(* *********************************************************************) (* *) (* 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 proof for ARM code generation: main proof. *) Require Import Coqlib. Require Import Maps. Require Import Errors. Require Import AST. Require Import Integers. Require Import Floats. Require Import Values. Require Import Memory. Require Import Events. Require Import Globalenvs. Require Import Smallstep. Require Import Op. Require Import Locations. Require Import Conventions. Require Import Mach. Require Import Asm. Require Import Asmgen. Require Import Asmgenproof0. Require Import Asmgenproof1. Section PRESERVATION. Variable prog: Mach.program. Variable tprog: Asm.program. Hypothesis TRANSF: transf_program prog = Errors.OK tprog. Let ge := Genv.globalenv prog. Let tge := Genv.globalenv tprog. Lemma symbols_preserved: forall id, Genv.find_symbol tge id = Genv.find_symbol ge id. Proof. intros. unfold ge, tge. apply Genv.find_symbol_transf_partial with transf_fundef. exact TRANSF. Qed. Lemma functions_translated: forall b f, Genv.find_funct_ptr ge b = Some f -> exists tf, Genv.find_funct_ptr tge b = Some tf /\ transf_fundef f = Errors.OK tf. Proof (Genv.find_funct_ptr_transf_partial transf_fundef _ TRANSF). Lemma functions_transl: forall f b tf, Genv.find_funct_ptr ge b = Some (Internal f) -> transf_function f = OK tf -> Genv.find_funct_ptr tge b = Some (Internal tf). Proof. intros. destruct (functions_translated _ _ H) as [tf' [A B]]. rewrite A. monadInv B. f_equal. congruence. Qed. Lemma varinfo_preserved: forall b, Genv.find_var_info tge b = Genv.find_var_info ge b. Proof. intros. unfold ge, tge. apply Genv.find_var_info_transf_partial with transf_fundef. exact TRANSF. Qed. (** * Properties of control flow *) Lemma transf_function_no_overflow: forall f tf, transf_function f = OK tf -> list_length_z (fn_code tf) <= Int.max_unsigned. Proof. intros. monadInv H. destruct (zlt Int.max_unsigned (list_length_z (fn_code x))); inv EQ0. omega. Qed. Lemma exec_straight_exec: forall f c ep tf tc c' rs m rs' m', transl_code_at_pc ge (rs PC) f c ep tf tc -> exec_straight tge tf tc rs m c' rs' m' -> plus step tge (State rs m) E0 (State rs' m'). Proof. intros. inv H. eapply exec_straight_steps_1; eauto. eapply transf_function_no_overflow; eauto. eapply functions_transl; eauto. Qed. Lemma exec_straight_at: forall f c ep tf tc c' ep' tc' rs m rs' m', transl_code_at_pc ge (rs PC) f c ep tf tc -> transl_code f c' ep' = OK tc' -> exec_straight tge tf tc rs m tc' rs' m' -> transl_code_at_pc ge (rs' PC) f c' ep' tf tc'. Proof. intros. inv H. exploit exec_straight_steps_2; eauto. eapply transf_function_no_overflow; eauto. eapply functions_transl; eauto. intros [ofs' [PC' CT']]. rewrite PC'. constructor; auto. Qed. (** The [find_label] function returns the code tail starting at the given label. A connection with [code_tail] is then established. *) Fixpoint find_label (lbl: label) (c: code) {struct c} : option code := match c with | nil => None | instr :: c' => if is_label lbl instr then Some c' else find_label lbl c' end. Lemma label_pos_code_tail: forall lbl c pos c', find_label lbl c = Some c' -> exists pos', label_pos lbl pos c = Some pos' /\ code_tail (pos' - pos) c c' /\ pos < pos' <= pos + list_length_z c. Proof. induction c. simpl; intros. discriminate. simpl; intros until c'. case (is_label lbl a). intro EQ; injection EQ; intro; subst c'. exists (pos + 1). split. auto. split. replace (pos + 1 - pos) with (0 + 1) by omega. constructor. constructor. rewrite list_length_z_cons. generalize (list_length_z_pos c). omega. intros. generalize (IHc (pos + 1) c' H). intros [pos' [A [B C]]]. exists pos'. split. auto. split. replace (pos' - pos) with ((pos' - (pos + 1)) + 1) by omega. constructor. auto. rewrite list_length_z_cons. omega. Qed. (** The following lemmas show that the translation from Mach to ARM preserves labels, in the sense that the following diagram commutes: << translation Mach code ------------------------ ARM instr sequence | | | Mach.find_label lbl find_label lbl | | | v v Mach code tail ------------------- ARM instr seq tail translation >> The proof demands many boring lemmas showing that ARM constructor functions do not introduce new labels. *) Section TRANSL_LABEL. Variable lbl: label. Remark iterate_op_label: forall op1 op2 l k, (forall so, is_label lbl (op1 so) = false) -> (forall so, is_label lbl (op2 so) = false) -> find_label lbl (iterate_op op1 op2 l k) = find_label lbl k. Proof. intros. unfold iterate_op. destruct l as [ | hd tl]. simpl. rewrite H. auto. simpl. rewrite H. induction tl; simpl. auto. rewrite H0; auto. Qed. Hint Resolve iterate_op_label: labels. Remark loadimm_label: forall r n k, find_label lbl (loadimm r n k) = find_label lbl k. Proof. intros. unfold loadimm. destruct (NPeano.leb (length (decompose_int n)) (length (decompose_int (Int.not n)))); auto with labels. Qed. Hint Rewrite loadimm_label: labels. Remark addimm_label: forall r1 r2 n k, find_label lbl (addimm r1 r2 n k) = find_label lbl k. Proof. intros; unfold addimm. destruct (NPeano.leb (length (decompose_int n)) (length (decompose_int (Int.neg n)))); auto with labels. Qed. Hint Rewrite addimm_label: labels. Remark andimm_label: forall r1 r2 n k, find_label lbl (andimm r1 r2 n k) = find_label lbl k. Proof. intros; unfold andimm. destruct (is_immed_arith n). reflexivity. auto with labels. Qed. Hint Rewrite andimm_label: labels. Remark rsubimm_label: forall r1 r2 n k, find_label lbl (rsubimm r1 r2 n k) = find_label lbl k. Proof. intros; unfold rsubimm. auto with labels. Qed. Hint Rewrite rsubimm_label: labels. Remark orimm_label: forall r1 r2 n k, find_label lbl (orimm r1 r2 n k) = find_label lbl k. Proof. intros; unfold orimm. auto with labels. Qed. Hint Rewrite orimm_label: labels. Remark xorimm_label: forall r1 r2 n k, find_label lbl (xorimm r1 r2 n k) = find_label lbl k. Proof. intros; unfold xorimm. auto with labels. Qed. Hint Rewrite xorimm_label: labels. Remark loadind_int_label: forall base ofs dst k, find_label lbl (loadind_int base ofs dst k) = find_label lbl k. Proof. intros; unfold loadind_int. destruct (is_immed_mem_word ofs); autorewrite with labels; auto. Qed. Remark loadind_label: forall base ofs ty dst k c, loadind base ofs ty dst k = OK c -> find_label lbl c = find_label lbl k. Proof. intros. destruct ty; monadInv H. apply loadind_int_label. unfold loadind_float. destruct (is_immed_mem_float ofs); autorewrite with labels; auto. Qed. Remark storeind_int_label: forall base ofs src k, find_label lbl (storeind_int src base ofs k) = find_label lbl k. Proof. intros; unfold storeind_int. destruct (is_immed_mem_word ofs); autorewrite with labels; auto. Qed. Remark storeind_label: forall base ofs ty src k c, storeind src base ofs ty k = OK c -> find_label lbl c = find_label lbl k. Proof. intros. destruct ty; monadInv H. apply storeind_int_label. unfold storeind_float. destruct (is_immed_mem_float ofs); autorewrite with labels; auto. Qed. Hint Rewrite loadind_int_label loadind_label storeind_int_label storeind_label: labels. Ltac ArgsInv := repeat (match goal with | [ H: Error _ = OK _ |- _ ] => discriminate | [ H: match ?args with nil => _ | _ :: _ => _ end = OK _ |- _ ] => destruct args | [ H: bind _ _ = OK _ |- _ ] => monadInv H | [ H: assertion _ = OK _ |- _ ] => monadInv H end). Remark transl_cond_label: forall cond args k c, transl_cond cond args k = OK c -> find_label lbl c = find_label lbl k. Proof. unfold transl_cond; intros; destruct cond; ArgsInv; auto. destruct (is_immed_arith i); autorewrite with labels; auto. destruct (is_immed_arith i); autorewrite with labels; auto. Qed. Remark transl_op_label: forall op args r k c, transl_op op args r k = OK c -> find_label lbl c = find_label lbl k. Proof. unfold transl_op; intros; destruct op; ArgsInv; autorewrite with labels; auto. destruct (preg_of r); try discriminate; destruct (preg_of m); inv H; auto. destruct (ireg_eq x x0 || ireg_eq x x1); auto. simpl. autorewrite with labels; auto. erewrite transl_cond_label by eauto; auto. Qed. Remark transl_memory_access_label: forall (mk_instr_imm: ireg -> int -> instruction) (mk_instr_gen: option (ireg -> shift_addr -> instruction)) (is_immed: int -> bool) (addr: addressing) (args: list mreg) c k, transl_memory_access mk_instr_imm mk_instr_gen is_immed addr args k = OK c -> (forall r n, is_label lbl (mk_instr_imm r n) = false) -> (match mk_instr_gen with | None => True | Some f => forall r sa, is_label lbl (f r sa) = false end) -> find_label lbl c = find_label lbl k. Proof. unfold transl_memory_access; intros; destruct addr; ArgsInv; auto. destruct (is_immed i); autorewrite with labels; simpl; rewrite H0; auto. destruct mk_instr_gen. simpl. rewrite H1. auto. simpl. rewrite H0. auto. destruct mk_instr_gen. simpl. rewrite H1. auto. simpl. rewrite H0. auto. destruct (is_immed i); inv H; autorewrite with labels; simpl; rewrite H0; auto. Qed. Lemma transl_instr_label: forall f i ep k c, transl_instr f i ep k = OK c -> find_label lbl c = if Mach.is_label lbl i then Some k else find_label lbl k. Proof. unfold transl_instr, Mach.is_label; intros. destruct i; try (monadInv H). eapply loadind_label; eauto. eapply storeind_label; eauto. destruct ep; autorewrite with labels; eapply loadind_label; eauto. eapply transl_op_label; eauto. destruct m; simpl in H; monadInv H; eapply transl_memory_access_label; eauto; simpl; auto. destruct m; simpl in H; monadInv H; eapply transl_memory_access_label; eauto; simpl; auto. destruct s0; monadInv H; auto. destruct s0; monadInv H; autorewrite with labels; auto. auto. auto. simpl. auto. auto. erewrite transl_cond_label. 2: eauto. auto. auto. autorewrite with labels; auto. Qed. Lemma transl_code_label: forall f c ep tc, transl_code f c ep = OK tc -> match Mach.find_label lbl c with | None => find_label lbl tc = None | Some c' => exists tc', find_label lbl tc = Some tc' /\ transl_code f c' false = OK tc' end. Proof. induction c; simpl; intros. inv H. auto. monadInv H. rewrite (transl_instr_label _ _ _ _ _ EQ0). generalize (Mach.is_label_correct lbl a). destruct (Mach.is_label lbl a); intros. subst a. simpl in EQ. exists x; auto. eapply IHc; eauto. Qed. Lemma transl_find_label: forall f tf, transf_function f = OK tf -> match Mach.find_label lbl f.(Mach.fn_code) with | None => find_label lbl (fn_code tf) = None | Some c => exists tc, find_label lbl (fn_code tf) = Some tc /\ transl_code f c false = OK tc end. Proof. intros. monadInv H. destruct (zlt Int.max_unsigned (list_length_z (fn_code x))); inv EQ0. monadInv EQ. simpl. eapply transl_code_label; eauto. Qed. End TRANSL_LABEL. (** A valid branch in a piece of Mach code translates to a valid ``go to'' transition in the generated ARM code. *) (** A valid branch in a piece of Mach code translates to a valid ``go to'' transition in the generated PPC code. *) Lemma find_label_goto_label: forall f tf lbl rs m c' b ofs, Genv.find_funct_ptr ge b = Some (Internal f) -> transf_function f = OK tf -> rs PC = Vptr b ofs -> Mach.find_label lbl f.(Mach.fn_code) = Some c' -> exists tc', exists rs', goto_label tf lbl rs m = Next rs' m /\ transl_code_at_pc ge (rs' PC) f c' false tf tc' /\ forall r, r <> PC -> rs'#r = rs#r. Proof. intros. exploit (transl_find_label lbl f tf); eauto. rewrite H2. intros [tc [A B]]. exploit label_pos_code_tail; eauto. instantiate (1 := 0). intros [pos' [P [Q R]]]. exists tc; exists (rs#PC <- (Vptr b (Int.repr pos'))). split. unfold goto_label. rewrite P. rewrite H1. auto. split. rewrite Pregmap.gss. constructor; auto. rewrite Int.unsigned_repr. replace (pos' - 0) with pos' in Q. auto. omega. generalize (transf_function_no_overflow _ _ H0). omega. intros. apply Pregmap.gso; auto. Qed. (** * Proof of semantic preservation *) (** Semantic preservation is proved using simulation diagrams of the following form. << st1 --------------- st2 | | t| *|t | | v v st1'--------------- st2' >> The invariant is the [match_states] predicate below, which includes: - The ARM code pointed by the PC register is the translation of the current Mach code sequence. - Mach register values and ARM register values agree. *) Inductive match_states: Mach.state -> Asm.state -> Prop := | match_states_intro: forall s f sp c ep ms m m' rs tf tc ra (STACKS: match_stack ge s m m' ra sp) (MEXT: Mem.extends m m') (AT: transl_code_at_pc ge (rs PC) f c ep tf tc) (AG: agree ms (Vptr sp Int.zero) rs) (RSA: retaddr_stored_at m m' sp (Int.unsigned f.(fn_retaddr_ofs)) ra) (DXP: ep = true -> rs#IR10 = parent_sp s), match_states (Mach.State s f (Vptr sp Int.zero) c ms m) (Asm.State rs m') | match_states_call: forall s fd ms m m' rs fb (STACKS: match_stack ge s m m' rs#(IR IR14) (Mem.nextblock m)) (MEXT: Mem.extends m m') (AG: agree ms (parent_sp s) rs) (ATPC: rs PC = Vptr fb Int.zero) (FUNCT: Genv.find_funct_ptr ge fb = Some fd) (WTRA: Val.has_type rs#(IR IR14) Tint), match_states (Mach.Callstate s fd ms m) (Asm.State rs m') | match_states_return: forall s ms m m' rs (STACKS: match_stack ge s m m' (rs PC) (Mem.nextblock m)) (MEXT: Mem.extends m m') (AG: agree ms (parent_sp s) rs), match_states (Mach.Returnstate s ms m) (Asm.State rs m'). Lemma exec_straight_steps: forall s f rs1 i c ep tf tc m1' m2 m2' sp ms2 ra, match_stack ge s m2 m2' ra sp -> Mem.extends m2 m2' -> retaddr_stored_at m2 m2' sp (Int.unsigned f.(fn_retaddr_ofs)) ra -> transl_code_at_pc ge (rs1 PC) f (i :: c) ep tf tc -> (forall k c (TR: transl_instr f i ep k = OK c), exists rs2, exec_straight tge tf c rs1 m1' k rs2 m2' /\ agree ms2 (Vptr sp Int.zero) rs2 /\ (r10_is_parent ep i = true -> rs2#IR10 = parent_sp s)) -> exists st', plus step tge (State rs1 m1') E0 st' /\ match_states (Mach.State s f (Vptr sp Int.zero) c ms2 m2) st'. Proof. intros. inversion H2; subst. monadInv H7. exploit H3; eauto. intros [rs2 [A [B C]]]. exists (State rs2 m2'); split. eapply exec_straight_exec; eauto. econstructor; eauto. eapply exec_straight_at; eauto. Qed. Lemma exec_straight_steps_goto: forall s f rs1 i c ep tf tc m1' m2 m2' sp ms2 lbl c' ra, match_stack ge s m2 m2' ra sp -> Mem.extends m2 m2' -> retaddr_stored_at m2 m2' sp (Int.unsigned f.(fn_retaddr_ofs)) ra -> Mach.find_label lbl f.(Mach.fn_code) = Some c' -> transl_code_at_pc ge (rs1 PC) f (i :: c) ep tf tc -> r10_is_parent ep i = false -> (forall k c (TR: transl_instr f i ep k = OK c), exists jmp, exists k', exists rs2, exec_straight tge tf c rs1 m1' (jmp :: k') rs2 m2' /\ agree ms2 (Vptr sp Int.zero) rs2 /\ exec_instr tge tf jmp rs2 m2' = goto_label tf lbl rs2 m2') -> exists st', plus step tge (State rs1 m1') E0 st' /\ match_states (Mach.State s f (Vptr sp Int.zero) c' ms2 m2) st'. Proof. intros. inversion H3; subst. monadInv H9. exploit H5; eauto. intros [jmp [k' [rs2 [A [B C]]]]]. generalize (functions_transl _ _ _ H7 H8); intro FN. generalize (transf_function_no_overflow _ _ H8); intro NOOV. exploit exec_straight_steps_2; eauto. intros [ofs' [PC2 CT2]]. exploit find_label_goto_label; eauto. intros [tc' [rs3 [GOTO [AT' OTH]]]]. exists (State rs3 m2'); split. eapply plus_right'. eapply exec_straight_steps_1; eauto. econstructor; eauto. eapply find_instr_tail. eauto. rewrite C. eexact GOTO. traceEq. econstructor; eauto. apply agree_exten with rs2; auto with asmgen. congruence. Qed. (** We need to show that, in the simulation diagram, we cannot take infinitely many Mach transitions that correspond to zero transitions on the ARM side. Actually, all Mach transitions correspond to at least one ARM transition, except the transition from [Machsem.Returnstate] to [Machsem.State]. So, the following integer measure will suffice to rule out the unwanted behaviour. *) Definition measure (s: Mach.state) : nat := match s with | Mach.State _ _ _ _ _ _ => 0%nat | Mach.Callstate _ _ _ _ => 0%nat | Mach.Returnstate _ _ _ => 1%nat end. Remark preg_of_not_R10: forall r, negb (mreg_eq r IT1) = true -> IR IR10 <> preg_of r. Proof. intros. change (IR IR10) with (preg_of IT1). red; intros. exploit preg_of_injective; eauto. intros; subst r. unfold proj_sumbool in H; rewrite dec_eq_true in H; discriminate. Qed. (** This is the simulation diagram. We prove it by case analysis on the Mach transition. *) Theorem step_simulation: forall S1 t S2, Mach.step ge S1 t S2 -> forall S1' (MS: match_states S1 S1'), (exists S2', plus step tge S1' t S2' /\ match_states S2 S2') \/ (measure S2 < measure S1 /\ t = E0 /\ match_states S2 S1')%nat. Proof. induction 1; intros; inv MS. - (* Mlabel *) left; eapply exec_straight_steps; eauto; intros. monadInv TR. econstructor; split. apply exec_straight_one. simpl; eauto. auto. split. apply agree_nextinstr; auto. simpl; congruence. - (* Mgetstack *) unfold load_stack in H. exploit Mem.loadv_extends; eauto. intros [v' [A B]]. rewrite (sp_val _ _ _ AG) in A. left; eapply exec_straight_steps; eauto. intros. simpl in TR. exploit loadind_correct; eauto with asmgen. intros [rs' [P [Q R]]]. exists rs'; split. eauto. split. eapply agree_set_mreg; eauto with asmgen. congruence. simpl; congruence. - (* Msetstack *) unfold store_stack in H. assert (Val.lessdef (rs src) (rs0 (preg_of src))). eapply preg_val; eauto. exploit Mem.storev_extends; eauto. intros [m2' [A B]]. left; eapply exec_straight_steps; eauto. eapply match_stack_storev; eauto. eapply retaddr_stored_at_storev; eauto. rewrite (sp_val _ _ _ AG) in A. intros. simpl in TR. exploit storeind_correct; eauto with asmgen. intros [rs' [P Q]]. exists rs'; split. eauto. split. change (undef_setstack rs) with rs. apply agree_exten with rs0; auto with asmgen. simpl; intros. rewrite Q; auto with asmgen. - (* Mgetparam *) unfold load_stack in *. exploit Mem.loadv_extends. eauto. eexact H. auto. intros [parent' [A B]]. rewrite (sp_val _ _ _ AG) in A. exploit lessdef_parent_sp; eauto. clear B; intros B; subst parent'. exploit Mem.loadv_extends. eauto. eexact H0. auto. intros [v' [C D]]. Opaque loadind. left; eapply exec_straight_steps; eauto; intros. destruct ep; monadInv TR. (* R10 contains parent *) exploit loadind_correct. eexact EQ. instantiate (2 := rs0). rewrite DXP; eauto. intros [rs1 [P [Q R]]]. exists rs1; split. eauto. split. eapply agree_set_mreg. eapply agree_set_mreg; eauto. congruence. auto with asmgen. simpl; intros. rewrite R; auto with asmgen. apply preg_of_not_R10; auto. (* GPR11 does not contain parent *) exploit loadind_int_correct. eexact A. instantiate (1 := IR10). intros [rs1 [P [Q R]]]. exploit loadind_correct. eexact EQ. instantiate (2 := rs1). rewrite Q. eauto. intros [rs2 [S [T U]]]. exists rs2; split. eapply exec_straight_trans; eauto. split. eapply agree_set_mreg. eapply agree_set_mreg. eauto. eauto. instantiate (1 := rs1#IR10 <- (rs2#IR10)). intros. rewrite Pregmap.gso; auto with asmgen. congruence. intros. unfold Pregmap.set. destruct (PregEq.eq r' IR10). congruence. auto with asmgen. simpl; intros. rewrite U; auto with asmgen. apply preg_of_not_R10; auto. - (* Mop *) assert (eval_operation tge (Vptr sp0 Int.zero) op rs##args m = Some v). rewrite <- H. apply eval_operation_preserved. exact symbols_preserved. exploit eval_operation_lessdef. eapply preg_vals; eauto. eauto. eexact H0. intros [v' [A B]]. rewrite (sp_val _ _ _ AG) in A. left; eapply exec_straight_steps; eauto; intros. simpl in TR. exploit transl_op_correct; eauto. intros [rs2 [P [Q R]]]. assert (S: Val.lessdef v (rs2 (preg_of res))) by (eapply Val.lessdef_trans; eauto). exists rs2; split. eauto. split. assert (agree (Regmap.set res v (undef_temps rs)) (Vptr sp0 Int.zero) rs2). eapply agree_set_undef_mreg; eauto with asmgen. unfold undef_op; destruct op; auto. change (undef_move rs) with rs. eapply agree_set_mreg; eauto. simpl. destruct op; try congruence. destruct ep; simpl; try congruence. intros. rewrite R; auto. apply preg_of_not_R10; auto. - (* Mload *) assert (eval_addressing tge (Vptr sp0 Int.zero) addr rs##args = Some a). rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved. exploit eval_addressing_lessdef. eapply preg_vals; eauto. eexact H1. intros [a' [A B]]. rewrite (sp_val _ _ _ AG) in A. exploit Mem.loadv_extends; eauto. intros [v' [C D]]. left; eapply exec_straight_steps; eauto; intros. simpl in TR. exploit transl_load_correct; eauto. intros [rs2 [P [Q R]]]. exists rs2; split. eauto. split. eapply agree_set_undef_mreg; eauto. congruence. simpl; congruence. - (* Mstore *) assert (eval_addressing tge (Vptr sp0 Int.zero) addr rs##args = Some a). rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved. exploit eval_addressing_lessdef. eapply preg_vals; eauto. eexact H1. intros [a' [A B]]. rewrite (sp_val _ _ _ AG) in A. assert (Val.lessdef (rs src) (rs0 (preg_of src))). eapply preg_val; eauto. exploit Mem.storev_extends; eauto. intros [m2' [C D]]. left; eapply exec_straight_steps; eauto. eapply match_stack_storev; eauto. eapply retaddr_stored_at_storev; eauto. intros. simpl in TR. exploit transl_store_correct; eauto. intros [rs2 [P Q]]. exists rs2; split. eauto. split. eapply agree_exten_temps; eauto. simpl; congruence. - (* Mcall *) inv AT. assert (NOOV: list_length_z (fn_code tf) <= Int.max_unsigned). eapply transf_function_no_overflow; eauto. destruct ros as [rf|fid]; simpl in H; monadInv H3. + (* Indirect call *) exploit Genv.find_funct_inv; eauto. intros [bf EQ2]. rewrite EQ2 in H; rewrite Genv.find_funct_find_funct_ptr in H. assert (rs0 x0 = Vptr bf Int.zero). exploit ireg_val; eauto. rewrite EQ2; intros LD; inv LD; auto. generalize (code_tail_next_int _ _ _ _ NOOV H4). intro CT1. assert (TCA: transl_code_at_pc ge (Vptr b (Int.add ofs Int.one)) f c false tf x). econstructor; eauto. left; econstructor; split. apply plus_one. eapply exec_step_internal. eauto. eapply functions_transl; eauto. eapply find_instr_tail; eauto. simpl. eauto. econstructor; eauto. econstructor; eauto. Simpl. rewrite <- H0; eexact TCA. change (Mem.valid_block m sp0). eapply retaddr_stored_at_valid; eauto. simpl. eapply agree_exten; eauto. intros. Simpl. Simpl. rewrite <- H0. exact I. + (* Direct call *) destruct (Genv.find_symbol ge fid) as [bf|] eqn:FS; try discriminate. generalize (code_tail_next_int _ _ _ _ NOOV H4). intro CT1. assert (TCA: transl_code_at_pc ge (Vptr b (Int.add ofs Int.one)) f c false tf x). econstructor; eauto. left; econstructor; split. apply plus_one. eapply exec_step_internal. eauto. eapply functions_transl; eauto. eapply find_instr_tail; eauto. simpl. unfold symbol_offset. rewrite symbols_preserved. rewrite FS. eauto. econstructor; eauto. econstructor; eauto. rewrite <- H0. eexact