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:                                           *)
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   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_drops.ma".
  16 include "basic_2/s_computation/fqup_weight.ma".
  17 include "basic_2/s_computation/fqup_drops.ma".
  18 include "basic_2/static/aaa.ma".
  19
  20 (* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
  21
  22 (* Advanced properties ******************************************************)
  23
  24 (* Basic_2A1: was: aaa_lref *)
  25 lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
  26 #I #G #K #V #B #i elim i -i
  27 [ #L #H lapply (drops_fwd_isid … H ?) -H //
  28   #H destruct /2 width=1 by aaa_zero/
  29 | #i #IH #L <uni_succ #H #HB lapply (drops_inv_bind2_isuni_next … H) -H // *
  30   #Z #Y #HY #H destruct /3 width=1 by aaa_lref/
  31 ]
  32 qed.
  33
  34 (* Advanced inversion lemmas ************************************************)
  35
  36 (* Basic_2A1: was: aaa_inv_lref *)
  37 lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
  38                           ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
  39 #G #A #i elim i -i
  40 [ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
  41 | #i #IH #L #H elim (aaa_inv_lref … H) -H
  42   #I #K #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
  43 ]
  44 qed-.
  45
  46 (* Properties with generic slicing for local environments *******************)
  47
  48 (* Basic_2A1: includes: aaa_lift *)
  49 (* Note: it should use drops_split_trans_pair2 *)
  50 lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 →
  51                  ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
  52 @(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
  53 [ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
  54   lapply (aaa_inv_sort … H) -H #H destruct
  55   >(lifts_inv_sort1 … HX) -HX //
  56 | #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
  57   elim (aaa_inv_lref_drops … H) -H #J #K1 #V1 #HLK1 #HA
  58   elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
  59   lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
  60   elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK2 #HY
  61   lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
  62   elim (drops_inv_skip2 … HY) -HY #Z #K2 #HK21 #HZ #H destruct
  63   elim (liftsb_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
  64   /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
  65 | #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
  66   elim (aaa_inv_gref … H)
  67 | #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
  68   [ elim (aaa_inv_abbr … H) -H #B #HB #HA
  69     elim (lifts_inv_bind1 …  HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  70     /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
  71   | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
  72     elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  73     /5 width=8 by aaa_abst, drops_skip, ext2_pair/
  74   ]
  75 | * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
  76   [ elim (aaa_inv_appl … H) -H #B #HB #HA
  77     elim (lifts_inv_flat1 …  HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  78     /3 width=10 by aaa_appl/
  79   | elim (aaa_inv_cast … H) -H #H1A #H2A
  80     elim (lifts_inv_flat1 …  HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  81     /3 width=8 by aaa_cast/
  82   ]
  83 ]
  84 qed-.
  85
  86 (* Inversion lemmas with generic slicing for local environments *************)
  87
  88 (* Basic_2A1: includes: aaa_inv_lift *)
  89 lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b, f] L2 ≘ L1 →
  90                      ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
  91 @(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
  92 [ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
  93   lapply (aaa_inv_sort … H) -H #H destruct
  94   >(lifts_inv_sort2 … HX) -HX //
  95 | #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
  96   elim (aaa_inv_lref_drops … H) -H #J #K2 #V2 #HLK2 #HA
  97   elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
  98   lapply (drops_split_div … HL21 (𝐔❴i1❵) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
  99   lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by after_uni_dx/ ] #HY
 100   lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
 101   elim (drops_inv_skip1 … HY) -HY #Z #K1 #HK21 #HZ #H destruct
 102   elim (liftsb_inv_pair_dx … HZ) -HZ #V1 #HV12 #H destruct
 103   /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
 104 | #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
 105   elim (aaa_inv_gref … H)
 106 | #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
 107   [ elim (aaa_inv_abbr … H) -H #B #HB #HA
 108     elim (lifts_inv_bind2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 109     /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
 110   | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
 111     elim (lifts_inv_bind2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 112     /5 width=8 by aaa_abst, drops_skip, ext2_pair/
 113   ]
 114 | * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
 115   [ elim (aaa_inv_appl … H) -H #B #HB #HA
 116     elim (lifts_inv_flat2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 117     /3 width=10 by aaa_appl/
 118   | elim (aaa_inv_cast … H) -H #H1A #H2A
 119     elim (lifts_inv_flat2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 120     /3 width=8 by aaa_cast/
 121   ]
 122 ]
 123 qed-.