matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_lift.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/unfold/tpss_lift.ma".
  16 include "Basic_2/reducibility/tpr_lift.ma".
  17 include "Basic_2/reducibility/cpr.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
  24                   ↓[0, i] L ≡ K. 𝕓{Abbr} V1 → K ⊢ V1 [0, |L| - i - 1] ≫* W1 →
  25                   ↑[0, i + 1] W1 ≡ W2 → L ⊢ #i ⇒ W2.
  26 #L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
  27 lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
  28 @ex2_1_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
  29 qed.
  30
  31 (* Advanced inversion lemmas ************************************************)
  32
  33 (* Basic_1: was: pr2_gen_lref *)
  34 lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ⇒ T2 →
  35                      T2 = #i ∨
  36                      ∃∃K,V1,T1. ↓[0, i] L ≡ K. 𝕓{Abbr} V1 &
  37                                 K ⊢ V1 [0, |L| - i - 1] ≫* T1 &
  38                                 ↑[0, i + 1] T1 ≡ T2 &
  39                                 i < |L|.
  40 #L #T2 #i * #X #H
  41 >(tpr_inv_atom1 … H) -H #H
  42 elim (tpss_inv_lref1 … H) -H /2 width=1/
  43 * /3 width=6/
  44 qed-.
  45
  46 (* Basic_1: was: pr2_gen_abst *)
  47 lemma cpr_inv_abst1: ∀V1,T1,U2. 𝕔{Abst} V1. T1 ⇒ U2 →
  48                      ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Abst} V2. T2.
  49 /2 width=3/ qed-.
  50
  51 (* Relocation properties ****************************************************)
  52
  53 (* Basic_1: was: pr2_lift *)
  54 lemma cpr_lift: ∀L,K,d,e. ↓[d, e] L ≡ K →
  55                 ∀T1,U1. ↑[d, e] T1 ≡ U1 → ∀T2,U2. ↑[d, e] T2 ≡ U2 →
  56                 K ⊢ T1 ⇒ T2 → L ⊢ U1 ⇒ U2.
  57 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
  58 elim (lift_total T d e) #U #HTU
  59 lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
  60 elim (lt_or_ge (|K|) d) #HKd
  61 [ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=1/ | /3 width=4/ ]
  62 | lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
  63 ]
  64 qed.
  65
  66 (* Basic_1: was: pr2_gen_lift *)
  67 lemma cpr_inv_lift: ∀L,K,d,e. ↓[d, e] L ≡ K →
  68                     ∀T1,U1. ↑[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ⇒ U2 →
  69                     ∃∃T2. ↑[d, e] T2 ≡ U2 & K ⊢ T1 ⇒ T2.
  70 #L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
  71 elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
  72 elim (lt_or_ge (|L|) d) #HLd
  73 [ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=1/ ]
  74 | elim (lt_or_ge (|L|) (d + e)) #HLde
  75   [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // [ /5 width=4/ | /2 width=1/ ]
  76   | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK // /5 width=4/
  77   ]
  78 ]
  79 qed.