TCA. change (Mem.valid_block m sp0). eapply retaddr_stored_at_valid; eauto. simpl. eapply agree_exten; eauto. intros. Simpl. auto. rewrite <- H0. exact I. - (* Mtailcall *) inversion AT; subst. assert (NOOV: list_length_z (fn_code tf) <= Int.max_unsigned). eapply transf_function_no_overflow; eauto. rewrite (sp_val _ _ _ AG) in *. unfold load_stack in *. exploit Mem.loadv_extends. eauto. eexact H0. auto. simpl. intros [parent' [A B]]. exploit lessdef_parent_sp; eauto. intros. subst parent'. clear B. assert (C: Mem.loadv Mint32 m'0 (Val.add (rs0 SP) (Vint (fn_retaddr_ofs f))) = Some ra). Opaque Int.repr. erewrite agree_sp; eauto. simpl. rewrite Int.add_zero_l. eapply rsa_contains; eauto. exploit retaddr_stored_at_can_free; eauto. intros [m2' [E F]]. assert (M: match_stack ge s m'' m2' ra (Mem.nextblock m'')). apply match_stack_change_bound with stk. eapply match_stack_free_left; eauto. eapply match_stack_free_left; eauto. eapply match_stack_free_right; eauto. omega. apply Z.lt_le_incl. change (Mem.valid_block m'' stk). eapply Mem.valid_block_free_1; eauto. eapply Mem.valid_block_free_1; eauto. eapply retaddr_stored_at_valid; eauto. assert (X: forall k, exists rs2, exec_straight tge tf (loadind_int IR13 (fn_retaddr_ofs f) IR14 (Pfreeframe (fn_stacksize f) (fn_link_ofs f) :: k)) rs0 m'0 k rs2 m2' /\ rs2#SP = parent_sp s /\ rs2#RA = ra /\ forall r, r <> PC -> r <> SP -> r <> IR14 -> rs2#r = rs0#r). { intros. exploit loadind_int_correct. eexact C. intros [rs1 [P [Q R]]]. econstructor; split. eapply exec_straight_trans. eexact P. apply exec_straight_one. simpl. rewrite R; auto with asmgen. rewrite A. rewrite <- (sp_val _ _ _ AG). rewrite E. eauto. auto. split. Simpl. split. Simpl. intros. Simpl. } destruct ros as [rf|fid]; simpl in H; monadInv H6. + (* Indirect call *) exploit Genv.find_funct_inv; eauto. intros [bf EQ2]. rewrite EQ2 in H; rewrite Genv.find_funct_find_funct_ptr in H. assert (rs0 x0 = Vptr bf Int.zero). exploit ireg_val; eauto. rewrite EQ2; intros LD; inv LD; auto. destruct (X (Pbreg x0 sig :: x)) as [rs2 [P [Q [R S]]]]. exploit exec_straight_steps_2. eexact P. eauto. eauto. eapply functions_transl; eauto. eauto. intros [ofs' [Y Z]]. left; econstructor; split. eapply plus_right'. eapply exec_straight_exec; eauto. econstructor. eauto. eapply functions_transl; eauto. eapply find_instr_tail; eauto. simpl. reflexivity. traceEq. econstructor; eauto. Simpl. rewrite R; auto. constructor; intros. Simpl. Simpl. rewrite S; auto with asmgen. eapply preg_val; eauto. Simpl. rewrite S; auto with asmgen. rewrite <- (ireg_of_eq _ _ EQ1); auto with asmgen. rewrite <- (ireg_of_eq _ _ EQ1); auto with asmgen. Simpl. rewrite R. eapply retaddr_stored_at_type; eauto. + (* Direct call *) destruct (Genv.find_symbol ge fid) as [bf|] eqn:FS; try discriminate. destruct (X (Pbsymb fid sig :: x)) as [rs2 [P [Q [R S]]]]. exploit exec_straight_steps_2. eexact P. eauto. eauto. eapply functions_transl; eauto. eauto. intros [ofs' [Y Z]]. left; econstructor; split. eapply plus_right'. eapply exec_straight_exec; eauto. econstructor. eauto. eapply functions_transl; eauto. eapply find_instr_tail; eauto. simpl. unfold symbol_offset. rewrite symbols_preserved. rewrite FS. reflexivity. traceEq. econstructor; eauto. Simpl. rewrite R; auto. constructor; intros. Simpl. Simpl. rewrite S; auto with asmgen. eapply preg_val; eauto. Simpl. Simpl. rewrite R. eapply retaddr_stored_at_type; eauto. - (* Mbuiltin *) inv AT. monadInv H3. exploit functions_transl; eauto. intro FN. generalize (transf_function_no_overflow _ _ H2); intro NOOV. exploit external_call_mem_extends; eauto. eapply preg_vals; eauto. intros [vres' [m2' [A [B [C D]]]]]. left. econstructor; split. apply plus_one. eapply exec_step_builtin. eauto. eauto. eapply find_instr_tail; eauto. eapply external_call_symbols_preserved; eauto. exact symbols_preserved. exact varinfo_preserved. econstructor; eauto. eapply match_stack_extcall; eauto. intros; eapply external_call_max_perm; eauto. instantiate (2 := tf); instantiate (1 := x). Simpl. rewrite <- H0. simpl. econstructor; eauto. eapply code_tail_next_int; eauto. apply agree_nextinstr. eapply agree_set_undef_mreg; eauto. rewrite Pregmap.gss. auto. intros. Simpl. eapply retaddr_stored_at_extcall; eauto. intros; eapply external_call_max_perm; eauto. congruence. - (* Mannot *) inv AT. monadInv H4. exploit functions_transl; eauto. intro FN. generalize (transf_function_no_overflow _ _ H3); intro NOOV. exploit annot_arguments_match; eauto. intros [vargs' [P Q]]. exploit external_call_mem_extends; eauto. intros [vres' [m2' [A [B [C D]]]]]. left. econstructor; split. apply plus_one. eapply exec_step_annot. eauto. eauto. eapply find_instr_tail; eauto. eauto. eapply external_call_symbols_preserved; eauto. exact symbols_preserved. exact varinfo_preserved. eapply match_states_intro with (ep := false); eauto with coqlib. eapply match_stack_extcall; eauto. intros; eapply external_call_max_perm; eauto. unfold nextinstr. rewrite Pregmap.gss. rewrite <- H1; simpl. econstructor; eauto. eapply code_tail_next_int; eauto. apply agree_nextinstr. auto. eapply retaddr_stored_at_extcall; eauto. intros; eapply external_call_max_perm; eauto. congruence. - (* Mgoto *) inv AT. monadInv H3. exploit find_label_goto_label; eauto. intros [tc' [rs' [GOTO [AT2 INV]]]]. left; exists (State rs' m'); split. apply plus_one. econstructor; eauto. eapply functions_transl; eauto. eapply find_instr_tail; eauto. simpl; eauto. econstructor; eauto. eapply agree_exten; eauto with asmgen. congruence. - (* Mcond true *) exploit eval_condition_lessdef. eapply preg_vals; eauto. eauto. eauto. intros EC. left; eapply exec_straight_steps_goto; eauto. intros. simpl in TR. destruct (transl_cond_correct tge tf cond args _ rs0 m' _ TR) as [rs' [A [B C]]]. rewrite EC in B. econstructor; econstructor; econstructor; split. eexact A. split. eapply agree_exten_temps; eauto with asmgen. simpl. rewrite B. reflexivity. - (* Mcond false *) exploit eval_condition_lessdef. eapply preg_vals; eauto. eauto. eauto. intros EC. left; eapply exec_straight_steps; eauto. intros. simpl in TR. destruct (transl_cond_correct tge tf cond args _ rs0 m' _ TR) as [rs' [A [B C]]]. rewrite EC in B. econstructor; split. eapply exec_straight_trans. eexact A. apply exec_straight_one. simpl. rewrite B. reflexivity. auto. split. eapply agree_exten_temps; eauto with asmgen. intros; Simpl. simpl. congruence. - (* Mjumptable *) inv AT. monadInv H5. exploit functions_transl; eauto. intro FN. generalize (transf_function_no_overflow _ _ H4); intro NOOV. exploit find_label_goto_label. eauto. eauto. instantiate (2 := rs0#IR14 <- Vundef). Simpl. eauto. eauto. intros [tc' [rs' [A [B C]]]]. exploit ireg_val; eauto. rewrite H. intros LD; inv LD. left; econstructor; split. apply plus_one. econstructor; eauto. eapply find_instr_tail; eauto. simpl. rewrite <- H8. unfold Mach.label in H0; unfold label; rewrite H0. eexact A. econstructor; eauto. eapply agree_exten_temps; eauto. intros. rewrite C; auto with asmgen. Simpl. congruence. - (* Mreturn *) inversion AT; subst. assert (NOOV: list_length_z (fn_code tf) <= Int.max_unsigned). eapply transf_function_no_overflow; eauto. rewrite (sp_val _ _ _ AG) in *. unfold load_stack in *. exploit Mem.loadv_extends. eauto. eexact H. auto. simpl. intros [parent' [A B]]. exploit lessdef_parent_sp; eauto. intros. subst parent'. clear B. assert (C: Mem.loadv Mint32 m'0 (Val.add (rs0 SP) (Vint (fn_retaddr_ofs f))) = Some ra). Opaque Int.repr. erewrite agree_sp; eauto. simpl. rewrite Int.add_zero_l. eapply rsa_contains; eauto. exploit retaddr_stored_at_can_free; eauto. intros [m2' [E F]]. assert (M: match_stack ge s m'' m2' ra (Mem.nextblock m'')). apply match_stack_change_bound with stk. eapply match_stack_free_left; eauto. eapply match_stack_free_left; eauto. eapply match_stack_free_right; eauto. omega. apply Z.lt_le_incl. change (Mem.valid_block m'' stk). eapply Mem.valid_block_free_1; eauto. eapply Mem.valid_block_free_1; eauto. eapply retaddr_stored_at_valid; eauto. monadInv H5. assert (X: forall k, exists rs2, exec_straight tge tf (loadind_int IR13 (fn_retaddr_ofs f) IR14 (Pfreeframe (fn_stacksize f) (fn_link_ofs f) :: k)) rs0 m'0 k rs2 m2' /\ rs2#SP = parent_sp s /\ rs2#RA = ra /\ forall r, r <> PC -> r <> SP -> r <> IR14 -> rs2#r = rs0#r). { intros. exploit loadind_int_correct. eexact C. intros [rs1 [P [Q R]]]. econstructor; split. eapply exec_straight_trans. eexact P. apply exec_straight_one. simpl. rewrite R; auto with asmgen. rewrite A. rewrite <- (sp_val _ _ _ AG). rewrite E. eauto. auto. split. Simpl. split. Simpl. intros. Simpl. } destruct (X (Pbreg IR14 (Mach.fn_sig f) :: x)) as [rs2 [P [Q [R S]]]]. exploit exec_straight_steps_2. eexact P. eauto. eauto. eapply functions_transl; eauto. eauto. intros [ofs' [Y Z]]. left; econstructor; split. eapply plus_right'. eapply exec_straight_exec; eauto. econstructor. eauto. eapply functions_transl; eauto. eapply