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14
15 include "basic_2/rt_transition/cnr_tdeq.ma".
16 include "basic_2/rt_transition/cnu_drops.ma".
17 include "basic_2/rt_transition/cnu_cnr.ma".
18 include "basic_2/rt_transition/cnu_cnr_simple.ma".
19
20 (* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
21
22 (* Properties with context-free sort-irrelevant equivalence for terms *******)
23
24 lemma cnu_dec_tdeq (h) (G) (L):
25       ∀T1. ∨∨ ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
26             | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
27 #h #G #L #T1
28 @(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
29 [ #s #HG #HL #HT destruct -IH
30   /3 width=5 by cnu_sort, or_introl/
31 | #i #HG #HL #HT destruct -IH
32   elim (drops_F_uni L i)
33   [ /3 width=7 by cnu_atom_drops, or_introl/
34   | * * [ #I | * #V ] #K #HLK
35     [ /3 width=8 by cnu_unit_drops, or_introl/
36     | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
37       @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_delta_drops/ ] #H
38       lapply (tdeq_inv_lref1 … H) -H #H destruct
39       /2 width=5 by lifts_inv_lref2_uni_lt/
40     | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
41       @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_ell_drops/ ] #H
42       lapply (tdeq_inv_lref1 … H) -H #H destruct
43       /2 width=5 by lifts_inv_lref2_uni_lt/
44     ]
45   ]
46 | #l #HG #HL #HT destruct -IH
47   /3 width=5 by cnu_gref, or_introl/
48 | #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
49   [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
50     elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
51     [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
52       @or_intror @(ex2_2_intro … X1) [1,2: /2 width=4 by cpm_zeta/ ] #H
53       /2 width=7 by tdeq_lifts_inv_pair_sn/
54     | @or_intror @(ex2_2_intro … (+ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
55       elim (tdeq_inv_pair … H) -H /2 width=1 by/
56     ]
57   | elim (cnr_dec_tdeq h G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * ] -IH
58     [ #HT1 #HV1 /3 width=7 by cnu_abbr_neg, or_introl/
59     | #n #T2 #HT12 #HnT12 #_
60       @or_intror @(ex2_2_intro … (-ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
61       elim (tdeq_inv_pair … H) -H /2 width=1 by/
62     | #V2 #HV12 #HnV12
63       @or_intror @(ex2_2_intro … (-ⓓV2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
64       elim (tdeq_inv_pair … H) -H /2 width=1 by/
65     ]
66   | elim (cnr_dec_tdeq h G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * ] -IH
67     [ #HT1 #HV1 /3 width=7 by cnu_abst, or_introl/
68     | #n #T2 #HT12 #HnT12 #_
69       @or_intror @(ex2_2_intro … (ⓛ{p}V1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
70       elim (tdeq_inv_pair … H) -H /2 width=1 by/
71     | #V2 #HV12 #HnV12
72       @or_intror @(ex2_2_intro … (ⓛ{p}V2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
73       elim (tdeq_inv_pair … H) -H /2 width=1 by/
74     ]
75   ]
76 | * #V1 #T1 #HG #HL #HT destruct [| -IH ]
77   [ elim (cnr_dec_tdeq h G L V1) [ elim (IH G L T1) [| * | // ] | * ] -IH
78     [ #HT1 #HV1
79       elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
80       [ /3 width=7 by cnu_appl_simple, or_introl/
81       | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
82         @or_intror @(ex2_2_intro … (ⓓ{p}W1.ⓐX1.U1)) [1,2: /2 width=3 by cpm_theta/ ] #H
83         elim (tdeq_inv_pair … H) -H #H destruct
84       | @or_intror @(ex2_2_intro … (ⓓ{p}ⓝW1.V1.U1)) [1,2: /2 width=2 by cpm_beta/ ] #H
85         elim (tdeq_inv_pair … H) -H #H destruct
86       ]
87     | #n #T2 #HT12 #HnT12 #_
88       @or_intror @(ex2_2_intro … (ⓐV1.T2)) [1,2: /2 width=2 by cpm_appl/ ] #H
89       elim (tdeq_inv_pair … H) -H /2 width=1 by/
90     | #V2 #HV12 #HnV12
91       @or_intror @(ex2_2_intro … (ⓐV2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
92       elim (tdeq_inv_pair … H) -H /2 width=1 by/
93     ]
94   | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpm_eps/ ] #H
95     /2 width=4 by tdeq_inv_pair_xy_y/
96   ]
97 ]
98 qed-.