include "basic_2/rt_computation/csx_lpx.ma".
include "basic_2/rt_computation/csx_vector.ma".
-(* STRONGLY NORMALIZING TERM VECTORS FOR UNBOUND PARALLEL RT-TRANSITION *****)
+(* STRONGLY NORMALIZING TERM VECTORS FOR EXTENDED PARALLEL RT-TRANSITION ****)
(* Advanced properties ************************************* ****************)
(* Basic_1: was just: sn3_appls_beta *)
-lemma csx_applv_beta (h) (G):
- ∀p,L,Vs,V,W,T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⒶVs.ⓓ[p]ⓝW.V.T❫ →
- â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\92¶Vs.â\93\90V.â\93\9b[p]W.Tâ\9d«.
-#h #G #p #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+lemma csx_applv_beta (G) (L):
+ ∀p,Vs,V,W,T. ❨G,L❩ ⊢ ⬈*𝐒 ⒶVs.ⓓ[p]ⓝW.V.T →
+ â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\92¶Vs.â\93\90V.â\93\9b[p]W.T.
+#G #L #p #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
#V0 #Vs #IHV #V #W #T #H1T
lapply (csx_fwd_pair_sn … H1T) #HV0
lapply (csx_fwd_flat_dx … H1T) #H2T
]
qed.
-lemma csx_applv_delta_drops (h) (G):
- ∀I,L,K,V1,i. ⇩*[i] L ≘ K.ⓑ[I]V1 →
- ∀V2. ⇧*[↑i] V1 ≘ V2 →
- â\88\80Vs. â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\92¶Vs.V2â\9d« â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\92¶Vs.#iâ\9d«.
-#h #G #I #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+lemma csx_applv_delta_drops (G) (L):
+ ∀I,K,V1,i. ⇩[i] L ≘ K.ⓑ[I]V1 →
+ ∀V2. ⇧[↑i] V1 ≘ V2 →
+ â\88\80Vs. â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\92¶Vs.V2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\92¶Vs.#i.
+#G #L #I #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
[ /4 width=11 by csx_inv_lifts, csx_lref_pair_drops, drops_isuni_fwd_drop2/
| #V #Vs #IHV #H1T
lapply (csx_fwd_pair_sn … H1T) #HV
qed.
(* Basic_1: was just: sn3_appls_abbr *)
-lemma csx_applv_theta (h) (G):
- ∀p,L,V1b,V2b. ⇧*[1] V1b ≘ V2b →
- â\88\80V,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\93\93[p]V.â\92¶V2b.Tâ\9d« â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\92¶V1b.â\93\93[p]V.Tâ\9d«.
-#h #G #p #L #V1b #V2b * -V1b -V2b /2 width=1 by/
+lemma csx_applv_theta (G) (L):
+ ∀p,V1b,V2b. ⇧[1] V1b ≘ V2b →
+ â\88\80V,T. â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\93\93[p]V.â\92¶V2b.T â\86\92 â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\92¶V1b.â\93\93[p]V.T.
+#G #L #p #V1b #V2b * -V1b -V2b /2 width=1 by/
#V1b #V2b #V1 #V2 #HV12 #H
generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
elim H -V1b -V2b /2 width=3 by csx_appl_theta/
qed.
(* Basic_1: was just: sn3_appls_cast *)
-lemma csx_applv_cast (h) (G):
- ∀L,Vs,U. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⒶVs.U❫ →
- â\88\80T. â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\92¶Vs.Tâ\9d« â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªâ\92¶Vs.â\93\9dU.Tâ\9d«.
-#h #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+lemma csx_applv_cast (G) (L):
+ ∀Vs,U. ❨G,L❩ ⊢ ⬈*𝐒 ⒶVs.U →
+ â\88\80T. â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\92¶Vs.T â\86\92 â\9d¨G,Lâ\9d© â\8a¢ â¬\88*ð\9d\90\92 â\92¶Vs.â\93\9dU.T.
+#G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
#V #Vs #IHV #U #H1U #T #H1T
lapply (csx_fwd_pair_sn … H1U) #HV
lapply (csx_fwd_flat_dx … H1U) #H2U