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

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  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_trans_down_strap1: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
  23                               ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
  24                               ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫* T2.
  25 /3/ qed.
  26
  27 lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 →
  28                      ∀i. d ≤ i → i ≤ d + e →
  29                      ∃∃T. L ⊢ T1 [d, i - d] ≫* T & L ⊢ T [i, d + e - i] ≫* T2.
  30 #L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -H T2
  31 [ /2/
  32 | #T #T2 #_ #HT12 * #T3 #HT13 #HT3
  33   elim (tps_split_up … HT12 … Hdi Hide) -HT12 Hide #T0 #HT0 #HT02
  34   elim (tpss_trans_down_strap1 … HT3 … HT0 ?) -T [2: <plus_minus_m_m_comm // ]
  35   /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *)
  36 ]
  37 qed.
  38
  39 lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 →
  40                          ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
  41                          d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
  42                          ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫* T2 & ↑[d, e] T2 ≡ U2.
  43 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
  44 elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
  45 lapply (tpss_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
  46 lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
  47 elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
  48 qed.
  49
  50 (* Main properties **********************************************************)
  51
  52 theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 →
  53                       ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 →
  54                       ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T2 [d1, e1] ≫* T.
  55 /3/ qed.
  56
  57 theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 →
  58                        ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫* T2 →
  59                        (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
  60                        ∃∃T. L2 ⊢ T1 [d2, e2] ≫* T & L1 ⊢ T2 [d1, e1] ≫* T.
  61 /3/ qed.
  62
  63 theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
  64                        L ⊢ T1 [d, e] ≫* T → L ⊢ T [d, e] ≫* T2 →
  65                        L ⊢ T1 [d, e] ≫* T2.
  66 /2/ qed.
  67
  68 theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
  69                          ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 →
  70                          ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫* T2.
  71 /3/ qed.