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Diffstat (limited to 'mppa_k1c/lib/Machblockgenproof.v')
| -rw-r--r-- | mppa_k1c/lib/Machblockgenproof.v | 810 |
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diff --git a/mppa_k1c/lib/Machblockgenproof.v b/mppa_k1c/lib/Machblockgenproof.v new file mode 100644 index 00000000..91be5e2e --- /dev/null +++ b/mppa_k1c/lib/Machblockgenproof.v @@ -0,0 +1,810 @@ +Require Import Coqlib. +Require Import Maps. +Require Import AST. +Require Import Integers. +Require Import Values. +Require Import Memory. +Require Import Globalenvs. +Require Import Events. +Require Import Smallstep. +Require Import Op. +Require Import Locations. +Require Import Conventions. +Require Stacklayout. +Require Import Mach. +Require Import Linking. +Require Import Machblock. +Require Import Machblockgen. +Require Import ForwardSimulationBlock. + +Ltac subst_is_trans_code H := + rewrite is_trans_code_inv in H; + rewrite <- H in * |- *; + rewrite <- is_trans_code_inv in H. + +Definition inv_trans_rao (rao: function -> code -> ptrofs -> Prop) (f: Mach.function) (c: Mach.code) := + rao (transf_function f) (trans_code c). + +Definition match_prog (p: Mach.program) (tp: Machblock.program) := + match_program (fun _ f tf => tf = transf_fundef f) eq p tp. + +Lemma transf_program_match: forall p tp, transf_program p = tp -> match_prog p tp. +Proof. + intros. rewrite <- H. eapply match_transform_program; eauto. +Qed. + +Definition trans_stackframe (msf: Mach.stackframe) : stackframe := + match msf with + | Mach.Stackframe f sp retaddr c => Stackframe f sp retaddr (trans_code c) + end. + +Fixpoint trans_stack (mst: list Mach.stackframe) : list stackframe := + match mst with + | nil => nil + | msf :: mst0 => (trans_stackframe msf) :: (trans_stack mst0) + end. + +Definition trans_state (ms: Mach.state): state := + match ms with + | Mach.State s f sp c rs m => State (trans_stack s) f sp (trans_code c) rs m + | Mach.Callstate s f rs m => Callstate (trans_stack s) f rs m + | Mach.Returnstate s rs m => Returnstate (trans_stack s) rs m + end. + +Section PRESERVATION. + +Local Open Scope nat_scope. + +Variable prog: Mach.program. +Variable tprog: Machblock.program. +Hypothesis TRANSF: match_prog prog tprog. +Let ge := Genv.globalenv prog. +Let tge := Genv.globalenv tprog. + + +Variable rao: function -> code -> ptrofs -> Prop. + +Definition match_states: Mach.state -> state -> Prop + := ForwardSimulationBlock.match_states (Mach.semantics (inv_trans_rao rao) prog) (Machblock.semantics rao tprog) trans_state. + +Lemma match_states_trans_state s1: match_states s1 (trans_state s1). +Proof. + apply match_states_trans_state. +Qed. + +Local Hint Resolve match_states_trans_state. + +Lemma symbols_preserved: + forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s. +Proof (Genv.find_symbol_match TRANSF). + +Lemma senv_preserved: + Senv.equiv ge tge. +Proof (Genv.senv_match TRANSF). + +Lemma init_mem_preserved: + forall m, + Genv.init_mem prog = Some m -> + Genv.init_mem tprog = Some m. +Proof (Genv.init_mem_transf TRANSF). + +Lemma prog_main_preserved: + prog_main tprog = prog_main prog. +Proof (match_program_main TRANSF). + +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 = tf. +Proof. + intros. + exploit (Genv.find_funct_ptr_match TRANSF); eauto. intro. + destruct H0 as (cunit & tf & A & B & C). + eapply ex_intro. intuition; eauto. subst. eapply A. +Qed. + +Lemma find_function_ptr_same: + forall s rs, + Mach.find_function_ptr ge s rs = find_function_ptr tge s rs. +Proof. + intros. unfold Mach.find_function_ptr. unfold find_function_ptr. + destruct s; auto. + rewrite symbols_preserved; auto. +Qed. + +Lemma find_funct_ptr_same: + forall f f0, + Genv.find_funct_ptr ge f = Some (Internal f0) -> + Genv.find_funct_ptr tge f = Some (Internal (transf_function f0)). +Proof. + intros. exploit (Genv.find_funct_ptr_transf TRANSF); eauto. +Qed. + +Lemma find_funct_ptr_same_external: + forall f f0, + Genv.find_funct_ptr ge f = Some (External f0) -> + Genv.find_funct_ptr tge f = Some (External f0). +Proof. + intros. exploit (Genv.find_funct_ptr_transf TRANSF); eauto. +Qed. + +Lemma parent_sp_preserved: + forall s, + Mach.parent_sp s = parent_sp (trans_stack s). +Proof. + unfold parent_sp. unfold Mach.parent_sp. destruct s; simpl; auto. + unfold trans_stackframe. destruct s; simpl; auto. +Qed. + +Lemma parent_ra_preserved: + forall s, + Mach.parent_ra s = parent_ra (trans_stack s). +Proof. + unfold parent_ra. unfold Mach.parent_ra. destruct s; simpl; auto. + unfold trans_stackframe. destruct s; simpl; auto. +Qed. + +Lemma external_call_preserved: + forall ef args m t res m', + external_call ef ge args m t res m' -> + external_call ef tge args m t res m'. +Proof. + intros. eapply external_call_symbols_preserved; eauto. + apply senv_preserved. +Qed. + +Lemma Mach_find_label_split l i c c': + Mach.find_label l (i :: c) = Some c' -> + (i=Mlabel l /\ c' = c) \/ (i <> Mlabel l /\ Mach.find_label l c = Some c'). +Proof. + intros H. + destruct i; try (constructor 2; split; auto; discriminate ). + destruct (peq l0 l) as [P|P]. + - constructor. subst l0; split; auto. + revert H. unfold Mach.find_label. simpl. rewrite peq_true. + intros H; injection H; auto. + - constructor 2. split. + + intro F. injection F. intros. contradict P; auto. + + revert H. unfold Mach.find_label. simpl. rewrite peq_false; auto. +Qed. + +Lemma find_label_is_end_block_not_label i l c bl: + is_end_block (trans_inst i) bl -> + is_trans_code c bl -> + i <> Mlabel l -> find_label l (add_to_new_bblock (trans_inst i) :: bl) = find_label l bl. +Proof. + intros H H0 H1. + unfold find_label. + remember (is_label l _) as b. + cutrewrite (b = false); auto. + subst; unfold is_label. + destruct i; simpl in * |- *; try (destruct (in_dec l nil); intuition). + inversion H. + destruct (in_dec l (l0::nil)) as [H6|H6]; auto. + simpl in H6; intuition try congruence. +Qed. + +Lemma find_label_at_begin l bh bl: + In l (header bh) + -> find_label l (bh :: bl) = Some (bh::bl). +Proof. + unfold find_label; rewrite is_label_correct_true; intro H; rewrite H; simpl; auto. +Qed. + +Lemma find_label_add_label_diff l bh bl: + ~(In l (header bh)) -> + find_label l (bh::bl) = find_label l bl. +Proof. + unfold find_label; rewrite is_label_correct_false; intro H; rewrite H; simpl; auto. +Qed. + +Definition concat (h: list label) (c: code): code := + match c with + | nil => {| header := h; body := nil; exit := None |}::nil + | b::c' => {| header := h ++ (header b); body := body b; exit := exit b |}::c' + end. + +Lemma find_label_transcode_preserved: + forall l c c', + Mach.find_label l c = Some c' -> + exists h, In l h /\ find_label l (trans_code c) = Some (concat h (trans_code c')). +Proof. + intros l c. remember (trans_code _) as bl. + rewrite <- is_trans_code_inv in * |-. + induction Heqbl. + + (* Tr_nil *) + intros; exists (l::nil); simpl in * |- *; intuition. + discriminate. + + (* Tr_end_block *) + intros. + exploit Mach_find_label_split; eauto. + clear H0; destruct 1 as [(H0&H2)|(H0&H2)]. + - subst. rewrite find_label_at_begin; simpl; auto. + inversion H as [mbi H1 H2| | ]. + subst. + inversion Heqbl. + subst. + exists (l :: nil); simpl; eauto. + - exploit IHHeqbl; eauto. + destruct 1 as (h & H3 & H4). + exists h. + split; auto. + erewrite find_label_is_end_block_not_label;eauto. + + (* Tr_add_label *) + intros. + exploit Mach_find_label_split; eauto. + clear H0; destruct 1 as [(H0&H2)|(H0&H2)]. + - subst. + inversion H0 as [H1]. + clear H0. + erewrite find_label_at_begin; simpl; eauto. + subst_is_trans_code Heqbl. + exists (l :: nil); simpl; eauto. + - subst; assert (H: l0 <> l); try congruence; clear H0. + exploit IHHeqbl; eauto. + clear IHHeqbl Heqbl. + intros (h & H3 & H4). + simpl; unfold is_label, add_label; simpl. + destruct (in_dec l (l0::header bh)) as [H5|H5]; simpl in H5. + * destruct H5; try congruence. + exists (l0::h); simpl; intuition. + rewrite find_label_at_begin in H4; auto. + apply f_equal. inversion H4 as [H5]. clear H4. + destruct (trans_code c'); simpl in * |- *; + inversion H5; subst; simpl; auto. + * exists h. intuition. + erewrite <- find_label_add_label_diff; eauto. + + (* Tr_add_basic *) + intros. + exploit Mach_find_label_split; eauto. + destruct 1 as [(H2&H3)|(H2&H3)]. + rewrite H2 in H. unfold trans_inst in H. congruence. + exploit IHHeqbl; eauto. + clear IHHeqbl Heqbl. + intros (h & H4 & H5). + rewrite find_label_add_label_diff; auto. + rewrite find_label_add_label_diff in H5; eauto. + rewrite H0; auto. +Qed. + +Lemma find_label_preserved: + forall l f c, + Mach.find_label l (Mach.fn_code f) = Some c -> + exists h, In l h /\ find_label l (fn_code (transf_function f)) = Some (concat h (trans_code c)). +Proof. + intros. cutrewrite ((fn_code (transf_function f)) = trans_code (Mach.fn_code f)); eauto. + apply find_label_transcode_preserved; auto. +Qed. + +Lemma mem_free_preserved: + forall m stk f, + Mem.free m stk 0 (Mach.fn_stacksize f) = Mem.free m stk 0 (fn_stacksize (transf_function f)). +Proof. + intros. auto. +Qed. + +Local Hint Resolve symbols_preserved senv_preserved init_mem_preserved prog_main_preserved functions_translated + parent_sp_preserved. + + +Definition dist_end_block_code (c: Mach.code) := + match trans_code c with + | nil => 0 + | bh::_ => (size bh-1)%nat + end. + +Definition dist_end_block (s: Mach.state): nat := + match s with + | Mach.State _ _ _ c _ _ => dist_end_block_code c + | _ => 0 + end. + +Local Hint Resolve exec_nil_body exec_cons_body. +Local Hint Resolve exec_MBgetstack exec_MBsetstack exec_MBgetparam exec_MBop exec_MBload exec_MBstore. + +Lemma size_add_label l bh: size (add_label l bh) = size bh + 1. +Proof. + unfold add_label, size; simpl; omega. +Qed. + +Lemma size_add_basic bi bh: header bh = nil -> size (add_basic bi bh) = size bh + 1. +Proof. + intro H. unfold add_basic, size; rewrite H; simpl. omega. +Qed. + + +Lemma size_add_to_newblock i: size (add_to_new_bblock i) = 1. +Proof. + destruct i; auto. +Qed. + + +Lemma dist_end_block_code_simu_mid_block i c: + dist_end_block_code (i::c) <> 0 -> + (dist_end_block_code (i::c) = Datatypes.S (dist_end_block_code c)). +Proof. + unfold dist_end_block_code. + remember (trans_code (i::c)) as bl. + rewrite <- is_trans_code_inv in Heqbl. + inversion Heqbl as [|bl0 H| |]; subst; clear Heqbl. + - rewrite size_add_to_newblock; omega. + - rewrite size_add_label; + subst_is_trans_code H. + omega. + - rewrite size_add_basic; auto. + subst_is_trans_code H. + omega. +Qed. + +Local Hint Resolve dist_end_block_code_simu_mid_block. + + +Lemma