matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "ground_2/ynat/ynat_max.ma".
  16 include "basic_2/substitution/lift_lift.ma".
  17 include "basic_2/multiple/mr2_minus.ma".
  18 include "basic_2/multiple/lifts.ma".
  19
  20 (* GENERIC TERM RELOCATION **************************************************)
  21
  22 (* Properties concerning basic term relocation ******************************)
  23
  24 (* Basic_1: was: lift1_xhg (right to left) *)
  25 lemma lifts_lift_trans_le: ∀T1,T,cs. ⬆*[cs] T1 ≡ T → ∀T2. ⬆[0, 1] T ≡ T2 →
  26                            ∃∃T0. ⬆[0, 1] T1 ≡ T0 & ⬆*[cs + 1] T0 ≡ T2.
  27 #T1 #T #cs #H elim H -T1 -T -cs
  28 [ /2 width=3 by lifts_nil, ex2_intro/
  29 | #T1 #T3 #T #cs #l #m #HT13 #_ #IHT13 #T2 #HT2
  30   elim (IHT13 … HT2) -T #T #HT3 #HT2
  31   elim (lift_trans_le … HT13 … HT3) -T3 /3 width=5 by lifts_cons, ex2_intro/
  32 ]
  33 qed-.
  34
  35 (* Basic_1: was: lift1_free (right to left) *)
  36 lemma lifts_lift_trans: ∀cs,i,i0. @⦃i, cs⦄ ≡ i0 →
  37                         ∀cs0. cs + 1 ▭ i + 1 ≡ cs0 + 1 →
  38                         ∀T1,T0. ⬆*[cs0] T1 ≡ T0 →
  39                         ∀T2. ⬆[O, i0 + 1] T0 ≡ T2 →
  40                         ∃∃T. ⬆[0, i + 1] T1 ≡ T & ⬆*[cs] T ≡ T2.
  41 #cs elim cs -cs
  42 [ #i #x #H1 #cs0 #H2 #T1 #T0 #HT10 #T2
  43   <(at_inv_nil … H1) -x #HT02
  44   lapply (minuss_inv_nil1 … H2) -H2 #H
  45   >(pluss_inv_nil2 … H) in HT10; -cs0 #H
  46   >(lifts_inv_nil … H) -T1 /2 width=3 by lifts_nil, ex2_intro/
  47 | #l #m #cs #IHcs #i #i0 #H1 #cs0 #H2 #T1 #T0 #HT10 #T2 #HT02
  48   elim (at_inv_cons … H1) -H1 * #Hil #Hi0
  49   [ elim (minuss_inv_cons1_lt … H2) -H2 /2 width=1 by ylt_succ/ #cs1 #Hcs1
  50     <yplus_inj >yplus_SO2 >yplus_SO2 >yminus_succ #H
  51     elim (pluss_inv_cons2 … H) -H #cs2 #H1 #H2 destruct
  52     elim (lifts_inv_cons … HT10) -HT10 #T #HT1 #HT0
  53     elim (IHcs … Hi0 … Hcs1 … HT0 … HT02) -IHcs -Hi0 -Hcs1 -T0 #T0 #HT0 #HT02
  54     elim (lift_trans_le … HT1 … HT0) -T /2 width=1 by/ #T #HT1
  55     <yminus_plus2 <yplus_inj >yplus_SO2 >ymax_pre_sn
  56     /3 width=5 by lifts_cons, ex2_intro, ylt_fwd_le_succ1/
  57   | >commutative_plus in Hi0; #Hi0
  58     lapply (minuss_inv_cons1_ge … H2 ?) -H2 /2 width=1 by yle_succ/ <associative_plus #Hcs0
  59     elim (IHcs … Hi0 … Hcs0 … HT10 … HT02) -IHcs -Hi0 -Hcs0 -T0 #T0 #HT0 #HT02
  60     elim (lift_split … HT0 l (i+1)) -HT0 /3 width=5 by lifts_cons, yle_succ, yle_pred_sn, ex2_intro/
  61   ]
  62 ]
  63 qed-.