matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx.ma

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  14
  15 include "basic_2A/notation/relations/predsn_5.ma".
  16 include "basic_2A/reduction/lpr.ma".
  17 include "basic_2A/reduction/cpx.ma".
  18
  19 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
  20
  21 definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
  22                 λh,g,G. lpx_sn (cpx h g G).
  23
  24 interpretation "extended parallel reduction (local environment, sn variant)"
  25    'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
  26
  27 (* Basic inversion lemmas ***************************************************)
  28
  29 lemma lpx_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
  30 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
  31
  32 lemma lpx_inv_pair1: ∀h,g,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
  33                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
  34                               L2 = K2. ⓑ{I} V2.
  35 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
  36
  37 lemma lpx_inv_atom2: ∀h,g,G,L1.  ⦃G, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
  38 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
  39
  40 lemma lpx_inv_pair2: ∀h,g,I,G,L1,K2,V2.  ⦃G, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
  41                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
  42                              L1 = K1. ⓑ{I} V1.
  43 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
  44
  45 lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2.  ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
  46                     ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
  47 /2 width=1 by lpx_sn_inv_pair/ qed-.
  48
  49 (* Basic properties *********************************************************)
  50
  51 lemma lpx_refl: ∀h,g,G,L.  ⦃G, L⦄ ⊢ ➡[h, g] L.
  52 /2 width=1 by lpx_sn_refl/ qed.
  53
  54 lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
  55                 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
  56 /2 width=1 by lpx_sn_pair/ qed.
  57
  58 lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
  59 #h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
  60 qed.
  61
  62 (* Basic forward lemmas *****************************************************)
  63
  64 lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
  65 /2 width=2 by lpx_sn_fwd_length/ qed-.