(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)
(** Correctness of instruction selection for operators *)
Local Open Scope cminorsel_scope.
(** * 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 InvEval1 :=
match goal with
| [ H: (eval_expr _ _ _ _ _ (Eop _ Enil) _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_expr _ _ _ _ _ (Eop _ (_ ::: Enil)) _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_expr _ _ _ _ _ (Eop _ (_ ::: _ ::: Enil)) _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_exprlist _ _ _ _ _ Enil _) |- _ ] =>
inv H; InvEval1
| [ H: (eval_exprlist _ _ _ _ _ (_ ::: _) _) |- _ ] =>
inv H; InvEval1
| _ =>
idtac
end.
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.
(** * Correctness of the smart constructors *)
Section CMCONSTR.
Variable ge: genv.
Variable sp: val.
Variable e: env.
Variable m: mem.
(** 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.

Theorem eval_mulimm:
forall n, unary_constructor_sound (mulimm n) (fun x => Val.mul x (Vint n)).