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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/reducibility/cnf.ma".
17 (* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
19 definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …).
22 "context-sensitive strong normalization (term)"
25 (* Basic eliminators ********************************************************)
27 lemma csn_ind: ∀L. ∀R:predicate term.
29 (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → R T2) →
33 #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
34 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
37 (* Basic properties *********************************************************)
39 (* Basic_1: was: sn3_pr2_intro *)
40 lemma csn_intro: ∀L,T1.
41 (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊* T2) → L ⊢ ⬊* T1.
44 (* Basic_1: was: sn3_nf2 *)
45 lemma csn_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ ⬊* T.
48 lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬊* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬊* T2.
49 #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
50 @csn_intro #T #HLT2 #HT2
51 elim (term_eq_dec T1 T2) #HT12
52 [ -IHT1 -HLT12 destruct /3 width=1/
53 | -HT1 -HT2 /3 width=4/
56 (* Basic_1: was: sn3_cast *)
57 lemma csn_cast: ∀L,W. L ⊢ ⬊* W → ∀T. L ⊢ ⬊* T → L ⊢ ⬊* ⓝW. T.
58 #L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
60 elim (cpr_inv_cast1 … H1) -H1
61 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
62 elim (eq_false_inv_tpair_sn … H2) -H2
64 | -HLW0 * #H destruct /3 width=1/
70 (* Basic forward lemmas *****************************************************)
72 fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬊* T.
73 #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
74 @csn_intro #T2 #HLT2 #HT2
75 @(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
78 (* Basic_1: was: sn3_gen_flat *)
79 lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬊* ⓕ{I} V. T → L ⊢ ⬊* T.
82 (* Basic_1: removed theorems 14:
84 sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
85 sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
86 sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind