(* *)
(**************************************************************************)
-include "basic_2/notation/relations/sn_5.ma".
-include "basic_2/reduction/cnx.ma".
+include "basic_2/notation/relations/predtystrong_4.ma".
+include "static_2/syntax/tdeq.ma".
+include "basic_2/rt_transition/cpx.ma".
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-definition csx: ∀h. sd h → relation3 genv lenv term ≝
- λh,o,G,L. SN … (cpx h o G L) (eq …).
+definition csx: ∀h. relation3 genv lenv term ≝
+ λh,G,L. SN … (cpx h G L) tdeq.
interpretation
- "context-sensitive extended strong normalization (term)"
- 'SN h o G L T = (csx h o G L T).
+ "strong normalization for unbound context-sensitive parallel rt-transition (term)"
+ 'PRedTyStrong h G L T = (csx h G L T).
(* Basic eliminators ********************************************************)
-lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
- (â\88\80T1. â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T1 →
- (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T2 â\86\92 (T1 = T2 â\86\92 â\8a¥) â\86\92 R T2) →
- R T1
+lemma csx_ind: ∀h,G,L. ∀Q:predicate term.
+ (â\88\80T1. â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83T1â¦\84 →
+ (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h] T2 â\86\92 (T1 â\89\9b T2 â\86\92 â\8a¥) â\86\92 Q T2) →
+ Q T1
) →
- â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T â\86\92 R T.
-#h #o #G #L #R #H0 #T1 #H elim H -T1
+ â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 â\86\92 Q T.
+#h #G #L #Q #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,o,G,L,T1.
- (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T2 â\86\92 (T1 = T2 â\86\92 â\8a¥) â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T2) →
- â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T1.
+lemma csx_intro: ∀h,G,L,T1.
+ (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h] T2 â\86\92 (T1 â\89\9b T2 â\86\92 â\8a¥) â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83T2â¦\84) →
+ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83T1â¦\84.
/4 width=1 by SN_intro/ qed.
-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,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,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,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
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ /3 width=3 by csx_cpx_trans/
- | -HLW0 * #H destruct /3 width=1 by/
- ]
-|2,3: /3 width=3 by csx_cpx_trans/
-]
-qed.
-
(* Basic forward lemmas *****************************************************)
-fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
- â\88\80I,V,T. U = â\91¡{I}V.T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] V.
-#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ â\88\80I,V,T. U = â\91¡{I}V.T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Vâ¦\84.
+#h #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/
+#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_head *)
-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.
+lemma csx_fwd_pair_sn: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 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] 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
+fact csx_fwd_bind_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #U #H elim H -H #U0 #_ #IH #p #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/
+@(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_bind *)
-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.
+lemma csx_fwd_bind_dx: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 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] U →
- â\88\80I,V,T. U = â\93\95{I}V.T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T.
-#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ â\88\80I,V,T. U = â\93\95{I}V.T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84.
+#h #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/
+#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+lemma csx_fwd_flat_dx: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
-lemma csx_fwd_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T →
- â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] V â\88§ â¦\83G, L.â\93\91{I}Vâ¦\84 â\8a¢ â¬\8a*[h, o] T.
+lemma csx_fwd_bind: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Vâ¦\84 â\88§ â¦\83G, L.â\93\91{I}Vâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84.
/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] ⓕ{I}V.T →
- â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] V â\88§ â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T.
+lemma csx_fwd_flat: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ →
+ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Vâ¦\84 â\88§ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84.
/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
+*)
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