(** The language of Bellantoni and Cook *)
(** * Syntax and induction principle  - [BC_ind2] is a handmade induction principle because the one generated automatically by Coq ignores the types [list BC] in the constructor [comp]. *)
Inductive BC : Set :=
| zero : BC
| proj : nat -> nat -> nat -> BC
| succ : bool -> BC
| pred : BC
| cond : BC
| rec : BC -> BC -> BC -> BC
| comp : nat -> nat -> BC -> list BC -> list BC -> BC.
(** * Arities  - [arities e] returns the arities (i.e., the number of normal and safe variables) of the expression [e] if it is well formed, or an error including the arities of the subterms. *)
Inductive Arities : Set :=
| error_rec : Arities -> Arities -> Arities -> Arities
| error_comp : Arities -> list Arities -> list Arities -> Arities
| error_proj : nat -> nat -> nat -> Arities
| ok_arities : nat -> nat -> Arities.
Definition aeq (x y:Arities) : bool :=
match x, y with
| ok_arities xn xs, ok_arities yn ys => beq_nat xn yn && beq_nat xs ys
| _, _ => false
end.
Fixpoint arities (e:BC) : Arities :=
match e with
| zero => ok_arities 0 0
| proj n s i =>
if leb (S i) (n + s) then ok_arities n s
else error_proj n s i
| succ _ => ok_arities 0 1
| pred => ok_arities 0 1
| cond => ok_arities 0 4
| rec g h0 h1 =>
match arities g, arities h0, arities h1
with | ok_arities gn gs, ok_arities h0n h0s, ok_arities h1n h1s =>
if beq_nat (S gn) h0n && beq_nat h0n h1n &&
beq_nat (S gs) h0s && beq_nat h0s h1s
then ok_arities h0n gs
else error_rec (ok_arities gn gs) (ok_arities h0n h0s) (ok_arities h1n h1s)
| e1, e2, e3 => error_rec e1 e2 e3
end
| comp n s h nl sl =>
match arities h, map arities nl, map arities sl with
| ok_arities hn hs, anl, asl =>
if beq_nat hn (length nl) && beq_nat hs (length sl) &&
forallb (fun ne => aeq (arities ne) (ok_arities n 0)) nl &&
forallb (fun se => aeq (arities se) (ok_arities n s)) sl
then ok_arities n s
else error_comp (ok_arities hn hs) anl asl
| e , anl, asl => error_comp e anl asl
end
end.
Definition arities2 e :=
match arities e with
| ok_arities n s => (n, s)
| _ => (0, 0)
end.

Lemma BC_ind_inf :
forall (P : nat -> nat -> BC -> Prop),
P 0 0 zero ->
(forall n s i, i < n + s ->  P n s (proj n s i)) ->
(forall b, P 0 1 (succ b)) ->
P 0 1 pred ->
P 0 4 cond ->
(forall n s g h0 h1,
arities g = ok_arities n s ->
arities h0 = ok_arities (S n) (S s) ->
arities h1 = ok_arities (S n) (S s) ->
P n s g ->
P (S n) (S s) h0 ->
P (S n) (S s) h1 ->
P (S n) s (rec g h0 h1)) ->
(forall n s h rl tl,
arities h = ok_arities (length rl) (length tl) ->
(forall e, In e rl -> arities e = ok_arities n 0) ->
(forall e, In e tl -> arities e = ok_arities n s) ->
P (length rl) (length tl) h ->
(forall r, In r rl -> P n 0 r) ->
(forall r, In r tl -> P n s r) ->
P n s (comp n s h rl tl)) ->
forall e n s , arities e = ok_arities n s -> P n s e.