Require Equations.Prop.EqDec. (*TODO: Remove Import *)
From Undecidability.H10 Require Import H10 dio_single dio_logic.
From Undecidability.L.Datatypes Require Import LNat Lists LOptions LSum.
From Undecidability.L Require Import Tactics.LTactics Computability.MuRec Computability.Synthetic Tactics.GenEncode.
From Undecidability.Shared Require Import DLW.Utils.finite DLW.Vec.vec DLW.Vec.pos.
From Undecidability.MuRec Require Import recalg ra_bs ra_sem_eq.
Reserved Notation "  '[' f ';' v ';' min ';' c ']' '~~>' x " (at level 70).
(* Bigstep semantics for recursive algorithms *)
Inductive ra_bs_c : nat -> nat -> forall k, recalg k -> vec nat k -> nat -> Prop :=
| in_ra_bs_c_cst  : forall min c n v,             [ra_cst n;        v; min; S c] ~~> n
| in_ra_bs_c_zero : forall min c v,               [ra_zero;         v; min; S c] ~~> 0
| in_ra_bs_c_succ : forall min c v,               [ra_succ;         v; min; S c] ~~> S (vec_head v)
| in_ra_bs_c_proj : forall min c k v j,           [@ra_proj k j;    v; min; S c] ~~> vec_pos v j
| in_ra_bs_c_comp : forall min c k i f (gj : vec (recalg i) k) v w x,
(forall j, [vec_pos gj j;    v; min; c - pos2nat j] ~~> vec_pos w j)
->             [f;               w; min; S c] ~~> x
->             [ra_comp f gj;    v; min; S (S c)] ~~> x
| in_ra_bs_c_rec_0 : forall min c k f (g : recalg (S (S k))) v x,
[f;               v; min; c] ~~> x
->             [ra_rec f g;   0##v; min; S c] ~~> x
| in_ra_bs_c_rec_S : forall min c k f (g : recalg (S (S k))) v n x y,
[ra_rec f g;   n##v; min; c] ~~> x
->             [g;         n##x##v; min; c] ~~> y
->             [ra_rec f g; S n##v; min; S c] ~~> y
| in_ra_bs_c_min : forall min c k (f : recalg (S k)) v x w , x >= min ->
(forall j : pos x, pos2nat j >= min -> [f;    pos2nat j##v; 0; c - (pos2nat j - min)] ~~> S (vec_pos w j))
->             [f;            x##v; 0; c - (x - min)] ~~> 0
->             [ra_min f;        v; min; S c] ~~> x
where " [ f ; v ; min ; c ] ~~> x " := (@ra_bs_c min c _ f v x).
Local Hint Resolve ra_bs_from_c ra_bs_to_c : core.
Fact ra_bs_c_correct k (f : recalg k) v x :
[|f|] v x <-> exists c, [f ; v ; 0 ; c] ~~> x.
Inductive reccode :=
| rc_cst (n : nat)
| rc_zero
| rc_succ
| rc_proj (n : nat)
| rc_comp (f : reccode) (g : reccode)
| rc_cons (f : reccode) (g : reccode)
| rc_nil
| rc_rec (f : reccode) (g : reccode)
| rc_min (f : reccode).
Fixpoint eval (fuel : nat) (min : nat) (c : reccode) (v : list nat) : option (nat + list nat) :=
match fuel with
| 0 => None
| S fuel =>
match c with
| rc_cst n => Some (inl n)
| rc_zero => Some (inl 0)
| rc_succ => match v with
| x :: _ => Some (inl (S x))
| _ => None
end
| rc_proj n => match nth_error v n with Some r => Some (inl r) | None => None end
| rc_comp f g => match eval fuel min g v
with
| Some (inr g) => match eval fuel min f g with Some (inl f) => Some (inl f) | _ => None end
| _ => None
end
| rc_cons f g => match eval fuel min f v, eval fuel min g v with
| Some (inl f), Some (inr g) => Some (inr (f :: g))
| _, _ => None
end
| rc_nil => Some (inr nil)
| rc_rec f g => match v with
| 0 :: v => match eval fuel min f v with Some (inl r) => Some (inl r) | _ => None end
| S n :: v => match eval fuel min (rc_rec f g) (n :: v) with
| Some (inl y) => match eval fuel min g (n :: y :: v) with Some (inl r) => Some (inl r) | _ => None end
| _ => None
end
| _ => None
end
| rc_min f => match eval fuel 0 f (min :: v) with
| Some (inl 0) => Some (inl min)
| Some (inl _) => match eval fuel (S min) (rc_min f) v with Some (inl r) => Some (inl r) | _ => None end
| _ => None
end
end
end.
Definition rec_erase i (erase : forall i, recalg i -> reccode) := (fix rec k (v : vec (recalg i) k) := match v with vec_nil => rc_nil | x ## v => rc_cons (erase _ x) (rec _ v) end).
Fixpoint erase k (f : recalg k) : reccode :=
match f with
| ra_cst n => rc_cst n
| ra_zero => rc_zero
| ra_succ => rc_succ
| ra_proj _ p => rc_proj (pos2nat p)
| ra_comp _ _ f g => rc_comp (erase f) (rec_erase erase g)
| ra_rec _ f g => rc_rec (erase f) (erase g)
| ra_min _ f => rc_min (erase f)
end.
(* Require Import Equations.Prop.DepElim. *)

Lemma le_ind2 m (P : nat -> Prop) :
P m -> (forall n, P (S n) -> S n <= m -> P n) -> forall n, n <= m -> P n.