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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/reducibility/cpr.ma".
16 include "basic_2/reducibility/cnf.ma".
18 (* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
20 definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …).
23 "context-sensitive strong normalization (term)"
26 (* Basic eliminators ********************************************************)
28 lemma csn_ind: ∀L. ∀R:predicate term.
30 (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → R T2) →
34 #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: sn3_pr2_intro *)
41 lemma csn_intro: ∀L,T1.
42 (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬇* T2) → L ⊢ ⬇* T1.
45 (* Basic_1: was: sn3_nf2 *)
46 lemma csn_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ ⬇* T.
49 lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬇* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬇* T2.
50 #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
51 @csn_intro #T #HLT2 #HT2
52 elim (term_eq_dec T1 T2) #HT12
53 [ -IHT1 -HLT12 destruct /3 width=1/
54 | -HT1 -HT2 /3 width=4/
57 (* Basic_1: was: sn3_cast *)
58 lemma csn_cast: ∀L,W. L ⊢ ⬇* W → ∀T. L ⊢ ⬇* T → L ⊢ ⬇* ⓣW. T.
59 #L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
61 elim (cpr_inv_cast1 … H1) -H1
62 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
63 elim (eq_false_inv_tpair_sn … H2) -H2
65 | -HLW0 * #H destruct /3 width=1/
71 (* Basic forward lemmas *****************************************************)
73 fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬇* T.
74 #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
75 @csn_intro #T2 #HLT2 #HT2
76 @(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
79 (* Basic_1: was: sn3_gen_flat *)
80 lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬇* ⓕ{I} V. T → L ⊢ ⬇* T.
83 (* Basic_1: removed theorems 14:
85 sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
86 sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
87 sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind