(* Vector form of forward lemmas with head equivalence for terms ************)
-lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
+lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
#h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/
#V #Vs #IHVs #X2 #H
elim (cpxs_inv_appl1 … H) -H *
(* Basic_2A1: was: cpxs_fwd_delta_vector *)
lemma cpxs_fwd_delta_drops_vector: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
- ∨∨ ⒶVs.#i ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
+ ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
+ ∨∨ ⒶVs.#i ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs
elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
#V #Vs #IHVs #X2 #H -K -V1
qed-.
(* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
- ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
+lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+ ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
#h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
#V0 #Vs #IHVs #V #W #T #X2 #H
elim (cpxs_inv_appl1 … H) -H *
(* Basic_1: was just: pr3_iso_appls_abbr *)
lemma cpxs_fwd_theta_vector: ∀h,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀p,V,T,X2. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
- ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
+ ∀p,V,T,X2. ⦃G,L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
+ ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
#h #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
#V1b #V2b #V1a #V2a #HV12a #HV12b #p
generalize in match HV12a; -HV12a
qed-.
(* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
+lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
∨∨ ⒶVs. ⓝW. T ⩳ X2
- | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2
- | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
+ | ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2
+ | ⦃G,L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
#h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
#V #Vs #IHVs #W #T #X2 #H
elim (cpxs_inv_appl1 … H) -H *
qed-.
(* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
- ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
+lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
#h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
#V #Vs #IHVs #X2 #H
elim (cpxs_inv_appl1 … H) -H *