(* Vector form of forward lemmas involving same top term constructor ********)
-(* Basic_1: was just: nf2_iso_applv_lref *)
+(* Basic_1: was just: nf2_iso_appls_lref *)
lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≂ U.
#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
qed-.
-(* Basic_1: was just: pr3_iso_applv_beta *)
+(* Basic_1: was just: pr3_iso_appls_beta *)
lemma cpxs_fwd_beta_vector: ∀h,g,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
]
qed-.
-(* Basic_1: was just: pr3_iso_applv_abbr *)
+(* Basic_1: was just: pr3_iso_appls_abbr *)
lemma cpxs_fwd_theta_vector: ∀h,g,G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
ⒶV1s. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
]
qed-.
-(* Basic_1: was just: pr3_iso_applv_cast *)
+(* Basic_1: was just: pr3_iso_appls_cast *)
lemma cpxs_fwd_cast_vector: ∀h,g,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
∨∨ ⒶVs. ⓝW. T ≂ U
| ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U