(* Advanced properties ******************************************************)
-lemma csx_teqx_trans (h) (G):
- ∀L,T1. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- ∀T2. T1 ≛ T2 → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T2❫.
+lemma csx_teqx_trans (h) (G) (L):
+ ∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
+ ∀T2. T1 ≛ T2 → ❪G,L❫ ⊢ ⬈*𝐒[h] T2.
#h #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
@csx_intro #T1 #HT21 #HnT21 elim (teqx_cpx_trans … HT2 … HT21) -HT21
/4 width=5 by teqx_repl/
qed-.
-lemma csx_cpx_trans (h) (G):
- ∀L,T1. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- ∀T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T2❫.
+lemma csx_cpx_trans (h) (G) (L):
+ ∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
+ ∀T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → ❪G,L❫ ⊢ ⬈*𝐒[h] T2.
#h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
elim (teqx_dec T1 T2) /3 width=4 by csx_teqx_trans/
qed-.
(* Basic_1: was just: sn3_cast *)
-lemma csx_cast (h) (G):
- ∀L,W. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪W❫ →
- ∀T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫ → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪ⓝW.T❫.
+lemma csx_cast (h) (G) (L):
+ ∀W. ❪G,L❫ ⊢ ⬈*𝐒[h] W →
+ ∀T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓝW.T.
#h #G #L #W #HW @(csx_ind … HW) -W
#W #HW #IHW #T #HT @(csx_ind … HT) -T
#T #HT #IHT @csx_intro
(* Basic_1: was just: sn3_abbr *)
(* Basic_2A1: was: csx_lref_bind *)
-lemma csx_lref_pair_drops (h) (G):
- ∀I,L,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
- ❪G,K❫ ⊢ ⬈*[h] 𝐒❪V❫ → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪#i❫.
-#h #G #I #L #K #V #i #HLK #HV
+lemma csx_lref_pair_drops (h) (G) (L):
+ ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
+ ❪G,K❫ ⊢ ⬈*𝐒[h] V → ❪G,L❫ ⊢ ⬈*𝐒[h] #i.
+#h #G #L #I #K #V #i #HLK #HV
@csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
[ #H destruct elim Hi //
| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
(* Basic_1: was: sn3_gen_def *)
(* Basic_2A1: was: csx_inv_lref_bind *)
-lemma csx_inv_lref_pair_drops (h) (G):
- ∀I,L,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
- ❪G,L❫ ⊢ ⬈*[h] 𝐒❪#i❫ → ❪G,K❫ ⊢ ⬈*[h] 𝐒❪V❫.
-#h #G #I #L #K #V #i #HLK #Hi
+lemma csx_inv_lref_pair_drops (h) (G) (L):
+ ∀I,K,V,i. ⇩[i] L ≘ K.ⓑ[I]V →
+ ❪G,L❫ ⊢ ⬈*𝐒[h] #i → ❪G,K❫ ⊢ ⬈*𝐒[h] V.
+#h #G #L #I #K #V #i #HLK #Hi
elim (lifts_total V (𝐔❨↑i❩))
/4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
qed-.
-lemma csx_inv_lref_drops (h) (G):
- ∀L,i. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪#i❫ →
+lemma csx_inv_lref_drops (h) (G) (L):
+ ∀i. ❪G,L❫ ⊢ ⬈*𝐒[h] #i →
∨∨ ⇩*[Ⓕ,𝐔❨i❩] L ≘ ⋆
| ∃∃I,K. ⇩[i] L ≘ K.ⓤ[I]
- | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*[h] 𝐒❪V❫.
+ | ∃∃I,K,V. ⇩[i] L ≘ K.ⓑ[I]V & ❪G,K❫ ⊢ ⬈*𝐒[h] V.
#h #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
* * /4 width=9 by csx_inv_lref_pair_drops, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
qed-.