(* Advanced properties ******************************************************)
lemma cpts_delta (h) (n) (G):
- â\88\80K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬆*[h,n] V2 →
- â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\93V1â\9d« ⊢ #0 ⬆*[h,n] W2.
+ â\88\80K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬆*[h,n] V2 →
+ â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9d¨G,K.â\93\93V1â\9d© ⊢ #0 ⬆*[h,n] W2.
#h #n #G #K #V1 #V2 #H @(cpts_ind_dx … H) -V2
[ /3 width=3 by cpt_cpts, cpt_delta/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
qed.
lemma cpts_ell (h) (n) (G):
- â\88\80K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬆*[h,n] V2 →
- â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\9bV1â\9d« ⊢ #0 ⬆*[h,↑n] W2.
+ â\88\80K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬆*[h,n] V2 →
+ â\88\80W2. â\87§[1] V2 â\89\98 W2 â\86\92 â\9d¨G,K.â\93\9bV1â\9d© ⊢ #0 ⬆*[h,↑n] W2.
#h #n #G #K #V1 #V2 #H @(cpts_ind_dx … H) -V2
[ /3 width=3 by cpt_cpts, cpt_ell/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
qed.
lemma cpts_lref (h) (n) (I) (G):
- â\88\80K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬆*[h,n] T →
- â\88\80U. â\87§[1] T â\89\98 U â\86\92 â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬆*[h,n] U.
+ â\88\80K,T,i. â\9d¨G,Kâ\9d© ⊢ #i ⬆*[h,n] T →
+ â\88\80U. â\87§[1] T â\89\98 U â\86\92 â\9d¨G,K.â\93\98[I]â\9d© ⊢ #↑i ⬆*[h,n] U.
#h #n #I #G #K #T #i #H @(cpts_ind_dx … H) -T
[ /3 width=3 by cpt_cpts, cpt_lref/
| #n1 #n2 #T #T2 #_ #IH #HT2 #U2 #HTU2
qed.
lemma cpts_cast_sn (h) (n) (G) (L):
- â\88\80U1,U2. â\9dªG,Lâ\9d« ⊢ U1 ⬆*[h,n] U2 →
- â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝU1.T1 ⬆*[h,n] ⓝU2.T2.
+ â\88\80U1,U2. â\9d¨G,Lâ\9d© ⊢ U1 ⬆*[h,n] U2 →
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ ⓝU1.T1 ⬆*[h,n] ⓝU2.T2.
#h #n #G #L #U1 #U2 #H @(cpts_ind_sn … H) -U1 -n
[ /3 width=3 by cpt_cpts, cpt_cast/
| #n1 #n2 #U1 #U #HU1 #_ #IH #T1 #T2 #H
lemma cpts_delta_drops (h) (n) (G):
∀L,K,V,i. ⇩[i] L ≘ K.ⓓV →
- 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\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 #H @(cpts_ind_dx … H) -V2
[ /3 width=6 by cpt_cpts, cpt_delta_drops/
| #n1 #n2 #V1 #V2 #_ #IH #HV12 #W2 #HVW2
lemma cpts_ell_drops (h) (n) (G):
∀L,K,W,i. ⇩[i] L ≘ K.ⓛW →
- â\88\80W2. â\9dªG,Kâ\9d« ⊢ W ⬆*[h,n] W2 →
- â\88\80V2. â\87§[â\86\91i] W2 â\89\98 V2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬆*[h,↑n] V2.
+ â\88\80W2. â\9d¨G,Kâ\9d© ⊢ W ⬆*[h,n] W2 →
+ â\88\80V2. â\87§[â\86\91i] W2 â\89\98 V2 â\86\92 â\9d¨G,Lâ\9d© ⊢ #i ⬆*[h,↑n] V2.
#h #n #G #L #K #W #i #HLK #W2 #H @(cpts_ind_dx … H) -W2
[ /3 width=6 by cpt_cpts, cpt_ell_drops/
| #n1 #n2 #W1 #W2 #_ #IH #HW12 #V2 #HWV2
(* Advanced inversion lemmas ************************************************)
lemma cpts_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 #H @(cpts_ind_dx … H) -X2
[ /3 width=1 by or3_intro0, conj/
| #n1 #n2 #T #T2 #_ #IH #HT2 cases IH -IH *
qed-.
lemma cpts_inv_delta_sn (h) (n) (G) (K) (V):
- â\88\80X2. â\9dªG,K.â\93\93Vâ\9d« ⊢ #0 ⬆*[h,n] X2 →
+ â\88\80X2. â\9d¨G,K.â\93\93Vâ\9d© ⊢ #0 ⬆*[h,n] X2 →
∨∨ ∧∧ X2 = #0 & n = 0
- | â\88\83â\88\83V2. â\9dªG,Kâ\9d« ⊢ V ⬆*[h,n] V2 & ⇧[1] V2 ≘ X2.
+ | â\88\83â\88\83V2. â\9d¨G,Kâ\9d© ⊢ V ⬆*[h,n] V2 & ⇧[1] V2 ≘ X2.
#h #n #G #K #V #X2 #H
elim (cpts_inv_lref_sn_drops … H) -H *
[ /3 width=1 by or_introl, conj/
qed-.
lemma cpts_inv_ell_sn (h) (n) (G) (K) (V):
- â\88\80X2. â\9dªG,K.â\93\9bVâ\9d« ⊢ #0 ⬆*[h,n] X2 →
+ â\88\80X2. â\9d¨G,K.â\93\9bVâ\9d© ⊢ #0 ⬆*[h,n] X2 →
∨∨ ∧∧ X2 = #0 & n = 0
- | â\88\83â\88\83m,V2. â\9dªG,Kâ\9d« ⊢ V ⬆*[h,m] V2 & ⇧[1] V2 ≘ X2 & n = ↑m.
+ | â\88\83â\88\83m,V2. â\9d¨G,Kâ\9d© ⊢ V ⬆*[h,m] V2 & ⇧[1] V2 ≘ X2 & n = ↑m.
#h #n #G #K #V #X2 #H
elim (cpts_inv_lref_sn_drops … H) -H *
[ /3 width=1 by or_introl, conj/
qed-.
lemma cpts_inv_lref_sn (h) (n) (I) (G) (K) (i):
- â\88\80X2. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬆*[h,n] X2 →
+ â\88\80X2. â\9d¨G,K.â\93\98[I]â\9d© ⊢ #↑i ⬆*[h,n] X2 →
∨∨ ∧∧ X2 = #↑i & n = 0
- | â\88\83â\88\83T2. â\9dªG,Kâ\9d« ⊢ #i ⬆*[h,n] T2 & ⇧[1] T2 ≘ X2.
+ | â\88\83â\88\83T2. â\9d¨G,Kâ\9d© ⊢ #i ⬆*[h,n] T2 & ⇧[1] T2 ≘ X2.
#h #n #I #G #K #i #X2 #H
elim (cpts_inv_lref_sn_drops … H) -H *
[ /3 width=1 by or_introl, conj/
qed-.
lemma cpts_inv_succ_sn (h) (n) (G) (L):
- â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬆*[h,↑n] T2 →
- â\88\83â\88\83T. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\86*[h,1] T & â\9dªG,Lâ\9d« ⊢ T ⬆*[h,n] T2.
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬆*[h,↑n] T2 →
+ â\88\83â\88\83T. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\86*[h,1] T & â\9d¨G,Lâ\9d© ⊢ T ⬆*[h,n] T2.
#h #n #G #L #T1 #T2
@(insert_eq_0 … (↑n)) #m #H
@(cpts_ind_sn … H) -T1 -m