intros ni; elim ni; clear ni.
simpl in |- *; trivial.
intros x ni ih.
simpl in |- *.
rewrite ih; trivial.
Qed.
(************************************************************************)
(* Atomic implactions are stored in AVL Trees ai over lists of normal   *)
(* clauses.                                                             *)
(* For the node with the key field=i and with the data field=bs,        *)
(* we define that, for each b in bs, (Imp (Atom i) b) is in ai.         *)
Definition atomic_imps := AVL nf_list.
Definition AI_Nil := AVL_NIL nf_list.
(************************************************************************)
(* Atoms are stored in AVL Trees ai over unit.                          *)
(* For the node with the key field=i,                                   *)
(* we define that (Atom i) is in ai.                                    *)
(* (Definition in Kripke_Trees.v                                        *)
(************************************************************************)
(* The formulae on the left-hand side of the sequent are given by:      *)
Inductive in_ngamma (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a : atoms) : normal_form -> Set :=
| In_Work :
forall (n : nat) (c : normal_form),
my_nth normal_form n work c -> in_ngamma work ds ni ai a c
| In_Disjs :
forall (n : nat) (i j : Int),
my_nth disj n ds (i, j) -> in_ngamma work ds ni ai a (NDisj i j)
| In_Nested_Imps :
forall (n : nat) (x : nimp),
my_nth nimp n (nested_imps2nimps ni) x ->
in_ngamma work ds ni ai a (NImp_NF x)
| In_Atomic_Imps :
forall (i : Int) (b : normal_form) (n : nat) (bs : nf_list),
LOOKUP nf_list i ai bs ->
my_nth normal_form n bs b -> in_ngamma work ds ni ai a (AImp i b)
| In_Atoms :
forall i : Int,
LOOKUP unit i a tt -> in_ngamma work ds ni ai a (NAtom i).
(********************************************************************)

Lemma in_ngamma_cons_work_tail :
forall (c0 : normal_form) (work : nf_list) (ds : disjs)
(ni : nested_imps) (ai : atomic_imps) (a : atoms)
(c : normal_form),
in_ngamma work ds ni ai a c -> in_ngamma (c0 :: work) ds ni ai a c.