matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/cnf_cif.etc

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/reducibility/cif.ma".
  16 include "basic_2/reducibility/cnf_lift.ma".
  17
  18 (* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
  19
  20 (* Main properties **********************************************************)
  21
  22 lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
  23 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
  24 [ //
  25 | #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
  26   elim (cif_inv_delta … HLK ?) //
  27 | #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
  28   elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
  29   lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
  30 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
  31   elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
  32 ]
  33 qed.
  34
  35 lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
  36 #L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
  37 #T #T2 #_ #HT2 #IHT1 #HT1
  38 lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
  39 qed.
  40
  41 lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
  42 #T1 #T2 #H elim H -T1 -T2
  43 [ //
  44 | * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
  45   [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
  46     >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
  47   | elim (cif_inv_ri2 … H) /2 width=1/
  48   ]
  49 | #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
  50   elim (cif_inv_appl … H) -H #_ #_ #H
  51   elim (simple_inv_bind … H)
  52 | #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
  53   [ lapply (tps_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
  54     elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
  55     lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
  56     lapply (IHT1 … HT1) -IHT1 #H destruct
  57     lapply (tps_cif_eq … HT2 ?) -HT2 //
  58   | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
  59     elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
  60   ]
  61 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
  62   elim (cif_inv_appl … H) -H #_ #_ #H
  63   elim (simple_inv_bind … H)
  64 | #V1 #T1 #T #T2 #_ #_ #_ #L #H
  65   elim (cif_inv_ri2 … H) /2 width=1/
  66 | #V1 #T1 #T2 #_ #_ #L #H
  67   elim (cif_inv_ri2 … H) /2 width=1/
  68 ]
  69 qed.
  70
  71 lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
  72 #L #T1 #T2 * #T0 #HT10 #HT02 #HT1
  73 lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
  74 qed.
  75
  76 theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
  77 /3 width=3/ qed.
  78
  79 (* Note: this property is unusual *)
  80 lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
  81 #L #T #H elim H -L -T
  82 [ #L #K #V #i #HLK #H
  83   elim (cnf_inv_delta … HLK H)
  84 | #L #V #T #_ #IHV #H
  85   elim (cnf_inv_appl … H) -H /2 width=1/
  86 | #L #V #T #_ #IHT #H
  87   elim (cnf_inv_appl … H) -H /2 width=1/
  88 | #I #L #V #T * #H1 #H2 destruct
  89   [ elim (cnf_inv_zeta … H2)
  90   | elim (cnf_inv_tau … H2)
  91   ]
  92 |5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
  93   [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
  94   |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
  95   ]
  96 | #a #L #V #W #T #H
  97   elim (cnf_inv_appl … H) -H #_ #_ #H
  98   elim (simple_inv_bind … H)
  99 | #a #L #V #W #T #H
 100   elim (cnf_inv_appl … H) -H #_ #_ #H
 101   elim (simple_inv_bind … H)
 102 ]
 103 qed.
 104
 105 theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
 106 /2 width=4/ qed.