(* Advanced properties ******************************************************)
-(* Basic_1: was just: sn3_appls_lref *)
+(* Basic_1: was just: sn3_applv_lref *)
lemma csx_applv_cnx: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
]
qed.
-(* Basic_1: was just: sn3_appls_beta *)
+(* Basic_1: was just: sn3_applv_beta *)
lemma csx_applv_beta: ∀h,g,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓓ{a}ⓝW.V.T →
⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs. ⓐV.ⓛ{a}W.T.
#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
∀V2. ⇧[0, i + 1] V1 ≡ V2 →
∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
-[ /4 width=12 by csx_inv_lift, csx_lref_bind, ldrop_fwd_drop2/
+[ /4 width=12 by csx_inv_lift, csx_lref_bind, drop_fwd_drop2/
| #V #Vs #IHV #H1T
lapply (csx_fwd_pair_sn … H1T) #HV
lapply (csx_fwd_flat_dx … H1T) #H2T
]
qed.
-(* Basic_1: was just: sn3_appls_abbr *)
+(* Basic_1: was just: sn3_applv_abbr *)
lemma csx_applv_theta: ∀h,g,a,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
]
qed.
-(* Basic_1: was just: sn3_appls_cast *)
+(* Basic_1: was just: sn3_applv_cast *)
lemma csx_applv_cast: ∀h,g,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T →
⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓝW.T.
#h #g #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/