R T1
) →
∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+#h #g #G #L #R #H0 #T1 #H elim H -T1
+/5 width=1 by SN_intro/
qed-.
(* Basic properties *********************************************************)
lemma csx_intro: ∀h,g,G,L,T1.
(∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
⦃G, L⦄ ⊢ ⬊*[h, g] T1.
-/4 width=1/ qed.
+/4 width=1 by SN_intro/ qed.
lemma csx_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csx_intro #T #HLT2 #HT2
-elim (eq_term_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
+#h #g #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+elim (eq_term_dec T1 T2) #HT12 destruct /3 width=4 by/
qed-.
(* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-/2 width=1/ qed.
+lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/2 width=1 by NF_to_SN/ qed.
-lemma cnx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
+lemma csx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
#h #g #G #L #k elim (deg_total h g k)
-#l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=1/
+#l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=6 by cnx_csx, cnx_sort/
#l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
#Hkl @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
[ #H destruct elim HX //
-| -HX * #l0 #_ #H destruct -l0 /2 width=1/
+| -HX * #l0 #_ #H destruct -l0 /2 width=1 by/
]
qed.
[ * #W0 #T0 #HLW0 #HLT0 #H destruct
elim (eq_false_inv_tpair_sn … H2) -H2
[ /3 width=3 by csx_cpx_trans/
- | -HLW0 * #H destruct /3 width=1/
+ | -HLW0 * #H destruct /3 width=1 by/
]
|2,3: /3 width=3 by csx_cpx_trans/
]
∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
#h #g #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=1/ -HLV2 #H destruct /2 width=1/
+@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
+#H destruct /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_head *)
∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
#h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{a,I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+@(IH (ⓑ{a,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+#H destruct /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_bind *)
∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
#h #g #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=1/ -HLT2 #H destruct /2 width=1/
+@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
+#H destruct /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_flat *)
lemma csx_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
+lemma csx_fwd_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
+
+lemma csx_fwd_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
+
(* Basic_1: removed theorems 14:
sn3_cdelta
sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change