matita/matita/contribs/lambda-delta/Basic-2/unfold/tpss_tpss.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   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/substitution/tps_tps.ma".
  16 include "Basic-2/unfold/tpss_lift.ma".
  17
  18 (* PARTIAL UNFOLD ON TERMS **************************************************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 →
  23                      ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 →
  24                      ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T2 [d1, e1] ≫* T.
  25 /3/ qed.
  26
  27 lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 →
  28                       ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫ T2 →
  29                       (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
  30                       ∃∃T. L2 ⊢ T1 [d2, e2] ≫ T & L1 ⊢ T2 [d1, e1] ≫* T.
  31 /3/ qed.
  32
  33 lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
  34                         ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
  35                         ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫* T2.
  36 /3/ qed.
  37
  38 lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 →
  39                         ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 →
  40                         ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫ T2.
  41 /3/ qed.
  42
  43 lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 →
  44                      ∀i. d ≤ i → i ≤ d + e →
  45                      ∃∃T. L ⊢ T1 [d, i - d] ≫* T & L ⊢ T [i, d + e - i] ≫* T2.
  46 #L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -H T2
  47 [ /2/
  48 | #T #T2 #_ #HT12 * #T3 #HT13 #HT3
  49   elim (tps_split_up … HT12 … Hdi Hide) -HT12 Hide #T0 #HT0 #HT02
  50   elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: <plus_minus_m_m_comm // ]
  51   /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *)
  52 ]
  53 qed.
  54
  55 lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 →
  56                          ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
  57                          d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
  58                          ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫* T2 & ↑[d, e] T2 ≡ U2.
  59 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
  60 elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
  61 lapply (tpss_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
  62 lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
  63 elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
  64 qed.
  65
  66 (* Main properties **********************************************************)
  67
  68 theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 →
  69                       ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 →
  70                       ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T2 [d1, e1] ≫* T.
  71 /3/ qed.
  72
  73 theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 →
  74                        ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫* T2 →
  75                        (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
  76                        ∃∃T. L2 ⊢ T1 [d2, e2] ≫* T & L1 ⊢ T2 [d1, e1] ≫* T.
  77 /3/ qed.
  78
  79 theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
  80                        L ⊢ T1 [d, e] ≫* T → L ⊢ T [d, e] ≫* T2 →
  81                        L ⊢ T1 [d, e] ≫* T2.
  82 /2/ qed.
  83
  84 theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
  85                          ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 →
  86                          ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫* T2.
  87 /3/ qed.