(* Advanced properties ******************************************************)
lemma cpts_delta (h) (n) (G):
- â\88\80K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬆*[h,n] V2 →
- ∀W2. ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ⬆*[h,n] W2.
+ â\88\80K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬆*[h,n] V2 →
+ ∀W2. ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #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
- elim (lifts_total V (ð\9d\90\94â\9d´1â\9dµ)) #W #HVW
+ elim (lifts_total V (ð\9d\90\94â\9d¨1â\9d©)) #W #HVW
/5 width=11 by cpts_step_dx, cpt_lifts_bi, drops_refl, drops_drop/
]
qed.
lemma cpts_ell (h) (n) (G):
- â\88\80K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬆*[h,n] V2 →
- ∀W2. ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ⬆*[h,↑n] W2.
+ â\88\80K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬆*[h,n] V2 →
+ ∀W2. ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #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
- elim (lifts_total V (ð\9d\90\94â\9d´1â\9dµ)) #W #HVW >plus_S1
+ elim (lifts_total V (ð\9d\90\94â\9d¨1â\9d©)) #W #HVW >plus_S1
/5 width=11 by cpts_step_dx, cpt_lifts_bi, drops_refl, drops_drop/
]
qed.
lemma cpts_lref (h) (n) (I) (G):
- â\88\80K,T,i. â¦\83G,Kâ¦\84 ⊢ #i ⬆*[h,n] T →
- ∀U. ⇧*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬆*[h,n] U.
+ â\88\80K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬆*[h,n] T →
+ ∀U. ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑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
- elim (lifts_total T (ð\9d\90\94â\9d´1â\9dµ)) #U #TU
+ elim (lifts_total T (ð\9d\90\94â\9d¨1â\9d©)) #U #TU
/5 width=11 by cpts_step_dx, cpt_lifts_bi, drops_refl, drops_drop/
]
qed.
lemma cpts_cast_sn (h) (n) (G) (L):
- â\88\80U1,U2. â¦\83G,Lâ¦\84 ⊢ U1 ⬆*[h,n] U2 →
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓝU1.T1 ⬆*[h,n] ⓝU2.T2.
+ 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.
#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
qed.
lemma cpts_delta_drops (h) (n) (G):
- ∀L,K,V,i. ⇩*[i] L ≘ K.ⓓV →
- â\88\80V2. â¦\83G,Kâ¦\84 ⊢ V ⬆*[h,n] V2 →
- ∀W2. ⇧*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬆*[h,n] W2.
+ ∀L,K,V,i. ⇩[i] L ≘ K.ⓓV →
+ â\88\80V2. â\9dªG,Kâ\9d« ⊢ V ⬆*[h,n] V2 →
+ ∀W2. ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #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
lapply (drops_isuni_fwd_drop2 … HLK) -HLK // #HLK
- elim (lifts_total V1 (ð\9d\90\94â\9d´â\86\91iâ\9dµ)) #W1 #HVW1
+ elim (lifts_total V1 (ð\9d\90\94â\9d¨â\86\91iâ\9d©)) #W1 #HVW1
/3 width=11 by cpt_lifts_bi, cpts_step_dx/
]
qed.
lemma cpts_ell_drops (h) (n) (G):
- ∀L,K,W,i. ⇩*[i] L ≘ K.ⓛW →
- â\88\80W2. â¦\83G,Kâ¦\84 ⊢ W ⬆*[h,n] W2 →
- ∀V2. ⇧*[↑i] W2 ≘ V2 → ⦃G,L⦄ ⊢ #i ⬆*[h,↑n] V2.
+ ∀L,K,W,i. ⇩[i] L ≘ K.ⓛW →
+ â\88\80W2. â\9dªG,Kâ\9d« ⊢ W ⬆*[h,n] W2 →
+ ∀V2. ⇧[↑i] W2 ≘ V2 → ❪G,L❫ ⊢ #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
lapply (drops_isuni_fwd_drop2 … HLK) -HLK // #HLK
- elim (lifts_total W1 (ð\9d\90\94â\9d´â\86\91iâ\9dµ)) #V1 #HWV1 >plus_S1
+ elim (lifts_total W1 (ð\9d\90\94â\9d¨â\86\91iâ\9d©)) #V1 #HWV1 >plus_S1
/3 width=11 by cpt_lifts_bi, cpts_step_dx/
]
qed.
