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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/sn_5.ma".
16 include "basic_2/reduction/cnx.ma".
18 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
20 definition csx: ∀h. sd h → relation3 genv lenv term ≝
21 λh,g,G,L. SN … (cpx h g G L) (eq …).
24 "context-sensitive extended strong normalization (term)"
25 'SN h g G L T = (csx h g G L T).
27 (* Basic eliminators ********************************************************)
29 lemma csx_ind: ∀h,g,G,L. ∀R:predicate term.
30 (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
31 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
34 ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
35 #h #g #G #L #R #H0 #T1 #H elim H -T1
36 /5 width=1 by SN_intro/
39 (* Basic properties *********************************************************)
41 (* Basic_1: was just: sn3_pr2_intro *)
42 lemma csx_intro: ∀h,g,G,L,T1.
43 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
45 /4 width=1 by SN_intro/ qed.
47 lemma csx_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
48 ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
49 #h #g #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
50 elim (eq_term_dec T1 T2) #HT12 destruct /3 width=4 by/
53 (* Basic_1: was just: sn3_nf2 *)
54 lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
55 /2 width=1 by NF_to_SN/ qed.
57 lemma csx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
58 #h #g #G #L #k elim (deg_total h g k)
59 #l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=6 by cnx_csx, cnx_sort/
60 #l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
61 #Hkl @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
62 [ #H destruct elim HX //
63 | -HX * #l0 #_ #H destruct -l0 /2 width=1 by/
67 (* Basic_1: was just: sn3_cast *)
68 lemma csx_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
69 ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
70 #h #g #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
72 elim (cpx_inv_cast1 … H1) -H1
73 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
74 elim (eq_false_inv_tpair_sn … H2) -H2
75 [ /3 width=3 by csx_cpx_trans/
76 | -HLW0 * #H destruct /3 width=1 by/
78 |2,3: /3 width=3 by csx_cpx_trans/
82 (* Basic forward lemmas *****************************************************)
84 fact csx_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
85 ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
86 #h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
87 @csx_intro #V2 #HLV2 #HV2
88 @(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
89 #H destruct /2 width=1 by/
92 (* Basic_1: was just: sn3_gen_head *)
93 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.
94 /2 width=5 by csx_fwd_pair_sn_aux/ qed-.
96 fact csx_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
97 ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
98 #h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
99 @csx_intro #T2 #HLT2 #HT2
100 @(IH (ⓑ{a,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
101 #H destruct /2 width=1 by/
104 (* Basic_1: was just: sn3_gen_bind *)
105 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.
106 /2 width=4 by csx_fwd_bind_dx_aux/ qed-.
108 fact csx_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
109 ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
110 #h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
111 @csx_intro #T2 #HLT2 #HT2
112 @(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
113 #H destruct /2 width=1 by/
116 (* Basic_1: was just: sn3_gen_flat *)
117 lemma csx_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
118 /2 width=5 by csx_fwd_flat_dx_aux/ qed-.
120 lemma csx_fwd_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T →
121 ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
122 /3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
124 lemma csx_fwd_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T →
125 ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
126 /3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
128 (* Basic_1: removed theorems 14:
130 sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
131 sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
132 sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind