(* 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¦\83G,Kâ¦\84 ⊢ V ⬆[h,n] V2 →
- â\88\80W2. â\87§*[â\86\91i] V2 â\89\98 W2 â\86\92 â¦\83G,Lâ¦\84 ⊢ #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\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.
#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¦\83G,Kâ¦\84 ⊢ V ⬆[h,n] V2 →
- â\88\80W2. â\87§*[â\86\91i] V2 â\89\98 W2 â\86\92 â¦\83G,Lâ¦\84 ⊢ #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\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.
#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¦\83G,Lâ¦\84 â\8a¢ â\93ª{I} ⬆[h,n] X2 →
- ∨∨ ∧∧ X2 = ⓪{I} & n = 0
+ â\88\80X2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[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¦\83G,Kâ¦\84 ⊢ 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¦\83G,Kâ¦\84 ⊢ 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\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.
#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¦\83G,Lâ¦\84 ⊢ #i ⬆[h,n] X2 →
+ â\88\80X2. â\9dªG,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¦\83G,Kâ¦\84 ⊢ 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¦\83G,Kâ¦\84 ⊢ 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\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.
#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¦\83G,Lâ¦\84 ⊢ T1 ⬆[h,n] T2 → ∀n1,n2. n1+n2 = n →
- â\88\83â\88\83T. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\86[h,n1] T & â¦\83G,Lâ¦\84 ⊢ T ⬆[h,n2] T2.
+ 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.
#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
]
| #n #G #K #V1 #V2 #W2 #_ #IH #HVW2 #n1 #n2 #H destruct
elim IH [|*: // ] -IH #V #HV1 #HV2
- elim (lifts_total V ð\9d\90\94â\9d´â\86\91Oâ\9dµ) #W #HVW
+ elim (lifts_total V ð\9d\90\94â\9d¨â\86\91Oâ\9d©) #W #HVW
/5 width=11 by cpt_lifts_bi, cpt_delta, drops_refl, drops_drop, ex2_intro/
| #n #G #K #V1 #V2 #W2 #HV12 #IH #HVW2 #x1 #n2 #H
elim (plus_inv_S3_sn … H) -H *
[ #H1 #H2 destruct -IH /3 width=3 by cpt_ell, ex2_intro/
| #n1 #H1 #H2 destruct -HV12
elim (IH n1) [|*: // ] -IH #V #HV1 #HV2
- elim (lifts_total V ð\9d\90\94â\9d´â\86\91Oâ\9dµ) #W #HVW
+ elim (lifts_total V ð\9d\90\94â\9d¨â\86\91Oâ\9d©) #W #HVW
/5 width=11 by cpt_lifts_bi, cpt_ell, drops_refl, drops_drop, ex2_intro/
]
| #n #I #G #K #T2 #U2 #i #_ #IH #HTU2 #n1 #n2 #H destruct
elim IH [|*: // ] -IH #T #HT1 #HT2
- elim (lifts_total T ð\9d\90\94â\9d´â\86\91Oâ\9dµ) #U #HTU
+ elim (lifts_total T ð\9d\90\94â\9d¨â\86\91Oâ\9d©) #U #HTU
/5 width=11 by cpt_lifts_bi, cpt_lref, drops_refl, drops_drop, ex2_intro/
| #n #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ #IHT #n1 #n2 #H destruct
elim IHT [|*: // ] -IHT #T #HT1 #HT2
qed-.
lemma cpt_fwd_plus (h) (n1) (n2) (G) (L):
- â\88\80T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ⬆[h,n1+n2] T2 →
- â\88\83â\88\83T. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\86[h,n1] T & â¦\83G,Lâ¦\84 ⊢ T ⬆[h,n2] T2.
+ 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.
/2 width=3 by cpt_fwd_plus_aux/ qed-.