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

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  14
  15 include "basic_2A/notation/relations/predsn_3.ma".
  16 include "basic_2A/substitution/lpx_sn.ma".
  17 include "basic_2A/reduction/cpr.ma".
  18
  19 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
  20
  21 definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (cpr G).
  22
  23 interpretation "parallel reduction (local environment, sn variant)"
  24    'PRedSn G L1 L2 = (lpr G L1 L2).
  25
  26 (* Basic inversion lemmas ***************************************************)
  27
  28 (* Basic_1: includes: wcpr0_gen_sort *)
  29 lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
  30 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
  31
  32 (* Basic_1: includes: wcpr0_gen_head *)
  33 lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
  34                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
  35 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
  36
  37 lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
  38 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
  39
  40 lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
  41                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
  42 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
  43
  44 (* Basic properties *********************************************************)
  45
  46 (* Note: lemma 250 *)
  47 lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
  48 /2 width=1 by lpx_sn_refl/ qed.
  49
  50 lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
  51                 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
  52 /2 width=1 by lpx_sn_pair/ qed.
  53
  54 (* Basic forward lemmas *****************************************************)
  55
  56 lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
  57 /2 width=2 by lpx_sn_fwd_length/ qed-.
  58
  59 (* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
  60                                 pr0_subst1_back
  61 *)