matita/matita/contribs/lambdadelta/basic_2A/etc/cpys0/lpys.etc

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  14
  15 include "basic_2/notation/relations/psubstsnstar_3.ma".
  16 include "basic_2/relocation/lpx_sn.ma".
  17 include "basic_2/substitution/cpys.ma".
  18
  19 (* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
  20
  21 definition lpys: relation3 genv lenv lenv ≝ λG. lpx_sn (cpys G).
  22
  23 interpretation
  24    "extended multiple substitution (local environment, sn variant)"
  25    'PSubstSnStar G L1 L2 = (lpys G L1 L2).
  26
  27 (* Basic inversion lemmas ***************************************************)
  28
  29 lemma lpys_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ▶* L2 → L2 = ⋆.
  30 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
  31
  32 lemma lpys_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* L2 →
  33                       ∃∃K2,V2. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
  34                                L2 = K2. ⓑ{I} V2.
  35 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
  36
  37 lemma lpys_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ▶* ⋆ → L1 = ⋆.
  38 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
  39
  40 lemma lpys_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ▶* K2.ⓑ{I}V2 →
  41                       ∃∃K1,V1. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
  42                                L1 = K1. ⓑ{I} V1.
  43 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
  44
  45 lemma lpys_inv_pair: ∀I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ▶* L2.ⓑ{I2}V2 →
  46                      ∧∧ ⦃G, L1⦄ ⊢ ▶* L2 & ⦃G, L1⦄ ⊢ V1 ▶* V2 & I1 = I2.
  47 /2 width=1 by lpx_sn_inv_pair/ qed-.
  48
  49 (* Basic properties *********************************************************)
  50
  51 lemma lpys_refl: ∀G,L. ⦃G, L⦄ ⊢ ▶* L.
  52 /2 width=1 by lpx_sn_refl/ qed.
  53
  54 lemma lpys_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ▶* K2 → ⦃G, K1⦄ ⊢ V1 ▶* V2 →
  55                  ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* K2.ⓑ{I}V2.
  56 /2 width=1 by lpx_sn_pair/ qed.
  57
  58 (* Basic forward lemmas *****************************************************)
  59
  60 lemma lpys_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ▶* L2 → |L1| = |L2|.
  61 /2 width=2 by lpx_sn_fwd_length/ qed-.