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

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  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/notation/functions/apply_2.ma".
  16 include "ground/arith/pnat_le_plus.ma".
  17 include "ground/relocation/pstream_eq.ma".
  18 include "ground/relocation/rtmap_istot.ma".
  19
  20 (* RELOCATION N-STREAM ******************************************************)
  21
  22 rec definition apply (i: pnat) on i: rtmap → pnat.
  23 * #p #f cases i -i
  24 [ @p
  25 | #i lapply (apply i f) -apply -i -f
  26   #i @(i+p)
  27 ]
  28 defined.
  29
  30 interpretation "functional application (nstream)"
  31    'Apply f i = (apply i f).
  32
  33 (* Specific properties on at ************************************************)
  34
  35 lemma at_O1: ∀i2,f. @❪𝟏, i2⨮f❫ ≘ i2.
  36 #i2 elim i2 -i2 /2 width=5 by at_refl, at_next/
  37 qed.
  38
  39 lemma at_S1: ∀p,f,i1,i2. @❪i1, f❫ ≘ i2 → @❪↑i1, p⨮f❫ ≘ i2+p.
  40 #p elim p -p /3 width=7 by at_push, at_next/
  41 qed.
  42
  43 lemma at_total: ∀i1,f. @❪i1, f❫ ≘ f@❨i1❩.
  44 #i1 elim i1 -i1
  45 [ * // | #i #IH * /3 width=1 by at_S1/ ]
  46 qed.
  47
  48 lemma at_istot: ∀f. 𝐓❪f❫.
  49 /2 width=2 by ex_intro/ qed.
  50
  51 lemma at_plus2: ∀f,i1,i,p,q. @❪i1, p⨮f❫ ≘ i → @❪i1, (p+q)⨮f❫ ≘ i+q.
  52 #f #i1 #i #p #q #H elim q -q
  53 /2 width=5 by at_next/
  54 qed.
  55
  56 (* Specific inversion lemmas on at ******************************************)
  57
  58 lemma at_inv_O1: ∀f,p,i2. @❪𝟏, p⨮f❫ ≘ i2 → p = i2.
  59 #f #p elim p -p /2 width=6 by at_inv_ppx/
  60 #p #IH #i2 #H elim (at_inv_xnx … H) -H [|*: // ]
  61 #j2 #Hj * -i2 /3 width=1 by eq_f/
  62 qed-.
  63
  64 lemma at_inv_S1: ∀f,p,j1,i2. @❪↑j1, p⨮f❫ ≘ i2 →
  65                  ∃∃j2. @❪j1, f❫ ≘ j2 & j2+p = i2.
  66 #f #p elim p -p /2 width=5 by at_inv_npx/
  67 #p #IH #j1 #i2 #H elim (at_inv_xnx … H) -H [|*: // ]
  68 #j2 #Hj * -i2 elim (IH … Hj) -IH -Hj
  69 #i2 #Hi * -j2 /2 width=3 by ex2_intro/
  70 qed-.
  71
  72 lemma at_inv_total: ∀f,i1,i2. @❪i1, f❫ ≘ i2 → f@❨i1❩ = i2.
  73 /2 width=6 by at_mono/ qed-.
  74
  75 (* Spercific forward lemmas on at *******************************************)
  76
  77 lemma at_increasing_plus: ∀f,p,i1,i2. @❪i1, p⨮f❫ ≘ i2 → i1 + p ≤ ↑i2.
  78 #f #p *
  79 [ #i2 #H <(at_inv_O1 … H) -i2 //
  80 | #i1 #i2 #H elim (at_inv_S1 … H) -H
  81   #j1 #Ht * -i2 <pplus_succ_sn
  82   /4 width=2 by at_increasing, ple_plus_bi_dx, ple_succ_bi/
  83 ]
  84 qed-.
  85
  86 lemma at_fwd_id: ∀f,p,i. @❪i, p⨮f❫ ≘ i → 𝟏 = p.
  87 #f #p #i #H elim (at_fwd_id_ex … H) -H
  88 #g #H elim (push_inv_seq_dx … H) -H //
  89 qed-.
  90
  91 (* Basic properties *********************************************************)
  92
  93 lemma apply_O1: ∀p,f. (p⨮f)@❨𝟏❩ = p.
  94 // qed.
  95
  96 lemma apply_S1: ∀p,f,i. (p⨮f)@❨↑i❩ = f@❨i❩+p.
  97 // qed.
  98
  99 lemma apply_eq_repl (i): eq_repl … (λf1,f2. f1@❨i❩ = f2@❨i❩).
 100 #i elim i -i [2: #i #IH ] * #p1 #f1 * #p2 #f2 #H
 101 elim (eq_inv_seq_aux … H) -H #Hp #Hf //
 102 >apply_S1 >apply_S1 /3 width=1 by eq_f2/
 103 qed.
 104
 105 lemma apply_S2: ∀f,i. (↑f)@❨i❩ = ↑(f@❨i❩).
 106 * #p #f * //
 107 qed.
 108
 109 (* Main inversion lemmas ****************************************************)
 110
 111 theorem apply_inj: ∀f,i1,i2,j. f@❨i1❩ = j → f@❨i2❩ = j → i1 = i2.
 112 /2 width=4 by at_inj/ qed-.
 113
 114 corec theorem nstream_eq_inv_ext: ∀f1,f2. (∀i. f1@❨i❩ = f2@❨i❩) → f1 ≗ f2.
 115 * #p1 #f1 * #p2 #f2 #Hf @stream_eq_cons
 116 [ @(Hf (𝟏))
 117 | @nstream_eq_inv_ext -nstream_eq_inv_ext #i
 118   lapply (Hf (𝟏)) >apply_O1 >apply_O1 #H destruct
 119   lapply (Hf (↑i)) >apply_S1 >apply_S1 #H
 120   /3 width=2 by eq_inv_pplus_bi_dx, eq_inv_psucc_bi/
 121 ]
 122 qed-.