(* Advanced properties ************************************* ****************)
(* Basic_1: was just: sn3_appls_beta *)
-lemma csx_applv_beta: ∀h,p,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
+lemma csx_applv_beta: ∀h,p,G,L,Vs,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
#h #p #G #L #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
lemma csx_applv_delta: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄.
+ ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄.
#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
[ /4 width=11 by csx_inv_lifts, csx_lref_pair, drops_isuni_fwd_drop2/
| #V #Vs #IHV #H1T
(* Basic_1: was just: sn3_appls_abbr *)
lemma csx_applv_theta: ∀h,p,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
+ ∀V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
#h #p #G #L #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
qed.
(* Basic_1: was just: sn3_appls_cast *)
-lemma csx_applv_cast: ∀h,G,L,Vs,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄.
+lemma csx_applv_cast: ∀h,G,L,Vs,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ →
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄.
#h #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