(* Advanced inversion lemmas ************************************************)
lemma cpts_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
- | ∃∃K,V,V2. ⇩*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ⬆*[h,n] V2 & ⇧*[↑i] V2 ≘ X2
- | ∃∃m,K,V,V2. ⇩*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬆*[h,m] V2 & ⇧*[↑i] V2 ≘ X2 & n = ↑m.
+ | ∃∃K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ⬆*[h,n] V2 & ⇧[↑i] V2 ≘ X2
+ | ∃∃m,K,V,V2. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ 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¦\83G,K.â\93\93Vâ¦\84 ⊢ #0 ⬆*[h,n] X2 →
+ â\88\80X2. â\9dªG,K.â\93\93Vâ\9d« ⊢ #0 ⬆*[h,n] X2 →
∨∨ ∧∧ X2 = #0 & n = 0
- | â\88\83â\88\83V2. â¦\83G,Kâ¦\84 â\8a¢ V â¬\86*[h,n] V2 & â\87§*[1] V2 ≘ X2.
+ | â\88\83â\88\83V2. â\9dªG,Kâ\9d« â\8a¢ V â¬\86*[h,n] V2 & â\87§[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¦\83G,K.â\93\9bVâ¦\84 ⊢ #0 ⬆*[h,n] X2 →
+ â\88\80X2. â\9dªG,K.â\93\9bVâ\9d« ⊢ #0 ⬆*[h,n] X2 →
∨∨ ∧∧ X2 = #0 & n = 0
- | â\88\83â\88\83m,V2. â¦\83G,Kâ¦\84 â\8a¢ V â¬\86*[h,m] V2 & â\87§*[1] V2 ≘ X2 & n = ↑m.
+ | â\88\83â\88\83m,V2. â\9dªG,Kâ\9d« â\8a¢ V â¬\86*[h,m] V2 & â\87§[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¦\83G,K.â\93\98{I}â¦\84 ⊢ #↑i ⬆*[h,n] X2 →
+ â\88\80X2. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬆*[h,n] X2 →
∨∨ ∧∧ X2 = #↑i & n = 0
- | â\88\83â\88\83T2. â¦\83G,Kâ¦\84 â\8a¢ #i â¬\86*[h,n] T2 & â\87§*[1] T2 ≘ X2.
+ | â\88\83â\88\83T2. â\9dªG,Kâ\9d« â\8a¢ #i â¬\86*[h,n] T2 & â\87§[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/
| #L #V #V2 #H #HV2 #HVU2
lapply (drops_inv_drop1 … H) -H #HLK
- elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ)) -HVU2
+ elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©)) -HVU2
[| // ] #T2 #HVT2 #HTU2
/4 width=6 by cpts_delta_drops, ex2_intro, or_intror/
| #m #L #V #V2 #H #HV2 #HVU2 #H0 destruct
lapply (drops_inv_drop1 … H) -H #HLK
- elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ)) -HVU2
+ elim (lifts_split_trans â\80¦ HVU2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©)) -HVU2
[| // ] #T2 #HVT2 #HTU2
/4 width=6 by cpts_ell_drops, ex2_intro, or_intror/
]
qed-.
lemma cpts_inv_succ_sn (h) (n) (G) (L):
- â\88\80T1,T2. â¦\83G,Lâ¦\84 ⊢ T1 ⬆*[h,↑n] T2 →
- â\88\83â\88\83T. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\86*[h,1] T & â¦\83G,Lâ¦\84 ⊢ T ⬆*[h,n] T2.
+ 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.
#h #n #G #L #T1 #T2
@(insert_eq_0 … (↑n)) #m #H
@(cpts_ind_sn … H) -T1 -m