matita/matita/contribs/lambdadelta/basic_2A/etc/llpx/llpx.etc

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  14
  15 include "basic_2/notation/relations/lazypredsn_7.ma".
  16 include "basic_2/relocation/llpx_sn.ma".
  17 include "basic_2/reduction/cpx.ma".
  18
  19 (* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
  20
  21 definition llpx: ∀h. sd h → genv → relation4 ynat term lenv lenv ≝
  22                  λh,g,G. llpx_sn (cpx h g  G).
  23
  24 interpretation "lazy extended parallel reduction (local environment, sn variant)"
  25    'LazyPRedSn G L1 L2 h g T d = (llpx h g G d T L1 L2).
  26
  27 (* Basic inversion lemmas ***************************************************)
  28
  29 lemma llpx_inv_flat: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
  30                      ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
  31 /2 width=2 by llpx_sn_inv_flat/ qed-.
  32
  33 (* Basic forward lemmas *****************************************************)
  34
  35 lemma llpx_fwd_length: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → |L1| = |L2|.
  36 /2 width=4 by llpx_sn_fwd_length/ qed-.
  37
  38 lemma llpx_fwd_flat_dx: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
  39                      ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
  40 /2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
  41
  42 lemma llpx_fwd_pair_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ②{I}V.T, d] L2 →
  43                         ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
  44 /2 width=3 by llpx_sn_fwd_pair_sn/ qed-.
  45
  46 (* Note: this might be removed *)
  47 lemma llpx_fwd_bind_sn: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, d] L2 →
  48                         ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
  49 /2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
  50
  51 (* Note: this might be removed *)
  52 lemma llpx_fwd_bind_dx: ∀h,g,a,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓑ{a,I}V.T, d] L2 →
  53                         ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, g, T, ⫯d] L2.ⓑ{I}V.
  54 /2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
  55
  56 (* Note: this might be removed *)
  57 lemma llpx_fwd_flat_sn: ∀h,g,I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, ⓕ{I}V.T, d] L2 →
  58                         ⦃G, L1⦄ ⊢ ➡[h, g, V, d] L2.
  59 /2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
  60
  61 (* Basic properties *********************************************************)
  62
  63 lemma llpx_lref: ∀h,g,I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
  64                  ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
  65                  ⦃G, K1⦄ ⊢ ➡[h, g, V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L1⦄ ⊢ ➡[h, g, #i, d] L2.
  66 /2 width=9 by llpx_sn_lref/ qed.
  67
  68 lemma llpx_refl: ∀h,g,G,T,d. reflexive … (llpx h g G d T).
  69 /2 width=1 by llpx_sn_refl/ qed.