matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr_lift.ma

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  14
  15 include "Basic_2/unfold/tpss_lift.ma".
  16 include "Basic_2/reduction/tpr_lift.ma".
  17 include "Basic_2/reduction/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 @ex2_1_intro [2: // | skip | @tpss_subst /2 width=6/ ] (**) (* /4 width=6/ is too slow *)
  28 qed.
  29
  30 (* Advanced inversion lemmas ************************************************)
  31
  32 (* Basic_1: was: pr2_gen_lref *)
  33 lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ⇒ T2 →
  34                      T2 = #i ∨
  35                      ∃∃K,V1,T1. ↓[0, i] L ≡ K. 𝕓{Abbr} V1 &
  36                                 K ⊢ V1 [0, |L| - i - 1] ≫* T1 &
  37                                 ↑[0, i + 1] T1 ≡ T2 &
  38                                 i < |L|.
  39 #L #T2 #i * #X #H
  40 >(tpr_inv_atom1 … H) -H #H
  41 elim (tpss_inv_lref1 … H) -H /2/
  42 * /3 width=6/
  43 qed.
  44
  45 (* Basic_1: was: pr2_gen_abst *)
  46 lemma cpr_inv_abst1: ∀V1,T1,U2. 𝕔{Abst} V1. T1 ⇒ U2 →
  47                      ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Abst} V2. T2.
  48 /2/ qed.
  49
  50 (* Relocation properties ****************************************************)
  51
  52 (* Basic_1: was: pr2_lift *)
  53 lemma cpr_lift: ∀L,K,d,e. ↓[d, e] L ≡ K →
  54                 ∀T1,U1. ↑[d, e] T1 ≡ U1 → ∀T2,U2. ↑[d, e] T2 ≡ U2 →
  55                 K ⊢ T1 ⇒ T2 → L ⊢ U1 ⇒ U2.
  56 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
  57 elim (lift_total T d e) #U #HTU
  58 lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
  59 elim (lt_or_ge (|K|) d) #HKd
  60 [ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 T HLK [ /2/ | /3/ ]
  61 | lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 T HLK // /3/
  62 ]
  63 qed.
  64
  65 (* Basic_1: was: pr2_gen_lift *)
  66 lemma cpr_inv_lift: ∀L,K,d,e. ↓[d, e] L ≡ K →
  67                     ∀T1,U1. ↑[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ⇒ U2 →
  68                     ∃∃T2. ↑[d, e] T2 ≡ U2 & K ⊢ T1 ⇒ T2.
  69 #L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
  70 elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
  71 elim (lt_or_ge (|L|) d) #HLd
  72 [ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U HLK [ /5/ | /2/ ]
  73 | elim (lt_or_ge (|L|) (d + e)) #HLde
  74   [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U HLK // [ /5/ | /2/ ]
  75   | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U HLK // /5/
  76   ]
  77 ]
  78 qed.