matita/matita/contribs/lambda_delta/basic_2/reducibility/cnf.ma

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  14
  15 include "basic_2/reducibility/cpr.ma".
  16
  17 (* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
  18
  19 definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …).
  20
  21 interpretation
  22    "context-sensitive normality (term)"
  23    'Normal L T = (cnf L T).
  24
  25 (* Basic inversion lemmas ***************************************************)
  26
  27 lemma cnf_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
  28 #L #V1 #T1 #HVT1 @and3_intro
  29 [ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
  30 | #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
  31 | generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
  32   [ elim (lift_total V1 0 1) #V2 #HV12
  33     lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3/ -HV12 #H destruct
  34   | lapply (H (ⓓ{a}V1.U1) ?) -H /3 width=1/ #H destruct
  35 ]
  36 qed-.
  37
  38 lemma cnf_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
  39 #L #V #T #H elim (is_lift_dec T 0 1)
  40 [ * #U #HTU
  41   lapply (H U ?) -H /3 width=3 by cpr_tpr, tpr_zeta/ #H destruct (**) (* auto too slow without trace *)
  42   elim (lift_inv_pair_xy_y … HTU)
  43 | #HT
  44   elim (tps_full (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12
  45   lapply (H (+ⓓV.T2) ?) -H /3 width=3 by cpr_tpr, tpr_delta/ -HT2 #H destruct /3 width=2/ (**) (* auto too slow without trace *)
  46 ]
  47 qed.
  48
  49 lemma cnf_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
  50 #L #V #T #H lapply (H T ?) -H /2 width=1/ #H
  51 @discr_tpair_xy_y //
  52 qed-.
  53
  54 (* Basic properties *********************************************************)
  55
  56 (* Basic_1: was: nf2_sort *)
  57 lemma cnf_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄.
  58 #L #k #X #H
  59 >(cpr_inv_sort1 … H) //
  60 qed.
  61
  62 (* Basic_1: was: nf2_dec *)
  63 axiom cnf_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨
  64                ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
  65
  66 (* Basic_1: removed theorems 1: nf2_abst_shift *)