matita/matita/contribs/lambdadelta/basic_2/dynamic/shnv.ma

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  14
  15 include "basic_2/notation/relations/nativevalid_6.ma".
  16 include "basic_2/equivalence/scpes.ma".
  17 include "basic_2/dynamic/snv.ma".
  18
  19 (* STRATIFIED HIGHER NATIVE VALIDITY FOR TERMS ******************************)
  20
  21 inductive shnv (h) (g) (l1) (G) (L): predicate term ≝
  22 | shnv_cast: ∀U,T. ⦃G, L⦄ ⊢ U ¡[h, g] → ⦃G, L⦄ ⊢ T ¡[h, g] →
  23              (∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, l2, l2+1] T) →
  24              shnv h g l1 G L (ⓝU.T)
  25 .
  26
  27 interpretation "stratified higher native validity (term)"
  28    'NativeValid h g l G L T = (shnv h g l G L T).
  29
  30 (* Basic inversion lemmas ***************************************************)
  31
  32 fact shnv_inv_cast_aux: ∀h,g,G,L,X,l1. ⦃G, L⦄ ⊢ X ¡[h, g, l1] → ∀U,T. X = ⓝU.T →
  33                         ∧∧ ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g]
  34                          & (∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, l2, l2+1] T).
  35 #h #g #G #L #X #l1 * -X
  36 #U #T #HU #HT #HUT #U1 #T1 #H destruct /3 width=1 by and3_intro/
  37 qed-.
  38
  39 lemma shnv_inv_cast: ∀h,g,G,L,U,T,l1. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g, l1] →
  40                      ∧∧ ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g]
  41                       & (∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, l2, l2+1] T).
  42 /2 width=3 by shnv_inv_cast_aux/ qed-.
  43
  44 lemma shnv_inv_snv: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ¡[h, g, l] → ⦃G, L⦄ ⊢ T ¡[h, g].
  45 #h #g #G #L #T #l * -T
  46 #U #T #HU #HT #HUT elim (HUT 0) -HUT
  47 /3 width=3 by snv_cast, scpds_fwd_cprs/
  48 qed-.