matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_vector.ma

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  14
  15 include "basic_2A/grammar/term_vector.ma".
  16 include "basic_2A/substitution/lift.ma".
  17
  18 (* BASIC TERM VECTOR RELOCATION *********************************************)
  19
  20 inductive liftv (l,m:nat) : relation (list term) ≝
  21 | liftv_nil : liftv l m (ⓔ) (ⓔ)
  22 | liftv_cons: ∀T1s,T2s,T1,T2.
  23               ⬆[l, m] T1 ≡ T2 → liftv l m T1s T2s →
  24               liftv l m (T1 ⨮ T1s) (T2 ⨮ T2s)
  25 .
  26
  27 interpretation "relocation (vector)" 'RLift l m T1s T2s = (liftv l m T1s T2s).
  28
  29 (* Basic inversion lemmas ***************************************************)
  30
  31 fact liftv_inv_nil1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s → T1s = ⓔ → T2s = ⓔ.
  32 #T1s #T2s #l #m * -T1s -T2s //
  33 #T1s #T2s #T1 #T2 #_ #_ #H destruct
  34 qed-.
  35
  36 lemma liftv_inv_nil1: ∀T2s,l,m. ⬆[l, m] ⓔ ≡ T2s → T2s = ⓔ.
  37 /2 width=5 by liftv_inv_nil1_aux/ qed-.
  38
  39 fact liftv_inv_cons1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s →
  40                           ∀U1,U1s. T1s = U1 ⨮ U1s →
  41                           ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
  42                                     T2s = U2 ⨮ U2s.
  43 #T1s #T2s #l #m * -T1s -T2s
  44 [ #U1 #U1s #H destruct
  45 | #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
  46 ]
  47 qed-.
  48
  49 lemma liftv_inv_cons1: ∀U1,U1s,T2s,l,m. ⬆[l, m] U1 ⨮ U1s ≡ T2s →
  50                        ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
  51                                  T2s = U2 ⨮ U2s.
  52 /2 width=3 by liftv_inv_cons1_aux/ qed-.
  53
  54 (* Basic properties *********************************************************)
  55
  56 lemma liftv_total: ∀l,m. ∀T1s:list term. ∃T2s. ⬆[l, m] T1s ≡ T2s.
  57 #l #m #T1s elim T1s -T1s
  58 [ /2 width=2 by liftv_nil, ex_intro/
  59 | #T1 #T1s * #T2s #HT12s
  60   elim (lift_total T1 l m) /3 width=2 by liftv_cons, ex_intro/
  61 ]
  62 qed-.