size_nonzero c b bl: + is_trans_code c (b :: bl) -> size b <> 0. +Proof. + intros H; inversion H; subst. + - rewrite size_add_to_newblock; omega. + - rewrite size_add_label; omega. + - rewrite size_add_basic; auto; omega. +Qed. + +Inductive is_header: list label -> Mach.code -> Mach.code -> Prop := + | header_empty : is_header nil nil nil + | header_not_label i c: (forall l, i <> Mlabel l) -> is_header nil (i::c) (i::c) + | header_is_label l h c c0: is_header h c c0 -> is_header (l::h) ((Mlabel l)::c) c0 + . + +Inductive is_body: list basic_inst -> Mach.code -> Mach.code -> Prop := + | body_empty : is_body nil nil nil + | body_not_bi i c: (forall bi, (trans_inst i) <> (MB_basic bi)) -> is_body nil (i::c) (i::c) + | body_is_bi i lbi c0 c1 bi: (trans_inst i) = MB_basic bi -> is_body lbi c0 c1 -> is_body (bi::lbi) (i::c0) c1 + . + +Inductive is_exit: option control_flow_inst -> Mach.code -> Mach.code -> Prop := + | exit_empty: is_exit None nil nil + | exit_not_cfi i c: (forall cfi, (trans_inst i) <> MB_cfi cfi) -> is_exit None (i::c) (i::c) + | exit_is_cfi i c cfi: (trans_inst i) = MB_cfi cfi -> is_exit (Some cfi) (i::c) c + . + +Lemma Mlabel_is_not_basic i: + forall bi, trans_inst i = MB_basic bi -> forall l, i <> Mlabel l. +Proof. +intros. +unfold trans_inst in H. +destruct i; congruence. +Qed. + +Lemma Mlabel_is_not_cfi i: + forall cfi, trans_inst i = MB_cfi cfi -> forall l, i <> Mlabel l. +Proof. +intros. +unfold trans_inst in H. +destruct i; congruence. +Qed. + +Lemma MBbasic_is_not_cfi i: + forall cfi, trans_inst i = MB_cfi cfi -> forall bi, trans_inst i <> MB_basic bi. +Proof. +intros. +unfold trans_inst in H. +unfold trans_inst. +destruct i; congruence. +Qed. + + +Local Hint Resolve Mlabel_is_not_cfi. +Local Hint Resolve MBbasic_is_not_cfi. + +Lemma add_to_new_block_is_label i: + header (add_to_new_bblock (trans_inst i)) <> nil -> exists l, i = Mlabel l. +Proof. + intros. + unfold add_to_new_bblock in H. + destruct (trans_inst i) eqn : H1. + + exists lbl. + unfold trans_inst in H1. + destruct i; congruence. + + unfold add_basic in H; simpl in H; congruence. + + unfold cfi_bblock in H; simpl in H; congruence. +Qed. + +Local Hint Resolve Mlabel_is_not_basic. + +Lemma trans_code_decompose c: forall b bl, + is_trans_code c (b::bl) -> + exists c0 c1 c2, is_header (header b) c c0 /\ is_body (body b) c0 c1 /\ is_exit (exit b) c1 c2 /\ is_trans_code c2 bl. +Proof. + induction c as [|i c]. + { (* nil => absurd *) intros b bl H; inversion H. } + intros b bl H; remember (trans_inst i) as ti. + destruct ti as [lbl|bi|cfi]; + inversion H as [|d0 d1 d2 H0 H1| |]; subst; + try (rewrite <- Heqti in * |- *); simpl in * |- *; + try congruence. + + (* label at end block *) + inversion H1; subst. inversion H0; subst. + assert (X:i=Mlabel lbl). { destruct i; simpl in Heqti; congruence. } + subst. repeat econstructor; eauto. + + (* label at mid block *) + exploit IHc; eauto. + intros (c0 & c1 & c2 & H1 & H2 & H3 & H4). + repeat econstructor; eauto. + + (* basic at end block *) + inversion H1; subst. + lapply (Mlabel_is_not_basic i bi); auto. + intro H2. + - inversion H0; subst. + assert (X:(trans_inst i) = MB_basic bi ). { repeat econstructor; congruence. } + repeat econstructor; congruence. + - exists (i::c), c, c. + repeat econstructor; eauto; inversion H0; subst; repeat econstructor; simpl; try congruence. + * exploit (add_to_new_block_is_label i0); eauto. + intros (l & H8); subst; simpl; congruence. + * exploit H3; eauto. + * exploit (add_to_new_block_is_label