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(* *********************************************************************)
(* *)
(* 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. *)
(* *)
(* *********************************************************************)
(** Proof of type preservation for Reload and of
correctness of computation of stack bounds for Linear. *)
Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Op.
Require Import Locations.
Require Import LTLin.
Require Import LTLintyping.
Require Import Linear.
Require Import Lineartyping.
Require Import Conventions.
Require Import Parallelmove.
Require Import Reload.
Require Import Reloadproof.
(** * Typing Linear constructors *)
(** We show that the Linear constructor functions defined in [Reload]
generate well-typed instruction sequences,
given sufficient typing and well-formedness hypotheses over the locations involved. *)
Hint Resolve wt_Lgetstack wt_Lsetstack wt_Lopmove
wt_Lop wt_Lload wt_Lstore wt_Lcall wt_Ltailcall wt_Lalloc
wt_Llabel wt_Lgoto wt_Lcond wt_Lreturn: reloadty.
Remark wt_code_cons:
forall f i c, wt_instr f i -> wt_code f c -> wt_code f (i :: c).
Proof.
intros; red; simpl; intros. elim H1; intro.
subst i; auto. auto.
Qed.
Hint Resolve wt_code_cons: reloadty.
Definition loc_valid (f: function) (l: loc) :=
match l with R _ => True | S s => slot_valid f s end.
Lemma loc_acceptable_valid:
forall f l, loc_acceptable l -> loc_valid f l.
Proof.
destruct l; simpl; intro. auto.
destruct s; simpl. omega. tauto. tauto.
Qed.
Definition loc_writable (l: loc) :=
match l with R _ => True | S s => slot_writable s end.
Lemma loc_acceptable_writable:
forall l, loc_acceptable l -> loc_writable l.
Proof.
destruct l; simpl; intro. auto.
destruct s; simpl; tauto.
Qed.
Hint Resolve loc_acceptable_valid loc_acceptable_writable: reloadty.
Definition locs_valid (f: function) (ll: list loc) :=
forall l, In l ll -> loc_valid f l.
Definition locs_writable (ll: list loc) :=
forall l, In l ll -> loc_writable l.
Lemma locs_acceptable_valid:
forall f ll, locs_acceptable ll -> locs_valid f ll.
Proof.
unfold locs_acceptable, locs_valid. auto with reloadty.
Qed.
Lemma locs_acceptable_writable:
forall ll, locs_acceptable ll -> locs_writable ll.
Proof.
unfold locs_acceptable, locs_writable. auto with reloadty.
Qed.
Hint Resolve locs_acceptable_valid locs_acceptable_writable: reloadty.
Lemma wt_add_reload:
forall f src dst k,
loc_valid f src -> Loc.type src = mreg_type dst ->
wt_code f k -> wt_code f (add_reload src dst k).
Proof.
intros; unfold add_reload.
destruct src; eauto with reloadty.
destruct (mreg_eq m dst); eauto with reloadty.
Qed.
Hint Resolve wt_add_reload: reloadty.
Lemma wt_add_reloads:
forall f srcs dsts k,
locs_valid f srcs -> map Loc.type srcs = map mreg_type dsts ->
wt_code f k -> wt_code f (add_reloads srcs dsts k).
Proof.
induction srcs; destruct dsts; simpl; intros; try congruence.
auto. inv H0. apply wt_add_reload; auto with coqlib reloadty.
apply IHsrcs; auto. red; intros; auto with coqlib.
Qed.
Hint Resolve wt_add_reloads: reloadty.
Lemma wt_add_spill:
forall f src dst k,
loc_valid f dst -> loc_writable dst -> Loc.type dst = mreg_type src ->
wt_code f k -> wt_code f (add_spill src dst k).
Proof.
intros; unfold add_spill.
destruct dst; eauto with reloadty.
destruct (mreg_eq src m); eauto with reloadty.
Qed.
Hint Resolve wt_add_spill: reloadty.
