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

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  14
  15 include "basic_2/reduction/lpr.ma".
  16 include "basic_2/reduction/cpx.ma".
  17
  18 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
  19
  20 definition lpx: ∀h. sd h → relation lenv ≝ λh,g. lpx_sn (cpx h g).
  21
  22 interpretation "extended parallel reduction (local environment, sn variant)"
  23    'PRedSn h g L1 L2 = (lpx h g L1 L2).
  24
  25 (* Basic inversion lemmas ***************************************************)
  26
  27 lemma lpx_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡[g] L2 → L2 = ⋆.
  28 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
  29
  30 lemma lpx_inv_pair1: ∀h,g,I,K1,V1,L2. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡[g] L2 →
  31                      ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡[g] V2 &
  32                              L2 = K2. ⓑ{I} V2.
  33 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
  34
  35 lemma lpx_inv_atom2: ∀h,g,L1.  ⦃h, L1⦄ ⊢ ➡[g] ⋆ → L1 = ⋆.
  36 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
  37
  38 lemma lpx_inv_pair2: ∀h,g,I,L1,K2,V2.  ⦃h, L1⦄ ⊢ ➡[g] K2.ⓑ{I}V2 →
  39                      ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡[g] V2 &
  40                              L1 = K1. ⓑ{I} V1.
  41 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
  42
  43 (* Basic properties *********************************************************)
  44
  45 lemma lpx_refl: ∀h,g,L.  ⦃h, L⦄ ⊢ ➡[g] L.
  46 /2 width=1 by lpx_sn_refl/ qed.
  47
  48 lemma lpx_pair: ∀h,g,I,K1,K2,V1,V2. ⦃h, K1⦄ ⊢ ➡[g] K2 → ⦃h, K1⦄ ⊢ V1 ➡[g] V2 →
  49                 ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡[g] K2.ⓑ{I}V2.
  50 /2 width=1/ qed.
  51
  52 lemma lpx_append: ∀h,g,K1,K2. ⦃h, K1⦄ ⊢ ➡[g] K2 → ∀L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 →
  53                   ⦃h, L1 @@ K1⦄ ⊢ ➡[g] L2 @@ K2.
  54 /3 width=1 by lpx_sn_append, cpx_append/ qed.
  55 (*
  56 lamma lpr_lpx: ∀h,g,L1,L2. L1 ⊢ ➡ L2 → ⦃h, L1⦄ ⊢ ➡[g] L2.
  57 #h #g #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
  58 qed.
  59 *)
  60 (* Basic forward lemmas *****************************************************)
  61
  62 lemma lpx_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 → |L1| = |L2|.
  63 /2 width=2 by lpx_sn_fwd_length/ qed-.
  64
  65 (* Advanced forward lemmas **************************************************)
  66
  67 lemma lpx_fwd_append1: ∀h,g,K1,L1,L. ⦃h, K1 @@ L1⦄ ⊢ ➡[g] L →
  68                        ∃∃K2,L2. ⦃h, K1⦄ ⊢ ➡[g] K2 & L = K2 @@ L2.
  69 /2 width=2 by lpx_sn_fwd_append1/ qed-.
  70
  71 lemma lpx_fwd_append2: ∀h,g,L,K2,L2. ⦃h, L⦄ ⊢ ➡[g] K2 @@ L2 →
  72                        ∃∃K1,L1. ⦃h, K1⦄ ⊢ ➡[g] K2 & L = K1 @@ L1.
  73 /2 width=2 by lpx_sn_fwd_append2/ qed-.