matita/matita/contribs/lambdadelta/static_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 "static_2/relocation/drops_drops.ma".
  16 include "static_2/s_computation/fqup_weight.ma".
  17 include "static_2/s_computation/fqup_drops.ma".
  18 include "static_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 <pr_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 lemma aaa_pair_inv_lref (G) (L) (i):
  47       ∀A. ❨G,L❩ ⊢ #i ⁝ A → ∀I,K,V. ⇩[i] L ≘ K.ⓑ[I]V → ❨G,K❩ ⊢ V ⁝ A.
  48 #G #L #i #A #H #I #K #V #HLK
  49 elim (aaa_inv_lref_drops … H) -H #J #Y #X #HLY #HX
  50 lapply (drops_mono … HLY … HLK) -L -i #H destruct //
  51 qed-.
  52
  53 (* Properties with generic slicing for local environments *******************)
  54
  55 (* Basic_2A1: includes: aaa_lift *)
  56 (* Note: it should use drops_split_trans_pair2 *)
  57 lemma aaa_lifts: ∀G,L1,T1,A. ❨G,L1❩ ⊢ T1 ⁝ A → ∀b,f,L2. ⇩*[b,f] L2 ≘ L1 →
  58                  ∀T2. ⇧*[f] T1 ≘ T2 → ❨G,L2❩ ⊢ T2 ⁝ A.
  59 @(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
  60 [ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
  61   lapply (aaa_inv_sort … H) -H #H destruct
  62   >(lifts_inv_sort1 … HX) -HX //
  63 | #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
  64   elim (aaa_inv_lref_drops … H) -H #J #K1 #V1 #HLK1 #HA
  65   elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
  66   lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
  67   elim (drops_split_trans … H) -H [ |*: /2 width=6 by pr_after_nat_uni/ ] #Y #HLK2 #HY
  68   lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
  69   elim (drops_inv_skip2 … HY) -HY #Z #K2 #HK21 #HZ #H destruct
  70   elim (liftsb_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
  71   /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
  72 | #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
  73   elim (aaa_inv_gref … H)
  74 | #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
  75   [ elim (aaa_inv_abbr … H) -H #B #HB #HA
  76     elim (lifts_inv_bind1 …  HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  77     /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
  78   | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
  79     elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  80     /5 width=8 by aaa_abst, drops_skip, ext2_pair/
  81   ]
  82 | * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
  83   [ elim (aaa_inv_appl … H) -H #B #HB #HA
  84     elim (lifts_inv_flat1 …  HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  85     /3 width=10 by aaa_appl/
  86   | elim (aaa_inv_cast … H) -H #H1A #H2A
  87     elim (lifts_inv_flat1 …  HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
  88     /3 width=8 by aaa_cast/
  89   ]
  90 ]
  91 qed-.
  92
  93 (* Inversion lemmas with generic slicing for local environments *************)
  94
  95 (* Basic_2A1: includes: aaa_inv_lift *)
  96 lemma aaa_inv_lifts: ∀G,L2,T2,A. ❨G,L2❩ ⊢ T2 ⁝ A → ∀b,f,L1. ⇩*[b,f] L2 ≘ L1 →
  97                      ∀T1. ⇧*[f] T1 ≘ T2 → ❨G,L1❩ ⊢ T1 ⁝ A.
  98 @(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
  99 [ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
 100   lapply (aaa_inv_sort … H) -H #H destruct
 101   >(lifts_inv_sort2 … HX) -HX //
 102 | #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
 103   elim (aaa_inv_lref_drops … H) -H #J #K2 #V2 #HLK2 #HA
 104   elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
 105   lapply (drops_split_div … HL21 (𝐔❨i1❩) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
 106   lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by pr_after_nat_uni/ ] #HY
 107   lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
 108   elim (drops_inv_skip1 … HY) -HY #Z #K1 #HK21 #HZ #H destruct
 109   elim (liftsb_inv_pair_dx … HZ) -HZ #V1 #HV12 #H destruct
 110   /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
 111 | #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
 112   elim (aaa_inv_gref … H)
 113 | #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
 114   [ elim (aaa_inv_abbr … H) -H #B #HB #HA
 115     elim (lifts_inv_bind2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 116     /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
 117   | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
 118     elim (lifts_inv_bind2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 119     /5 width=8 by aaa_abst, drops_skip, ext2_pair/
 120   ]
 121 | * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
 122   [ elim (aaa_inv_appl … H) -H #B #HB #HA
 123     elim (lifts_inv_flat2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 124     /3 width=10 by aaa_appl/
 125   | elim (aaa_inv_cast … H) -H #H1A #H2A
 126     elim (lifts_inv_flat2 …  HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
 127     /3 width=8 by aaa_cast/
 128   ]
 129 ]
 130 qed-.