(* STRATIFIED HIGHER NATIVE VALIDITY FOR TERMS ******************************)
-inductive shnv (h) (g) (l1) (G) (L): predicate term ≝
-| shnv_cast: ∀U,T. ⦃G, L⦄ ⊢ U ¡[h, g] → ⦃G, L⦄ ⊢ T ¡[h, g] →
- (∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, l2, l2+1] T) →
- shnv h g l1 G L (ⓝU.T)
+inductive shnv (h) (o) (d1) (G) (L): predicate term ≝
+| shnv_cast: ∀U,T. ⦃G, L⦄ ⊢ U ¡[h, o] → ⦃G, L⦄ ⊢ T ¡[h, o] →
+ (∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ U •*⬌*[h, o, d2, d2+1] T) →
+ shnv h o d1 G L (ⓝU.T)
.
interpretation "stratified higher native validity (term)"
- 'NativeValid h g l G L T = (shnv h g l G L T).
+ 'NativeValid h o d G L T = (shnv h o d G L T).
(* Basic inversion lemmas ***************************************************)
-fact shnv_inv_cast_aux: ∀h,g,G,L,X,l1. ⦃G, L⦄ ⊢ X ¡[h, g, l1] → ∀U,T. X = ⓝU.T →
- ∧∧ ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g]
- & (∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, l2, l2+1] T).
-#h #g #G #L #X #l1 * -X
+fact shnv_inv_cast_aux: ∀h,o,G,L,X,d1. ⦃G, L⦄ ⊢ X ¡[h, o, d1] → ∀U,T. X = ⓝU.T →
+ ∧∧ ⦃G, L⦄ ⊢ U ¡[h, o] & ⦃G, L⦄ ⊢ T ¡[h, o]
+ & (∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ U •*⬌*[h, o, d2, d2+1] T).
+#h #o #G #L #X #d1 * -X
#U #T #HU #HT #HUT #U1 #T1 #H destruct /3 width=1 by and3_intro/
qed-.
-lemma shnv_inv_cast: ∀h,g,G,L,U,T,l1. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g, l1] →
- ∧∧ ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g]
- & (∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, l2, l2+1] T).
+lemma shnv_inv_cast: ∀h,o,G,L,U,T,d1. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, o, d1] →
+ ∧∧ ⦃G, L⦄ ⊢ U ¡[h, o] & ⦃G, L⦄ ⊢ T ¡[h, o]
+ & (∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ U •*⬌*[h, o, d2, d2+1] T).
/2 width=3 by shnv_inv_cast_aux/ qed-.
-lemma shnv_inv_snv: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ¡[h, g, l] → ⦃G, L⦄ ⊢ T ¡[h, g].
-#h #g #G #L #T #l * -T
-#U #T #HU #HT #HUT elim (HUT 0) -HUT
-/3 width=3 by snv_cast, scpds_fwd_cprs/
+lemma shnv_inv_snv: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ¡[h, o, d] → ⦃G, L⦄ ⊢ T ¡[h, o].
+#h #o #G #L #T #d * -T
+#U #T #HU #HT #HUT elim (HUT 0) -HUT /2 width=3 by snv_cast/
qed-.
+
+(* Basic properties *********************************************************)
+
+lemma snv_shnv_cast: ∀h,o,G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, o] → ⦃G, L⦄ ⊢ ⓝU.T ¡[h, o, 0].
+#h #o #G #L #U #T #H elim (snv_inv_cast … H) -H
+#U0 #HU #HT #HU0 #HTU0 @shnv_cast // -HU -HT
+#d #H <(le_n_O_to_eq … H) -d /2 width=3 by scpds_div/
+qed.