(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
definition csx: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L. SN … (cpx h g G L) (eq …).
+ λh,o,G,L. SN … (cpx h o G L) (eq …).
interpretation
"context-sensitive extended strong normalization (term)"
- 'SN h g G L T = (csx h g G L T).
+ 'SN h o G L T = (csx h o G L T).
(* Basic eliminators ********************************************************)
-lemma csx_ind: ∀h,g,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
+lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) →
R T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H elim H -T1
/5 width=1 by SN_intro/
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was just: sn3_pr2_intro *)
-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.
+lemma csx_intro: ∀h,o,G,L,T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] T2) →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] T1.
/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 @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ⬊*[h, o] T2.
+#h #o #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⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
/2 width=1 by NF_to_SN/ qed.
-lemma csx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
-#h #g #G #L #k elim (deg_total h g k)
-#d generalize in match k; -k @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
-#d #IHd #k #Hkd lapply (deg_next_SO … Hkd) -Hkd
+lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬊*[h, o] ⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
#Hkd @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
[ #H destruct elim HX //
| -HX * #d0 #_ #H destruct -d0 /2 width=1 by/
qed.
(* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
- ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
-#h #g #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, o] W →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓝW.T.
+#h #o #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
elim (cpx_inv_cast1 … H1) -H1
[ * #W0 #T0 #HLW0 #HLT0 #H destruct
(* Basic forward lemmas *****************************************************)
-fact csx_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
- ∀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
+fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+ ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] 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
#H destruct /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
-fact csx_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
- ∀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
+fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+ ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+#h #o #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=3 by cpx_bind/ -HLT2
#H destruct /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
-fact csx_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
- ∀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
+fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] 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
#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.
+lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] 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.
+lemma csx_fwd_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] 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.
+lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] T.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
(* Basic_1: removed theorems 14: