matita/matita/contribs/lambdadelta/ground/relocation/tr_pat.ma

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  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "ground/notation/functions/apply_2.ma".
  16 include "ground/relocation/tr_map.ma".
  17 (*
  18 include "ground/arith/pnat_le_plus.ma".
  19 include "ground/relocation/pstream_eq.ma".
  20 include "ground/relocation/rtmap_istot.ma".
  21 *)
  22 (* POSITIVE APPLICATION FOR TOTAL RELOCATION MAPS ***************************)
  23
  24 rec definition tr_pat (i: pnat) on i: tr_map → pnat.
  25 * #p #f cases i -i
  26 [ @p
  27 | #i lapply (tr_pat i f) -tr_pat -i -f
  28   #i @(i+p)
  29 ]
  30 defined.
  31
  32 interpretation
  33   "functional positive application (total relocation maps)"
  34   'apply f i = (tr_pat i f).
  35
  36 (* Properties on at (specific) ************************************************)
  37
  38 lemma at_O1: ∀i2,f. @❪𝟏, i2⨮f❫ ≘ i2.
  39 #i2 elim i2 -i2 /2 width=5 by gr_pat_refl, gr_pat_next/
  40 qed.
  41
  42 lemma at_S1: ∀p,f,i1,i2. @❪i1, f❫ ≘ i2 → @❪↑i1, p⨮f❫ ≘ i2+p.
  43 #p elim p -p /3 width=7 by gr_pat_push, gr_pat_next/
  44 qed.
  45
  46 lemma at_total: ∀i1,f. @❪i1, f❫ ≘ f@❨i1❩.
  47 #i1 elim i1 -i1
  48 [ * // | #i #IH * /3 width=1 by at_S1/ ]
  49 qed.
  50
  51 lemma at_istot: ∀f. 𝐓❪f❫.
  52 /2 width=2 by ex_intro/ qed.
  53
  54 lemma at_plus2: ∀f,i1,i,p,q. @❪i1, p⨮f❫ ≘ i → @❪i1, (p+q)⨮f❫ ≘ i+q.
  55 #f #i1 #i #p #q #H elim q -q
  56 /2 width=5 by gr_pat_next/
  57 qed.
  58
  59 (* Inversion lemmas on at (specific) ******************************************)
  60
  61 lemma at_inv_O1: ∀f,p,i2. @❪𝟏, p⨮f❫ ≘ i2 → p = i2.
  62 #f #p elim p -p /2 width=6 by gr_pat_inv_unit_push/
  63 #p #IH #i2 #H elim (gr_pat_inv_next … H) -H [|*: // ]
  64 #j2 #Hj * -i2 /3 width=1 by eq_f/
  65 qed-.
  66
  67 lemma at_inv_S1: ∀f,p,j1,i2. @❪↑j1, p⨮f❫ ≘ i2 →
  68                  ∃∃j2. @❪j1, f❫ ≘ j2 & j2+p = i2.
  69 #f #p elim p -p /2 width=5 by gr_pat_inv_succ_push/
  70 #p #IH #j1 #i2 #H elim (gr_pat_inv_next … H) -H [|*: // ]
  71 #j2 #Hj * -i2 elim (IH … Hj) -IH -Hj
  72 #i2 #Hi * -j2 /2 width=3 by ex2_intro/
  73 qed-.
  74
  75 lemma at_inv_total: ∀f,i1,i2. @❪i1, f❫ ≘ i2 → f@❨i1❩ = i2.
  76 /2 width=6 by fr2_nat_mono/ qed-.
  77
  78 (* Forward lemmas on at (specific) *******************************************)
  79
  80 lemma at_increasing_plus: ∀f,p,i1,i2. @❪i1, p⨮f❫ ≘ i2 → i1 + p ≤ ↑i2.
  81 #f #p *
  82 [ #i2 #H <(at_inv_O1 … H) -i2 //
  83 | #i1 #i2 #H elim (at_inv_S1 … H) -H
  84   #j1 #Ht * -i2 <pplus_succ_sn
  85   /4 width=2 by gr_pat_increasing, ple_plus_bi_dx, ple_succ_bi/
  86 ]
  87 qed-.
  88
  89 lemma at_fwd_id: ∀f,p,i. @❪i, p⨮f❫ ≘ i → 𝟏 = p.
  90 #f #p #i #H elim (gr_pat_des_id … H) -H
  91 #g #H elim (push_inv_seq_dx … H) -H //
  92 qed-.
  93
  94 (* Basic properties *********************************************************)
  95
  96 lemma tr_pat_O1: ∀p,f. (p⨮f)@❨𝟏❩ = p.
  97 // qed.
  98
  99 lemma tr_pat_S1: ∀p,f,i. (p⨮f)@❨↑i❩ = f@❨i❩+p.
 100 // qed.
 101
 102 lemma tr_pat_eq_repl (i): gr_eq_repl … (λf1,f2. f1@❨i❩ = f2@❨i❩).
 103 #i elim i -i [2: #i #IH ] * #p1 #f1 * #p2 #f2 #H
 104 elim (eq_inv_seq_aux … H) -H #Hp #Hf //
 105 >tr_pat_S1 >tr_pat_S1 /3 width=1 by eq_f2/
 106 qed.
 107
 108 lemma tr_pat_S2: ∀f,i. (↑f)@❨i❩ = ↑(f@❨i❩).
 109 * #p #f * //
 110 qed.
 111
 112 (* Main inversion lemmas ****************************************************)
 113
 114 theorem tr_pat_inj: ∀f,i1,i2,j. f@❨i1❩ = j → f@❨i2❩ = j → i1 = i2.
 115 /2 width=4 by gr_pat_inj/ qed-.
 116
 117 corec theorem nstream_eq_inv_ext: ∀f1,f2. (∀i. f1@❨i❩ = f2@❨i❩) → f1 ≗ f2.
 118 * #p1 #f1 * #p2 #f2 #Hf @stream_eq_cons
 119 [ @(Hf (𝟏))
 120 | @nstream_eq_inv_ext -nstream_eq_inv_ext #i
 121   ltr_pat (Hf (𝟏)) >tr_pat_O1 >tr_pat_O1 #H destruct
 122   ltr_pat (Hf (↑i)) >tr_pat_S1 >tr_pat_S1 #H
 123   /3 width=2 by eq_inv_pplus_bi_dx, eq_inv_psucc_bi/
 124 ]
 125 qed-.