(* FROM BASIC_1

(* NOTE: This can be generalized removing the last premise *)
      Lemma ty3_gen_cvoid: (g:?; c:?; t1,t2:?) (ty3 g c t1 t2) ->
                           (e:?; u:?; d:?) (getl d c (CHead e (Bind Void) u)) ->
                           (a:?) (drop (1) d c a) ->
                           (EX y1 y2 | t1 = (lift (1) d y1) &
                                       t2 = (lift (1) d y2) &
                                       (ty3 g a y1 y2)
                           ).

Lemma ty3_gen_appl_nf2: (g:?; c:?; w,v,x:?) (ty3 g c (THead (Flat Appl) w v) x) ->
                        (EX u t | (pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x) &
                                  (ty3 g c v (THead (Bind Abst) u t)) &
                                  (ty3 g c w u) &
                                  (nf2 c (THead (Bind Abst) u t))
                        ).

Lemma ty3_arity: (g:?; c:?; t1,t2:?) (ty3 g c t1 t2) ->
                 (EX a1 | (arity g c t1 a1) &
                          (arity g c t2 (asucc g a1))
                 ).

Lemma ty3_acyclic: (g:?; c:?; t,u:?)
                   (ty3 g c t u) -> (pc3 c u t) -> (P:Prop) P.

Theorem pc3_abst_dec: (g:?; c:?; u1,t1:?) (ty3 g c u1 t1) ->
                      (u2,t2:?) (ty3 g c u2 t2) ->
                      (EX u v2 | (pc3 c u1 (THead (Bind Abst) u2 u)) &
                                 (ty3 g c (THead (Bind Abst) v2 u) t1) &
                                 (pr3 c u2 v2) & (nf2 c v2)
                      ) \/
                      ((u:?) (pc3 c u1 (THead (Bind Abst) u2 u)) -> False).

file ty3_nf2_gen

*)