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 *)
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15 include "basic_2/substitution/tps_lift.ma".
16 include "basic_2/unfold/tpss.ma".
17 include "basic_2/reducibility/cif.ma".
18 include "basic_2/reducibility/cnf_lift.ma".
20 (* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
22 (* Main properties **********************************************************)
24 lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
25 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
27 | #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
28 elim (cif_inv_delta … HLK ?) //
29 | #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
30 elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
31 lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
32 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
33 elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
37 lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
38 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
39 #T #T2 #_ #HT2 #IHT1 #HT1
40 lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
43 lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
44 #T1 #T2 #H elim H -T1 -T2
46 | * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
47 [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
48 >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
49 | elim (cif_inv_ri2 … H) /2 width=1/
51 | #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
52 elim (cif_inv_appl … H) -H #_ #_ #H
53 elim (simple_inv_bind … H)
54 | #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
55 [ lapply (tps_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
56 elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
57 lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
58 lapply (IHT1 … HT1) -IHT1 #H destruct
59 lapply (tps_cif_eq … HT2 ?) -HT2 //
60 | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
61 elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
63 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
64 elim (cif_inv_appl … H) -H #_ #_ #H
65 elim (simple_inv_bind … H)
66 | #V1 #T1 #T #T2 #_ #_ #_ #L #H
67 elim (cif_inv_ri2 … H) /2 width=1/
68 | #V1 #T1 #T2 #_ #_ #L #H
69 elim (cif_inv_ri2 … H) /2 width=1/
73 lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
74 #L #T1 #T2 * #T0 #HT10 #HT02 #HT1
75 lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
78 theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
81 (* Note: this property is unusual *)
82 lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
85 elim (cnf_inv_delta … HLK H)
87 elim (cnf_inv_appl … H) -H /2 width=1/
89 elim (cnf_inv_appl … H) -H /2 width=1/
90 | #I #L #V #T * #H1 #H2 destruct
91 [ elim (cnf_inv_zeta … H2)
92 | elim (cnf_inv_tau … H2)
94 |5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
95 [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
96 |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
99 elim (cnf_inv_appl … H) -H #_ #_ #H
100 elim (simple_inv_bind … H)
102 elim (cnf_inv_appl … H) -H #_ #_ #H
103 elim (simple_inv_bind … H)
107 theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.