(* Basic eliminators ********************************************************)
lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T1⦄ →
+ (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
(∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) →
R T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄ → R T.
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
#h #o #G #L #R #H0 #T1 #H elim H -T1
/5 width=1 by SN_intro/
qed-.
(* Basic_1: was just: sn3_pr2_intro *)
lemma csx_intro: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T1⦄.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
/4 width=1 by SN_intro/ qed.
-lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃⋆s⦄.
-#h #o #G #L #s elim (deg_total h o s)
-#d generalize in match s; -s elim d -d
-[ #s1 #Hs1 @csx_intro #X #H #HX elim HX -HX
- elim (cpx_inv_sort1 … H) -H #H destruct //
- /3 width=3 by tdeq_sort, deg_next/
-| #d #IH #s #Hsd lapply (deg_next_SO … Hsd) -Hsd
- #Hsd @csx_intro #X #H #HX
- elim (cpx_inv_sort1 … H) -H #H destruct /2 width=1 by/
- elim HX //
-]
-qed.
-
(* Basic forward lemmas *****************************************************)
-fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃U⦄ →
- ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄.
+fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #V2 #HLV2 #HV2
@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
qed-.
(* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄.
+lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
-fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃U⦄ →
- ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
+ ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
#h #o #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
@(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
qed-.
(* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
-fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃U⦄ →
- ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
qed-.
(* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
-lemma csx_fwd_bind: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
-lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃ⓕ{I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
(* Basic_1: removed theorems 14:
sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
*)
+(* Basic_2A1: removed theorems 6:
+ csxa_ind csxa_intro csxa_cpxs_trans csxa_intro_cpx
+ csx_csxa csxa_csx
+*)