(** * Useful lemmas and tactics *)

(** The following are trivial lemmas and custom tactics that help
  perform backward (inversion) and forward reasoning over the evaluation
  of operator applications. *)

Ltac EvalOp := eapply eval_Eop; eauto with evalexpr.Ltac InvEval2 :=
  match goal with
  | [ H: (eval_operation _ _ _ nil _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | [ H: (eval_operation _ _ _ (_ :: nil) _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | [ H: (eval_operation _ _ _ (_ :: _ :: nil) _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | [ H: (eval_operation _ _ _ (_ :: _ :: _ :: nil) _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | _ =>
      idtac
  end.Ltac InvEval := InvEval1; InvEval2; InvEval2; subst.Ltac TrivialExists :=
  match goal with
  | [ |- exists v, _ /\ Val.lessdef ?a v ] => exists a; split; [EvalOp | auto]
  end.(** We now show that the code generated by "smart constructor" functions
  such as [SelectOp.notint] behaves as expected.  Continuing the
  [notint] example, we show that if the expression [e]
  evaluates to some integer value [Vint n], then [SelectOp.notint e]
  evaluates to a value [Vint (Int.not n)] which is indeed the integer
  negation of the value of [e].

  All proofs follow a common pattern:
- Reasoning by case over the result of the classification functions
  (such as [add_match] for integer addition), gathering additional
  information on the shape of the argument expressions in the non-default
  cases.
- Inversion of the evaluations of the arguments, exploiting the additional
  information thus gathered.
- Equational reasoning over the arithmetic operations performed,
  using the lemmas from the [Int] and [Float] modules.
- Construction of an evaluation derivation for the expression returned
  by the smart constructor.
*)

Definition unary_constructor_sound (cstr: expr -> expr) (sem: val -> val) : Prop :=
  forall le a x,
  eval_expr ge sp e m le a x ->
  exists v, eval_expr ge sp e m le (cstr a) v /\ Val.lessdef (sem x) v.Definition binary_constructor_sound (cstr: expr -> expr -> expr) (sem: val -> val -> val) : Prop :=
  forall le a x b y,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  exists v, eval_expr ge sp e m le (cstr a b) v /\ Val.lessdef (sem x y) v.

(* Specifications: *)
Lemma eval_Olea_ptr: forall a el m, eval_operation ge sp (Olea_ptr a) el m = eval_addressing ge sp a el.
Theorem eval_addrsymbol: forall le id ofs, exists v, eval_expr ge sp e m le (addrsymbol id ofs) v /\ Val.lessdef (Genv.symbol_address ge id ofs) v.
Theorem eval_addrstack: forall le ofs, exists v, eval_expr ge sp e m le (addrstack ofs) v /\ Val.lessdef (Val.offset_ptr sp ofs) v.
Theorem eval_notint: unary_constructor_sound notint Val.notint.
Theorem eval_addimm: forall n, unary_constructor_sound (addimm n) (fun x => Val.add x (Vint n)).
Theorem eval_add: binary_constructor_sound add Val.add.
Theorem eval_sub: binary_constructor_sound sub Val.sub.
Theorem eval_negint: unary_constructor_sound negint Val.neg.
Theorem eval_shlimm: forall n, unary_constructor_sound (fun a => shlimm a n) (fun x => Val.shl x (Vint n)).
Theorem eval_shruimm: forall n, unary_constructor_sound (fun a => shruimm a n) (fun x => Val.shru x (Vint n)).
Theorem eval_shrimm: forall n, unary_constructor_sound (fun a => shrimm a n) (fun x => Val.shr x (Vint n)).
Lemma eval_mulimm_base: forall n, unary_constructor_sound (mulimm_base n) (fun x => Val.mul x (Vint n)).
Theorem eval_mulimm: forall n, unary_constructor_sound (mulimm n) (fun x => Val.mul x (Vint n)).
Theorem eval_mul: binary_constructor_sound mul Val.mul.
Theorem eval_mulhs: binary_constructor_sound mulhs Val.mulhs.
Theorem eval_mulhu: binary_constructor_sound mulhu Val.mulhu.
Theorem eval_andimm: forall n, unary_constructor_sound (andimm n) (fun x => Val.and x (Vint n)).
Theorem eval_and: binary_constructor_sound and Val.and.
Theorem eval_orimm: forall n, unary_constructor_sound (orimm n) (fun x => Val.or x (Vint n)).