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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/reduction/cnx.ma".
17 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
19 definition csn: ∀h. sd h → lenv → predicate term ≝
20 λh,g,L. SN … (cpx h g L) (eq …).
23 "context-sensitive extended strong normalization (term)"
24 'SN h g L T = (csn h g L T).
26 (* Basic eliminators ********************************************************)
28 lemma csn_ind: ∀h,g,L. ∀R:predicate term.
29 (∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
30 (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → R T2) →
33 ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → R T.
34 #h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
35 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
38 (* Basic properties *********************************************************)
40 (* Basic_1: was just: sn3_pr2_intro *)
41 lemma csn_intro: ∀h,g,L,T1.
42 (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] T2) →
46 (* Basic_1: was just: sn3_nf2 *)
47 lemma cnx_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⦃h, L⦄ ⊢ ⬊*[g] T.
50 lemma csn_cpx_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
51 ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ ⬊*[g] T2.
52 #h #g #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
53 @csn_intro #T #HLT2 #HT2
54 elim (term_eq_dec T1 T2) #HT12
55 [ -IHT1 -HLT12 destruct /3 width=1/
56 | -HT1 -HT2 /3 width=4/
59 (* Basic_1: was just: sn3_cast *)
60 lemma csn_cast: ∀h,g,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
61 ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓝW.T.
62 #h #g #L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
64 elim (cpx_inv_cast1 … H1) -H1
65 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
66 elim (eq_false_inv_tpair_sn … H2) -H2
67 [ /3 width=3 by csn_cpx_trans/
68 | -HLW0 * #H destruct /3 width=1/
70 | /3 width=3 by csn_cpx_trans/
74 (* Basic forward lemmas *****************************************************)
76 fact csn_fwd_pair_sn_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
77 ∀I,V,T. U = ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
78 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
79 @csn_intro #V2 #HLV2 #HV2
80 @(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
83 (* Basic_1: was just: sn3_gen_head *)
84 lemma csn_fwd_pair_sn: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
85 /2 width=5 by csn_fwd_pair_sn_aux/ qed-.
87 fact csn_fwd_bind_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
88 ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
89 #h #g #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
90 @csn_intro #T2 #HLT2 #HT2
91 @(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
94 (* Basic_1: was just: sn3_gen_bind *)
95 lemma csn_fwd_bind_dx: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
96 /2 width=4 by csn_fwd_bind_dx_aux/ qed-.
98 fact csn_fwd_flat_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
99 ∀I,V,T. U = ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
100 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
101 @csn_intro #T2 #HLT2 #HT2
102 @(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
105 (* Basic_1: was just: sn3_gen_flat *)
106 lemma csn_fwd_flat_dx: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
107 /2 width=5 by csn_fwd_flat_dx_aux/ qed-.
109 (* Basic_1: removed theorems 14:
111 sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
112 sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
113 sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind