matita/matita/contribs/lambdadelta/basic_2/static/aaa_drops.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/relocation/drops.ma".
  16 include "basic_2/static/aaa.ma".
  17
  18 (* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
  19
  20 (* Properties with generic slicing for local environments *******************)
  21
  22 (* Basic_2A1: was: aaa_lref *)
  23 lemma aaa_lref_gen: ∀I,G,K,V,B,i,L. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
  24 #I #G #K #V #B #i elim i -i
  25 [ #L #H lapply (drops_fwd_isid … H ?) -H //
  26   #H destruct /2 width=1 by aaa_zero/
  27 | #i #IH #L <uni_succ #H #HB lapply (drops_inv_pair2_isuni_next … H) -H // *
  28   #Z #Y #X #HY #H destruct /3 width=1 by aaa_lref/
  29 ]
  30 qed.
  31
  32 (* Basic_2A1: includes: aaa_lift *)
  33 lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,c,f. ⬇*[c, f] L2 ≡ L1 →
  34                  ∀T2. ⬆*[f] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
  35 #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
  36 [ #G #L1 #s #L2 #c #f #_ #T2 #H
  37   >(lifts_inv_sort1 … H) -H //
  38 | #I #G #L1 #V1 #B #_ #IHB #L2 #c #f #HL21 #T2 #H
  39   elim (lifts_inv_lref1 … H) -H #i2 #Hi #H destruct
  40   @aaa_lref_gen [5: @IHB ]
  41 | #I #G #L1 #V1 #B #i1 #_ #IHB #L2 #c #f #HL21 #T2 #H
  42   elim (lifts_inv_lref1 … H) -H #i2 #Hi #H destruct
  43   lapply (at_inv_nxx … H)
  44
  45
  46
  47
  48 | #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #c #l #k #HL21 #T2 #H
  49   elim (lift_inv_lref1 … H) -H * #Hil #H destruct
  50   [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
  51     elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
  52     /3 width=9 by aaa_lref/
  53   | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
  54     /3 width=9 by aaa_lref, drop_inv_gen/
  55   ]
  56 | #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
  57   elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  58   /4 width=5 by aaa_abbr, drop_skip/
  59 | #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
  60   elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  61   /4 width=5 by aaa_abst, drop_skip/
  62 | #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
  63   elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  64   /3 width=5 by aaa_appl/
  65 | #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #c #l #k #HL21 #X #H
  66   elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  67   /3 width=5 by aaa_cast/
  68 ]
  69 qed.
  70
  71 (* Inversion lemmas with generic slicing for local environments *************)
  72
  73 (* Basic_2A1: was: aaa_inv_lref *)
  74 lemma aaa_inv_lref_gen: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
  75                         ∃∃I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
  76 #G #A #i elim i -i
  77 [ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
  78 | #i #IH #L #H elim (aaa_inv_lref … H) -H
  79   #I #K #V #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
  80 ]
  81 qed-.
  82
  83 lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,c,l,k. ⬇[c, l, k] L2 ≡ L1 →
  84                     ∀T1. ⬆[l, k] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
  85 #G #L2 #T2 #A #H elim H -G -L2 -T2 -A
  86 [ #G #L2 #s #L1 #c #l #k #_ #T1 #H
  87   >(lift_inv_sort2 … H) -H //
  88 | #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #c #l #k #HL21 #T1 #H
  89   elim (lift_inv_lref2 … H) -H * #Hil #H destruct
  90   [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
  91   | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
  92   ]
  93 | #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
  94   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
  95   /4 width=5 by aaa_abbr, drop_skip/
  96 | #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
  97   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
  98   /4 width=5 by aaa_abst, drop_skip/
  99 | #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
 100   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
 101   /3 width=5 by aaa_appl/
 102 | #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #c #l #k #HL21 #X #H
 103   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
 104   /3 width=5 by aaa_cast/
 105 ]
 106 qed-.