matita/matita/contribs/lambdadelta/basic_2A/etc/fpn/fpns.etc

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  14
  15 include "basic_2/notation/relations/btpredsnstar_8.ma".
  16 include "basic_2/reduction/fpn.ma".
  17
  18 (* COMPUTATION FOR "BIG TREE" NORMAL FORMS **********************************)
  19
  20 definition fpns: ∀h. sd h → tri_relation genv lenv term ≝
  21                  λh,g. tri_TC … (fpn h g).
  22
  23 interpretation
  24    "computation for 'big tree' normal forms (closure)"
  25    'BTPRedSnStar h g G1 L1 T1 G2 L2 T2 = (fpns h g G1 L1 T1 G2 L2 T2).
  26
  27 (* Basic eliminators ********************************************************)
  28
  29 lemma fpns_ind: ∀h,g,G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
  30                 (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
  31                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
  32 #h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
  33 lapply (tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
  34 qed-.
  35
  36 lemma fpns_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
  37                    (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
  38                    ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
  39 #h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
  40 @(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
  41 qed-.
  42
  43 (* Basic_properties *********************************************************)
  44
  45 lemma fpns_refl: ∀h,g. tri_reflexive … (fpns h g).
  46 /2 width=1 by tri_inj/ qed.
  47
  48 lemma fpn_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ →
  49                 ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
  50 /2 width=1 by tri_inj/ qed.
  51
  52 lemma fpns_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ →
  53                    ⦃G, L, T⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
  54 /2 width=5 by tri_step/ qed-.
  55
  56 lemma fpns_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡[h, g] ⦃G, L, T⦄ →
  57                    ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄.
  58 /2 width=5 by tri_TC_strap/ qed-.