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

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  14
  15 include "basic_2A/notation/relations/suptermoptalt_6.ma".
  16 include "basic_2A/substitution/fquq.ma".
  17
  18 (* OPTIONAL SUPCLOSURE ******************************************************)
  19
  20 (* alternative definition of fquq *)
  21 definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
  22
  23 interpretation
  24    "optional structural successor (closure) alternative"
  25    'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
  26
  27 (* Basic properties *********************************************************)
  28
  29 lemma fquqa_refl: tri_reflexive … fquqa.
  30 // qed.
  31
  32 lemma fquqa_drop: ∀G,L,K,T,U,m.
  33                   ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
  34 #G #L #K #T #U #m #HLK #HTU elim (eq_or_gt m)
  35 /3 width=5 by fqu_drop_lt, or_introl/ #H destruct
  36 >(drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
  37 qed.
  38
  39 (* Main properties **********************************************************)
  40
  41 theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
  42 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  43 /2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
  44 qed.
  45
  46 (* Main inversion properties ************************************************)
  47
  48 theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
  49 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
  50 * #H1 #H2 #H3 destruct //
  51 qed-.
  52
  53 (* Advanced inversion lemmas ************************************************)
  54
  55 lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
  56                     ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
  57 #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
  58 /2 width=1 by or_introl/
  59 qed-.