include "basic_2/computation/acp_cr.ma".
include "basic_2/computation/cprs_tstc_vector.ma".
-include "basic_2/computation/csn_lcpr.ma".
+include "basic_2/computation/csn_lfpr.ma".
include "basic_2/computation/csn_vector.ma".
(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
(* Advanced properties ******************************************************)
(* Basic_1: was only: sn3_appls_lref *)
-lemma csn_applv_cnf: ∀L,T. 𝐒[T] → L ⊢ 𝐍[T] →
- â\88\80Vs. L â\8a¢ â¬\87* Vs â\86\92 L â\8a¢ â¬\87* ⒶVs.T.
+lemma csn_applv_cnf: ∀L,T. 𝐒⦃T⦄ → L ⊢ 𝐍⦃T⦄ →
+ â\88\80Vs. L â\8a¢ â¬\8a* Vs â\86\92 L â\8a¢ â¬\8a* ⒶVs.T.
#L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(csn_cnf … H2T) ] (**) (* /2 width=1/ does not work *)
#V #Vs #IHV #H
elim (csnv_inv_cons … H) -H #HV #HVs
@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -HVs
-[ #X #H #H0
- lapply (cprs_fwd_cnf_vector … H) -H // -H1T -H2T #H
- elim (H0 ?) -H0 //
-| -L -V elim Vs -Vs //
-]
+#X #H #H0
+lapply (cprs_fwd_cnf_vector … H) -H // -H1T -H2T #H
+elim (H0 ?) -H0 //
qed.
(* Basic_1: was: sn3_appls_beta *)
-lemma csn_applv_beta: ∀L,W. L ⊢ ⬇* W →
- â\88\80Vs,V,T. L â\8a¢ â¬\87* â\92¶Vs.â\93\93V.T →
- L â\8a¢ â¬\87* â\92¶Vs. â\93\90V.â\93\9bW. T.
-#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
+lemma csn_applv_beta: ∀a,L,W. L ⊢ ⬊* W →
+ â\88\80Vs,V,T. L â\8a¢ â¬\8a* â\92¶Vs.â\93\93{a}V.T →
+ L â\8a¢ â¬\8a* â\92¶Vs. â\93\90V.â\93\9b{a}W. T.
+#a #L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
#V0 #Vs #IHV #V #T #H1T
lapply (csn_fwd_pair_sn … H1T) #HV0
lapply (csn_fwd_flat_dx … H1T) #H2T
@csn_appl_simple_tstc // -HV0 /2 width=1/ -IHV -H2T
-[ #X #H #H0
- elim (cprs_fwd_beta_vector … H) -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
-| -H1T elim Vs -Vs //
+#X #H #H0
+elim (cprs_fwd_beta_vector … H) -H #H
+[ -H1T elim (H0 ?) -H0 //
+| -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
]
qed.
lemma csn_applv_delta: ∀L,K,V1,i. ⇩[0, i] L ≡ K. ⓓV1 →
∀V2. ⇧[0, i + 1] V1 ≡ V2 →
- â\88\80Vs.L â\8a¢ â¬\87* (â\92¶Vs. V2) â\86\92 L â\8a¢ â¬\87* (ⒶVs. #i).
+ â\88\80Vs.L â\8a¢ â¬\8a* (â\92¶Vs. V2) â\86\92 L â\8a¢ â¬\8a* (ⒶVs. #i).
#L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
[ #H
lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
lapply (csn_fwd_pair_sn … H1T) #HV
lapply (csn_fwd_flat_dx … H1T) #H2T
@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
- [ #X #H #H0
- elim (cprs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
- | -L -K -V -V1 -V2 elim Vs -Vs //
+ #X #H #H0
+ elim (cprs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
+ [ -H1T elim (H0 ?) -H0 //
+ | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
]
]
qed.
(* Basic_1: was: sn3_appls_abbr *)
-lemma csn_applv_theta: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- â\88\80V,T. L â\8a¢ â¬\87* â\93\93V. â\92¶V2s. T â\86\92 L â\8a¢ â¬\87* V →
- L â\8a¢ â¬\87* â\92¶V1s. â\93\93V. T.
-#L #V1s #V2s * -V1s -V2s /2 width=1/
+lemma csn_applv_theta: ∀a,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ â\88\80V,T. L â\8a¢ â¬\8a* â\93\93{a}V. â\92¶V2s. T â\86\92 L â\8a¢ â¬\8a* V →
+ L â\8a¢ â¬\8a* â\92¶V1s. â\93\93{a}V. T.
+#a #L #V1s #V2s * -V1s -V2s /2 width=1/
#V1s #V2s #V1 #V2 #HV12 #H
generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
elim H -V1s -V2s /2 width=3/
lapply (csn_fwd_pair_sn … H) #HW1
lapply (csn_fwd_flat_dx … H) #H1
@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -HV -H1 #X #H1 #H2
-elim (cprs_fwd_theta_vector … (V2::V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
+elim (cprs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
[ -H #H elim (H2 ?) -H2 //
| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/
]
qed.
(* Basic_1: was: sn3_appls_cast *)
-lemma csn_applv_tau: â\88\80L,W. L â\8a¢ â¬\87* W →
- â\88\80Vs,T. L â\8a¢ â¬\87* ⒶVs. T →
- L â\8a¢ â¬\87* â\92¶Vs. â\93£W. T.
+lemma csn_applv_tau: â\88\80L,W. L â\8a¢ â¬\8a* W →
+ â\88\80Vs,T. L â\8a¢ â¬\8a* ⒶVs. T →
+ L â\8a¢ â¬\8a* â\92¶Vs. â\93\9dW. T.
#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
#V #Vs #IHV #T #H1T
lapply (csn_fwd_pair_sn … H1T) #HV
lapply (csn_fwd_flat_dx … H1T) #H2T
@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
-[ #X #H #H0
- elim (cprs_fwd_tau_vector … H) -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
-| -H1T elim Vs -Vs //
+#X #H #H0
+elim (cprs_fwd_tau_vector … H) -H #H
+[ -H1T elim (H0 ?) -H0 //
+| -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
]
qed.
-theorem csn_acr: acr cpr (eq â\80¦) (csn â\80¦) (λL,T. L â\8a¢ â¬\87* T).
+theorem csn_acr: acr cpr (eq â\80¦) (csn â\80¦) (λL,T. L â\8a¢ â¬\8a* T).
@mk_acr //
[ /3 width=1/
| /2 width=1/
| /2 width=6/
-| #L #V1 #V2 #HV12 #V #T #H #HVT
+| #L #V1 #V2 #HV12 #a #V #T #H #HVT
@(csn_applv_theta … HV12) -HV12 //
@(csn_abbr) //
| /2 width=1/