(* 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❫ →
- ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⒶVs.ⓐV.ⓛ[p]W.T❫.
-#h #G #p #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+lemma csx_applv_beta (h) (G) (L):
+ ∀p,Vs,V,W,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⒶVs.ⓓ[p]ⓝW.V.T →
+ ❪G,L❫ ⊢ ⬈*𝐒[h] ⒶVs.ⓐV.ⓛ[p]W.T.
+#h #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 →
- ∀Vs. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⒶVs.V2❫ → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⒶVs.#i❫.
-#h #G #I #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+lemma csx_applv_delta_drops (h) (G) (L):
+ ∀I,K,V1,i. ⇩[i] L ≘ K.ⓑ[I]V1 →
+ ∀V2. ⇧[↑i] V1 ≘ V2 →
+ ∀Vs. ❪G,L❫ ⊢ ⬈*𝐒[h] ⒶVs.V2 → ❪G,L❫ ⊢ ⬈*𝐒[h] ⒶVs.#i.
+#h #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 →
- ∀V,T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⓓ[p]V.ⒶV2b.T❫ → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⒶV1b.ⓓ[p]V.T❫.
-#h #G #p #L #V1b #V2b * -V1b -V2b /2 width=1 by/
+lemma csx_applv_theta (h) (G) (L):
+ ∀p,V1b,V2b. ⇧[1] V1b ≘ V2b →
+ ∀V,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓓ[p]V.ⒶV2b.T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⒶV1b.ⓓ[p]V.T.
+#h #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❫ →
- ∀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❫ ⊢ ⬈*𝐒[h] ⒶVs.U →
+ ∀T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⒶVs.T → ❪G,L❫ ⊢ ⬈*𝐒[h] Ⓐ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