Lemma wt_add_move:
forall f src dst k,
loc_valid f src -> loc_valid f dst -> loc_writable dst ->
Loc.type dst = Loc.type src ->
wt_code f k -> wt_code f (add_move src dst k).
Proof.
intros. unfold add_move.
destruct (Loc.eq src dst); auto.
destruct src; auto with reloadty.
destruct dst; auto with reloadty.
set (tmp := match slot_type s with
| Tint => IT1
| Tfloat => FT1
end).
assert (mreg_type tmp = Loc.type (S s)).
simpl. destruct (slot_type s); reflexivity.
apply wt_add_reload; auto with reloadty.
apply wt_add_spill; auto. congruence.
Qed.
Hint Resolve wt_add_move: reloadty.
Lemma wt_add_moves:
forall f b moves,
(forall s d, In (s, d) moves ->
loc_valid f s /\ loc_valid f d /\ loc_writable d /\ Loc.type s = Loc.type d) ->
wt_code f b ->
wt_code f
(List.fold_right (fun p k => add_move (fst p) (snd p) k) b moves).
Proof.
induction moves; simpl; intros.
auto.
destruct a as [s d]. simpl.
destruct (H s d) as [A [B [C D]]]. auto.
auto with reloadty.
Qed.
Theorem wt_parallel_move:
forall f srcs dsts b,
List.map Loc.type srcs = List.map Loc.type dsts ->
locs_valid f srcs -> locs_valid f dsts -> locs_writable dsts ->
wt_code f b -> wt_code f (parallel_move srcs dsts b).
Proof.
intros. unfold parallel_move. apply wt_add_moves; auto.
intros.
elim (parmove_prop_2 _ _ _ _ H4); intros A B.
split. destruct A as [C|[C|C]]; auto; subst s; exact I.
split. destruct B as [C|[C|C]]; auto; subst d; exact I.
split. destruct B as [C|[C|C]]; auto; subst d; exact I.
eapply parmove_prop_3; eauto.
Qed.
Hint Resolve wt_parallel_move: reloadty.
Lemma wt_reg_for:
forall l, mreg_type (reg_for l) = Loc.type l.
Proof.
intros. destruct l; simpl. auto.
case (slot_type s); reflexivity.
Qed.
Hint Resolve wt_reg_for: reloadty.
Lemma wt_regs_for_rec:
forall locs itmps ftmps,
(List.length locs <= List.length itmps)%nat ->
(List.length locs <= List.length ftmps)%nat ->
(forall r, In r itmps -> mreg_type r = Tint) ->
(forall r, In r ftmps -> mreg_type r = Tfloat) ->
List.map mreg_type (regs_for_rec locs itmps ftmps) = List.map Loc.type locs.
Proof.
induction locs; intros.
simpl. auto.
destruct itmps; simpl in H. omegaContradiction.
destruct ftmps; simpl in H0. omegaContradiction.
simpl. apply (f_equal2 (@cons typ)).
destruct a. reflexivity. simpl. case (slot_type s).
apply H1; apply in_eq. apply H2; apply in_eq.
apply IHlocs. omega. omega.
intros; apply H1; apply in_cons; auto.
intros; apply H2; apply in_cons; auto.
Qed.
Lemma wt_regs_for:
forall locs,
(List.length locs <= 3)%nat ->
List.map mreg_type (regs_for locs) = List.map Loc.type locs.
Proof.
intros. unfold regs_for. apply wt_regs_for_rec.
simpl. auto. simpl. auto.
simpl; intros; intuition; subst r; reflexivity.
simpl; intros; intuition; subst r; reflexivity.
Qed.
Hint Resolve wt_regs_for: reloadty.
Hint Resolve length_op_args length_addr_args length_cond_args: reloadty.
Hint Extern 4 (_ = _) => congruence : reloadty.
Lemma wt_transf_instr:
forall f instr k,
LTLintyping.wt_instr (LTLin.fn_sig f) instr ->
wt_code (transf_function f) k ->
wt_code (transf_function f) (transf_instr f instr k).
