1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2/computation/cprs.ma".
16 include "basic_2/computation/csn.ma".
18 (* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
20 (* Properties concerning context-sensitive computation on terms *************)
22 definition csns: lenv → predicate term ≝ λL. SN … (cprs L) (eq …).
25 "context-sensitive strong normalization (term)"
26 'SNStar L T = (csns L T).
28 (* Basic eliminators ********************************************************)
30 lemma csns_ind: ∀L. ∀R:predicate term.
32 (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → R T2) → R T1
35 #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
36 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
39 (* Basic properties *********************************************************)
41 (* Basic_1: was: sn3_intro *)
42 lemma csns_intro: ∀L,T1.
43 (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → L ⊢ ⬇** T2) → L ⊢ ⬇** T1.
48 fact csns_intro_aux: ∀L,T1.
49 (∀T,T2. L ⊢ T ➡* T2 → T1 = T → (T1 = T2 → False) → L ⊢ ⬇** T2) → L ⊢ ⬇** T1.
52 (* Basic_1: was: sn3_pr3_trans (old version) *)
53 lemma csns_cprs_trans: ∀L,T1. L ⊢ ⬇** T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬇** T2.
54 #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
55 @csns_intro #T #HLT2 #HT2
56 elim (term_eq_dec T1 T2) #HT12
57 [ -IHT1 -HLT12 destruct /3 width=1/
58 | -HT1 -HT2 /3 width=4/
61 (* Basic_1: was: sn3_pr2_intro (old version) *)
62 lemma csns_intro_cpr: ∀L,T1.
63 (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇** T2) →
66 @csns_intro_aux #T #T2 #H @(cprs_ind_dx … H) -T
69 | #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
70 elim (term_eq_dec T0 T) #HT0
71 [ -HLT1 -HLT2 -H /3 width=1/
72 | -IHT -HT12 /4 width=3/
77 (* Main properties **********************************************************)
79 theorem csn_csns: ∀L,T. L ⊢ ⬇* T → L ⊢ ⬇** T.
80 #L #T #H @(csn_ind … H) -T /4 width=1/
83 theorem csns_csn: ∀L,T. L ⊢ ⬇** T → L ⊢ ⬇* T.
84 #L #T #H @(csns_ind … H) -T /4 width=1/
87 (* Basic_1: was: sn3_pr3_trans *)
88 lemma csn_cprs_trans: ∀L,T1. L ⊢ ⬇* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬇* T2.
91 (* Main eliminators *********************************************************)
93 lemma csn_ind_cprs: ∀L. ∀R:predicate term.
95 (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → R T2) → R T1
98 #L #R #H0 #T1 #H @(csns_ind … (csn_csns … H)) -T1 #T1 #HT1 #IHT1
99 @H0 -H0 /2 width=1/ -HT1 /3 width=1/