i0); eauto. + intros (l & H8); subst; simpl; congruence. + + (* basic at mid block *) + inversion H1; subst. + exploit IHc; eauto. + intros (c0 & c1 & c2 & H3 & H4 & H5 & H6). + exists (i::c0), c1, c2. + repeat econstructor; eauto. + rewrite H2 in H3. + inversion H3; econstructor; eauto. + + (* cfi at end block *) + inversion H1; subst; + repeat econstructor; eauto. +Qed. + + +Lemma step_simu_header st f sp rs m s c h c' t: + is_header h c c' -> + starN (Mach.step (inv_trans_rao rao)) (Genv.globalenv prog) (length h) (Mach.State st f sp c rs m) t s -> + s = Mach.State st f sp c' rs m /\ t = E0. +Proof. + induction 1; simpl; intros hs; try (inversion hs; tauto). + inversion hs as [|n1 s1 t1 t2 s2 t3 s3 H1]. inversion H1. subst. auto. +Qed. + + + +Lemma step_simu_basic_step (i: Mach.instruction) (bi: basic_inst) (c: Mach.code) s f sp rs m (t:trace) (s':Mach.state): + trans_inst i = MB_basic bi -> + Mach.step (inv_trans_rao rao) ge (Mach.State s f sp (i::c) rs m) t s' -> + exists rs' m', s'=Mach.State s f sp c rs' m' /\ t=E0 /\ basic_step tge (trans_stack s) f sp rs m bi rs' m'. +Proof. + destruct i; simpl in * |-; + (discriminate + || (intro H; inversion_clear H; intro X; inversion_clear X; eapply ex_intro; eapply ex_intro; intuition eauto)). + - eapply exec_MBgetparam; eauto. exploit (functions_translated); eauto. intro. + destruct H3 as (tf & A & B). subst. eapply A. + all: simpl; rewrite <- parent_sp_preserved; auto. + - eapply exec_MBop; eauto. rewrite <- H. destruct o; simpl; auto. destruct (rs ## l); simpl; auto. + unfold Genv.symbol_address; rewrite symbols_preserved; auto. + - eapply exec_MBload; eauto; rewrite <- H; destruct a; simpl; auto; destruct (rs ## l); simpl; auto; + unfold Genv.symbol_address; rewrite symbols_preserved; auto. + - eapply exec_MBload_notrap1; eauto; rewrite <- H; destruct a; simpl; auto; destruct (rs ## l); simpl; auto; + unfold Genv.symbol_address; rewrite symbols_preserved; auto. + - eapply exec_MBload_notrap2; eauto; rewrite <- H; destruct a; simpl; auto; destruct (rs ## l); simpl; auto; + unfold Genv.symbol_address; rewrite symbols_preserved; auto. + - eapply exec_MBstore; eauto; rewrite <- H; destruct a; simpl; auto; destruct (rs ## l); simpl; auto; + unfold Genv.symbol_address; rewrite symbols_preserved; auto. +Qed. + + +Lemma star_step_simu_body_step s f sp c bdy c': + is_body bdy c c' -> forall rs m t s', + starN (Mach.step (inv_trans_rao rao)) ge (length bdy) (Mach.State s f sp c rs m) t s' -> + exists rs' m', s'=Mach.State s f sp c' rs' m' /\ t=E0 /\ body_step tge (trans_stack s) f sp bdy rs m rs' m'. +Proof. + induction 1; simpl. + + intros. inversion H. exists rs. exists m. auto. + + intros. inversion H0. exists rs. exists m. auto. + + intros. inversion H1; subst. + exploit (step_simu_basic_step ); eauto. + destruct 1 as [ rs1 [ m1 Hs]]. + destruct Hs as [Hs1 [Hs2 Hs3]]. + destruct (IHis_body rs1 m1 t2 s') as [rs2 Hb]. rewrite <- Hs1; eauto. + destruct Hb as [m2 [Hb1 [Hb2 Hb3]]]. + exists rs2, m2. + rewrite Hs2, Hb2; eauto. + Qed. + +Local Hint Resolve exec_MBcall exec_MBtailcall exec_MBbuiltin exec_MBgoto exec_MBcond_true exec_MBcond_false exec_MBjumptable exec_MBreturn exec_Some_exit exec_None_exit. +Local Hint Resolve eval_builtin_args_preserved external_call_symbols_preserved find_funct_ptr_same. + + +Lemma match_states_concat_trans_code st f sp c rs m h: + match_states (Mach.State st f sp c rs m) (State (trans_stack st) f sp (concat h (trans_code c)) rs m). +Proof. + intros; constructor 