(* *)
(**************************************************************************)
+include "basic_2/notation/relations/normal_4.ma".
include "basic_2/reduction/cnr.ma".
include "basic_2/reduction/cpx.ma".
]
qed-.
-axiom cnx_inv_zeta: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃+ⓓV.T⦄ → ⊥.
-(*
+lemma cnx_inv_zeta: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃+ⓓV.T⦄ → ⊥.
#h #g #L #V #T #H elim (is_lift_dec T 0 1)
[ * #U #HTU
lapply (H U ?) -H /2 width=3/ #H destruct
elim (lift_inv_pair_xy_y … HTU)
| #HT
- elim (cpss_delift (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12
+ elim (cpr_delift (⋆) V T (⋆.ⓓV) 0) // #T2 #T1 #HT2 #HT12
lapply (H (+ⓓV.T2) ?) -H /5 width=1/ -HT2 #H destruct /3 width=2/
]
qed-.
-*)
+
lemma cnx_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓐV.T⦄ →
∧∧ ⦃h, L⦄ ⊢ 𝐍[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ & 𝐒⦃T⦄.
#h #g #L #V1 #T1 #HVT1 @and3_intro
qed-.
(* Basic forward lemmas *****************************************************)
-(*
-lamma cnx_fwd_cnr: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+
+lemma cnx_fwd_cnr: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → L ⊢ 𝐍⦃T⦄.
#h #g #L #T #H #U #HTU
@H /2 width=1/ (**) (* auto fails because a δ-expansion gets in the way *)
qed-.
-*)
+
(* Basic properties *********************************************************)
lemma cnx_sort: ∀h,g,L,k. deg h g k 0 → ⦃h, L⦄ ⊢ 𝐍[g]⦃⋆k⦄.
lemma cnx_appl_simple: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃V⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → 𝐒⦃T⦄ →
⦃h, L⦄ ⊢ 𝐍[g]⦃ⓐV.T⦄.
#h #g #L #V #T #HV #HT #HS #X #H
-elim (cpx_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct
+elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
>(HV … HV0) -V0 >(HT … HT0) -T0 //
qed.