find_instr_tail; eauto. simpl. reflexivity. traceEq. econstructor; eauto. Simpl. rewrite R; auto. constructor; intros. Simpl. Simpl. rewrite S; auto with asmgen. eapply preg_val; eauto. - (* internal function *) exploit functions_translated; eauto. intros [tf [A B]]. monadInv B. generalize EQ; intros EQ'. monadInv EQ'. destruct (zlt Int.max_unsigned (list_length_z (fn_code x0))); inversion EQ1. clear EQ1. monadInv EQ0. unfold store_stack in *. exploit Mem.alloc_extends. eauto. eauto. apply Zle_refl. apply Zle_refl. intros [m1' [C D]]. assert (E: Mem.extends m2 m1') by (eapply Mem.free_left_extends; eauto). exploit Mem.storev_extends. eexact E. eexact H1. eauto. eauto. intros [m2' [F G]]. exploit retaddr_stored_at_can_alloc. eexact H. eauto. eauto. eauto. eauto. auto. auto. auto. auto. eauto. intros [m3' [P [Q R]]]. (* Execution of function prologue *) set (rs2 := nextinstr (rs0#IR10 <- (parent_sp s) #IR13 <- (Vptr stk Int.zero))). set (rs3 := nextinstr rs2). assert (EXEC_PROLOGUE: exec_straight tge x (fn_code x) rs0 m' x1 rs3 m3'). rewrite <- H5 at 2; unfold fn_code. apply exec_straight_two with rs2 m2'. unfold exec_instr. rewrite C. fold sp. rewrite <- (sp_val _ _ _ AG). unfold chunk_of_type in F. rewrite F. auto. simpl. auto. simpl. unfold exec_store. change (rs2 IR14) with (rs0 IR14). rewrite Int.add_zero_l. simpl. rewrite P. auto. auto. auto. left; exists (State rs3 m3'); split. eapply exec_straight_steps_1; eauto. omega. constructor. econstructor; eauto. assert (STK: stk = Mem.nextblock m) by (eapply Mem.alloc_result; eauto). rewrite <- STK in STACKS. simpl in F. simpl in H1. eapply match_stack_invariant; eauto. intros. eapply Mem.perm_alloc_4; eauto. eapply Mem.perm_free_3; eauto. eapply Mem.perm_store_2; eauto. unfold block; omega. intros. eapply Mem.perm_store_1; eauto. eapply Mem.perm_store_1; eauto. eapply Mem.perm_alloc_1; eauto. intros. erewrite Mem.load_store_other. 2: eauto. erewrite Mem.load_store_other. 2: eauto. eapply Mem.load_alloc_other; eauto. left; unfold block; omega. left; unfold block; omega. change (rs3 PC) with (Val.add (Val.add (rs0 PC) Vone) Vone). rewrite ATPC. simpl. constructor; eauto. subst x. eapply code_tail_next_int. omega. eapply code_tail_next_int. omega. constructor. unfold rs3, rs2. apply agree_nextinstr. apply agree_nextinstr. eapply agree_change_sp. apply agree_exten_temps with rs0; eauto. intros. Simpl. congruence. - (* external function *) exploit functions_translated; eauto. intros [tf [A B]]. simpl in B. inv B. exploit extcall_arguments_match; eauto. intros [args' [C D]]. exploit external_call_mem_extends; eauto. intros [res' [m2' [P [Q [R S]]]]]. left; econstructor; split. apply plus_one. eapply exec_step_external; eauto. eapply external_call_symbols_preserved; eauto. exact symbols_preserved. exact varinfo_preserved. econstructor; eauto. rewrite Pregmap.gss. apply match_stack_change_bound with (Mem.nextblock m). eapply match_stack_extcall; eauto. intros. eapply external_call_max_perm; eauto. eapply external_call_nextblock; eauto. unfold loc_external_result. eapply agree_set_mreg; eauto. rewrite Pregmap.gso; auto with asmgen. rewrite Pregmap.gss. auto. intros; Simpl. - (* return *) inv STACKS. simpl in *. right. split. omega. split. auto. econstructor; eauto. congruence. Qed. Lemma transf_initial_states: forall st1, Mach.initial_state prog st1 -> exists st2, Asm.initial_state tprog st2 /\ match_states st1 st2. Proof. intros. inversion H. unfold ge0 in *. exploit functions_translated; eauto. intros [tf [A B]]. econstructor; split. econstructor. eapply Genv.init_mem_transf_partial; eauto. replace (symbol_offset (Genv.globalenv tprog) (prog_main tprog) Int.zero) with (Vptr b Int.zero). econstructor; eauto. constructor. apply Mem.extends_refl. split. auto. intros. rewrite Regmap.gi. auto. reflexivity. exact I. unfold symbol_offset. rewrite (transform_partial_program_main _ _ TRANSF). rewrite symbols_preserved. unfold ge; rewrite H1. auto. Qed. Lemma transf_final_states: forall st1 st2 r, match_states st1 st2 -> Mach.final_state st1 r -> Asm.final_state st2 r. Proof. intros. inv H0. inv H. inv STACKS. constructor. auto. compute in H1. generalize (preg_val _ _ _ R0 AG). rewrite H1. intros LD; inv LD. auto. Qed. Theorem transf_program_correct: forward_simulation (Mach.semantics prog) (Asm.semantics tprog). Proof. eapply forward_simulation_star with (measure := measure). eexact symbols_preserved. eexact transf_initial_states. eexact transf_final_states. exact step_simulation. Qed. End PRESERVATION.