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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 *)
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15 include "static_2/relocation/lifts_tdeq.ma".
16 include "basic_2/rt_transition/cpr_drops_basic.ma".
17 include "basic_2/rt_transition/cnr_simple.ma".
18 include "basic_2/rt_transition/cnr_drops.ma".
20 (* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
22 (* Properties with context-free sort-irrelevant equivalence for terms *******)
24 (* Basic_1: was: nf2_dec *)
25 (* Basic_2A1: uses: cnr_dec *)
26 lemma cnr_dec_tdeq (h) (G) (L):
27 ∀T1. ∨∨ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄
28 | ∃∃T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥).
30 @(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
31 [ #s #HG #HL #HT destruct -IH
32 /3 width=4 by cnr_sort, or_introl/
33 | #i #HG #HL #HT destruct -IH
34 elim (drops_F_uni L i)
35 [ /3 width=6 by cnr_lref_atom, or_introl/
36 | * * [ #I | * #V ] #K #HLK
37 [ /3 width=7 by cnr_lref_unit, or_introl/
38 | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
39 @or_intror @(ex2_intro … W) [ /2 width=6 by cpm_delta_drops/ ] #H
40 lapply (tdeq_inv_lref1 … H) -H #H destruct
41 /2 width=5 by lifts_inv_lref2_uni_lt/
42 | /3 width=7 by cnr_lref_abst, or_introl/
45 | #l #HG #HL #HT destruct -IH
46 /3 width=4 by cnr_gref, or_introl/
47 | #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
48 [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
49 elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
50 [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
51 @or_intror @(ex2_intro … X1) [ /2 width=3 by cpm_zeta/ ] #H
52 /2 width=7 by tdeq_lifts_inv_pair_sn/
53 | @or_intror @(ex2_intro … (+ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
54 elim (tdeq_inv_pair … H) -H /2 width=1 by/
56 | elim (IH G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * | // ] -IH
57 [ #HT1 #HV1 /3 width=6 by cnr_abbr_neg, or_introl/
59 @or_intror @(ex2_intro … (-ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
60 elim (tdeq_inv_pair … H) -H /2 width=1 by/
62 @or_intror @(ex2_intro … (-ⓓV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
63 elim (tdeq_inv_pair … H) -H /2 width=1 by/
65 | elim (IH G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * | // ] -IH
66 [ #HT1 #HV1 /3 width=6 by cnr_abst, or_introl/
68 @or_intror @(ex2_intro … (ⓛ{p}V1.T2)) [ /2 width=1 by cpm_bind/ ] #H
69 elim (tdeq_inv_pair … H) -H /2 width=1 by/
71 @or_intror @(ex2_intro … (ⓛ{p}V2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
72 elim (tdeq_inv_pair … H) -H /2 width=1 by/
75 | * #V1 #T1 #HG #HL #HT destruct [| -IH ]
76 [ elim (IH G L V1) [ elim (IH G L T1) [| * | // ] | * | // ] -IH
78 elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
79 [ /3 width=6 by cnr_appl_simple, or_introl/
80 | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
81 @or_intror @(ex2_intro … (ⓓ{p}W1.ⓐX1.U1)) [ /2 width=3 by cpm_theta/ ] #H
82 elim (tdeq_inv_pair … H) -H #H destruct
83 | @or_intror @(ex2_intro … (ⓓ{p}ⓝW1.V1.U1)) [ /2 width=1 by cpm_beta/ ] #H
84 elim (tdeq_inv_pair … H) -H #H destruct
87 @or_intror @(ex2_intro … (ⓐV1.T2)) [ /2 width=1 by cpm_appl/ ] #H
88 elim (tdeq_inv_pair … H) -H /2 width=1 by/
90 @or_intror @(ex2_intro … (ⓐV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
91 elim (tdeq_inv_pair … H) -H /2 width=1 by/
93 | @or_intror @(ex2_intro … T1) [ /2 width=1 by cpm_eps/ ] #H
94 /2 width=4 by tdeq_inv_pair_xy_y/