Definition reduction_ok (k: kind) (a: expr) (m: mem) (rd: reduction) : Prop :=
  match k, rd with
  | LV, Lred _ l' m' => lred ge e a m l' m'
  | RV, Rred _ r' m' t => rred ge a m t r' m' /\ exists w', possible_trace w t w'
  | RV, Callred _ fd args tyres m' => callred ge a m fd args tyres /\ m' = m
  | LV, Stuckred => ~imm_safe_t k a m
  | RV, Stuckred => ~imm_safe_t k a m
  | _, _ => False
  end.Definition reducts_ok (k: kind) (a: expr) (m: mem) (ll: reducts expr) : Prop :=
  (forall C rd,
      In (C, rd) ll ->
      exists a', exists k', context k' k C /\ a = C a' /\ reduction_ok k' a' m rd)
  /\ (ll = nil -> match k with LV => is_loc a <> None | RV => is_val a <> None end).Definition list_reducts_ok (al: exprlist) (m: mem) (ll: reducts exprlist) : Prop :=
  (forall C rd,
      In (C, rd) ll ->
      exists a', exists k', contextlist k' C /\ al = C a' /\ reduction_ok k' a' m rd)
  /\ (ll = nil -> is_val_list al <> None).Ltac monadInv :=
  match goal with
  | [ H: match ?x with Some _ => _ | None => None end = Some ?y |- _ ] =>
      destruct x eqn:?; [monadInv|discriminate]
  | [ H: match ?x with left _ => _ | right _ => None end = Some ?y |- _ ] =>
      destruct x; [monadInv|discriminate]
  | _ => idtac
  end.

(* Specifications: *)
Lemma sem_cast_arguments_sound: forall m rargs vtl tyargs vargs, is_val_list rargs = Some vtl -> sem_cast_arguments vtl tyargs m = Some vargs -> cast_arguments m rargs tyargs vargs.
Lemma sem_cast_arguments_complete: forall m al tyl vl, cast_arguments m al tyl vl -> exists vtl, is_val_list al = Some vtl /\ sem_cast_arguments vtl tyl m = Some vl.
Lemma topred_ok: forall k a m rd, reduction_ok k a m rd -> reducts_ok k a m (topred rd).
Lemma stuck_ok: forall k a m, ~imm_safe_t k a m -> reducts_ok k a m stuck.
Lemma wrong_kind_ok: forall k a m, k <> Cstrategy.expr_kind a -> reducts_ok k a m stuck.
Lemma not_invert_ok: forall k a m, match a with | Eloc _ _ _ => False | Eval _ _ => False | _ => invert_expr_prop a m -> False end -> reducts_ok k a m stuck.
Lemma incontext_ok: forall k a m C res k' a', reducts_ok k' a' m res -> a = C a' -> context k' k C -> match k' with LV => is_loc a' = None | RV => is_val a' = None end -> reducts_ok k a m (incontext C res).
Lemma incontext2_ok: forall k a m k1 a1 res1 k2 a2 res2 C1 C2, reducts_ok k1 a1 m res1 -> reducts_ok k2 a2 m res2 -> a = C1 a1 -> a = C2 a2 -> context k1 k C1 -> context k2 k C2 -> match k1 with LV => is_loc a1 = None | RV => is_val a1 = None end \/ match k2 with LV => is_loc a2 = None | RV => is_val a2 = None end -> reducts_ok k a m (incontext2 C1 res1 C2 res2).
Lemma incontext_list_ok: forall ef tyargs al ty m res, list_reducts_ok al m res -> is_val_list al = None -> reducts_ok RV (Ebuiltin ef tyargs al ty) m (incontext (fun x => Ebuiltin ef tyargs x ty) res).
Lemma incontext2_list_ok: forall a1 a2 ty m res1 res2, reducts_ok RV a1 m res1 -> list_reducts_ok a2 m res2 -> is_val a1 = None \/ is_val_list a2 = None -> reducts_ok RV (Ecall a1 a2 ty) m (incontext2 (fun x => Ecall x a2 ty) res1 (fun x => Ecall a1 x ty) res2).
Lemma incontext2_list_ok': forall a1 a2 m res1 res2, reducts_ok RV a1 m res1 -> list_reducts_ok a2 m res2 -> list_reducts_ok (Econs a1 a2) m (incontext2 (fun x => Econs x a2) res1 (fun x => Econs a1 x) res2).
Lemma is_val_list_all_values: forall al vtl, is_val_list al = Some vtl -> exprlist_all_values al.