Proof.
Opaque reg_for regs_for.
intros. inv H; simpl; auto with reloadty.
caseEq (is_move_operation op args); intros.
destruct (is_move_operation_correct _ _ H). congruence.
assert (map mreg_type (regs_for args) = map Loc.type args).
eauto with reloadty.
assert (mreg_type (reg_for res) = Loc.type res). eauto with reloadty.
auto with reloadty.
assert (map mreg_type (regs_for args) = map Loc.type args).
eauto with reloadty.
assert (mreg_type (reg_for dst) = Loc.type dst). eauto with reloadty.
auto with reloadty.
caseEq (regs_for (src :: args)); intros.
red; simpl; tauto.
assert (map mreg_type (regs_for (src :: args)) = map Loc.type (src :: args)).
apply wt_regs_for. simpl. apply le_n_S. eauto with reloadty.
rewrite H in H5. injection H5; intros.
auto with reloadty.
assert (locs_valid (transf_function f) (loc_arguments sig)).
red; intros. generalize (loc_arguments_acceptable sig l H).
destruct l; simpl; auto. destruct s; simpl; intuition.
assert (locs_writable (loc_arguments sig)).
red; intros. generalize (loc_arguments_acceptable sig l H7).
destruct l; simpl; auto. destruct s; simpl; intuition.
assert (map Loc.type args = map Loc.type (loc_arguments sig)).
rewrite loc_arguments_type; auto.
assert (Loc.type res = mreg_type (loc_result sig)).
rewrite H3. unfold loc_result.
destruct (sig_res sig); auto. destruct t; auto.
destruct ros; auto 10 with reloadty.
assert (locs_valid (transf_function f) (loc_arguments sig)).
red; intros. generalize (loc_arguments_acceptable sig l H).
destruct l; simpl; auto. destruct s; simpl; intuition.
assert (locs_writable (loc_arguments sig)).
red; intros. generalize (loc_arguments_acceptable sig l H7).
destruct l; simpl; auto. destruct s; simpl; intuition.
assert (map Loc.type args = map Loc.type (loc_arguments sig)).
rewrite loc_arguments_type; auto.
destruct ros; auto 10 with reloadty.
assert (map mreg_type (regs_for args) = map Loc.type args).
eauto with reloadty.
auto with reloadty.
destruct optres; simpl in *; auto with reloadty.
apply wt_add_reload; auto with reloadty.
unfold loc_result. rewrite <- H1.
destruct (Loc.type l); reflexivity.
Qed.
Lemma wt_transf_code:
forall f c,
LTLintyping.wt_code (LTLin.fn_sig f) c ->
Lineartyping.wt_code (transf_function f) (transf_code f c).
Proof.
induction c; simpl; intros.
red; simpl; tauto.
apply wt_transf_instr; auto with coqlib.
apply IHc. red; auto with coqlib.
Qed.
Lemma wt_transf_fundef:
forall fd,
LTLintyping.wt_fundef fd ->
Lineartyping.wt_fundef (transf_fundef fd).
Proof.
intros. destruct fd; simpl.
inv H. inv H1. constructor. unfold wt_function. simpl.
apply wt_parallel_move; auto with reloadty.
rewrite loc_parameters_type. auto.
unfold loc_parameters; red; intros.
destruct (list_in_map_inv _ _ _ H) as [r [A B]]. rewrite A.
generalize (loc_arguments_acceptable _ _ B).
destruct r; simpl; auto. destruct s; try tauto.
intros; simpl. split. omega.
apply loc_arguments_bounded; auto.
apply wt_transf_code; auto.
constructor.
Qed.
Lemma program_typing_preserved:
forall p,
LTLintyping.wt_program p ->
Lineartyping.wt_program (transf_program p).
Proof.
intros; red; intros.
destruct (transform_program_function _ _ _ _ H0) as [f0 [A B]].
subst f; apply wt_transf_fundef. eauto.
Qed.
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