆[h,n2] T2.
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬆[h,n] T2 → ∀n1,n2. n1+n2 = n →
+ â\88\83â\88\83T. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\86[h,n1] T & â\9d¨G,Lâ\9d© ⊢ T ⬆[h,n2] T2.
#h #n #G #L #T1 #T2 #H @(cpt_ind … H) -G -L -T1 -T2 -n
[ #I #G #L #n1 #n2 #H
elim (plus_inv_O3 … H) -H #H1 #H2 destruct
qed-.
lemma cpt_fwd_plus (h) (n1) (n2) (G) (L):
- â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬆[h,n1+n2] T2 →
- â\88\83â\88\83T. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\86[h,n1] T & â\9dªG,Lâ\9d« ⊢ T ⬆[h,n2] T2.
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬆[h,n1+n2] T2 →
+ â\88\83â\88\83T. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\86[h,n1] T & â\9d¨G,Lâ\9d© ⊢ T ⬆[h,n2] T2.
/2 width=3 by cpt_fwd_plus_aux/ qed-.