matita/matita/contribs/lambdadelta/basic_2/reduction/fpbc.ma

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  14
  15 include "basic_2/notation/relations/btpredproper_8.ma".
  16 include "basic_2/reduction/fpb.ma".
  17
  18 (* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
  19
  20 inductive fpbc (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
  21 | fpbc_fsup  : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpbc h g G1 L1 T1 G2 L2 T2
  22 | fpbc_cpx   : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpbc h g G1 L1 T1 G1 L1 T2
  23 .
  24
  25 interpretation
  26    "'big tree' proper parallel reduction (closure)"
  27    'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma fpbc_fpb: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
  32                ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
  33 #h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
  34 /2 width=1 by fpb_fsup, fpb_cpx/
  35 qed.
  36
  37 lemma cpr_fpbc: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
  38                ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
  39 /3 width=1 by fpbc_cpx, cpr_cpx/ qed.
  40
  41 (* Inversion lemmas on "big tree" parallel reduction for closures ***********)
  42
  43 lemma fpb_inv_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
  44                    ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ ∨
  45                    ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡[h, g] L2 & T1 = T2.
  46 #h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
  47 /3 width=1 by and3_intro, or_introl, or_intror, fpbc_fsup/
  48 #T2 #HT12 elim (term_eq_dec T1 T2) #H destruct
  49 /4 width=1 by and3_intro, or_introl, or_intror, fpbc_cpx/
  50 qed-.