1; simpl. + + intros (t0 & s1' & H0) t s'. + remember (trans_code _) as bl. + destruct bl as [|bh bl]. + { rewrite <- is_trans_code_inv in Heqbl; inversion Heqbl; inversion H0; congruence. } + clear H0. + simpl; constructor 1; + intros X; inversion X as [d1 d2 d3 d4 d5 d6 d7 rs' m' d10 d11 X1 X2| | | ]; subst; simpl in * |- *; + eapply exec_bblock; eauto; simpl; + inversion X2 as [cfi d1 d2 d3 H1|]; subst; eauto; + inversion H1; subst; eauto. + + intros H r; constructor 1; intro X; inversion X. +Qed. + +Lemma step_simu_cfi_step (i: Mach.instruction) (cfi: control_flow_inst) (c: Mach.code) (blc:code) stk f sp rs m (t:trace) (s':Mach.state) b: + trans_inst i = MB_cfi cfi -> + is_trans_code c blc -> + Mach.step (inv_trans_rao rao) ge (Mach.State stk f sp (i::c) rs m) t s' -> + exists s2, cfi_step rao tge cfi (State (trans_stack stk) f sp (b::blc) rs m) t s2 /\ match_states s' s2. +Proof. + destruct i; simpl in * |-; + (intro H; intro Htc;apply is_trans_code_inv in Htc;rewrite Htc;inversion_clear H;intro X; inversion_clear X). + * eapply ex_intro. + intuition auto. + eapply exec_MBcall;eauto. + rewrite <-H; exploit (find_function_ptr_same); eauto. + * eapply ex_intro. + intuition auto. + eapply exec_MBtailcall;eauto. + - rewrite <-H; exploit (find_function_ptr_same); eauto. + - simpl; rewrite <- parent_sp_preserved; auto. + - simpl; rewrite <- parent_ra_preserved; auto. + * eapply ex_intro. + intuition auto. + eapply exec_MBbuiltin ;eauto. + * exploit find_label_transcode_preserved; eauto. + intros (x & X1 & X2). + eapply ex_intro; constructor 1; [ idtac | eapply match_states_concat_trans_code ]; eauto. + * exploit find_label_transcode_preserved; eauto. + intros (x & X1 & X2). + eapply ex_intro; constructor 1; [ idtac | eapply match_states_concat_trans_code ]; eauto. + * eapply ex_intro; constructor 1; [ idtac | eapply match_states_trans_state ]; eauto. + eapply exec_MBcond_false; eauto. + * exploit find_label_transcode_preserved; eauto. intros (h & X1 & X2). + eapply ex_intro; constructor 1; [ idtac | eapply match_states_concat_trans_code ]; eauto. + * eapply ex_intro; constructor 1; [ idtac | eapply match_states_trans_state ]; eauto. + eapply exec_MBreturn; eauto. + rewrite parent_sp_preserved in H0; subst; auto. + rewrite parent_ra_preserved in H1; subst; auto. +Qed. + +Lemma step_simu_exit_step stk f sp rs m t s1 e c c' b blc: + is_exit e c c' -> is_trans_code c' blc -> + starN (Mach.step (inv_trans_rao rao)) (Genv.globalenv prog) (length_opt e) (Mach.State stk f sp c rs m) t s1 -> + exists s2, exit_step rao tge e (State (trans_stack stk) f sp (b::blc) rs m) t s2 /\ match_states s1 s2. +Proof. + destruct 1. + - (* None *) + intros H0 H1. inversion H1. exists (State (trans_stack stk) f sp blc rs m). + split; eauto. + apply is_trans_code_inv in H0. + rewrite H0. + apply match_states_trans_state. + - (* None *) + intros H0 H1. inversion H1. exists (State (trans_stack stk) f sp blc rs m). + split; eauto. + apply is_trans_code_inv in H0. + rewrite H0. + apply match_states_trans_state. + - (* Some *) + intros H0 H1. + inversion H1; subst. + exploit (step_simu_cfi_step); eauto. + intros [s2 [Hcfi1 Hcfi3]]. + inversion H4. subst; simpl. + autorewrite with trace_rewrite. + exists s2. + split;eauto. +Qed. + +Lemma simu_end_block: + forall s1 t s1', + starN (Mach.step (inv_trans_rao rao)) ge (Datatypes.S (dist_end_block s1)) s1 t s1' -> + exists s2', step rao tge (trans_state s1) t s2' /\ match_states s1' s2'. +Proof. + destruct s1; simpl. + + (* State *) + remember (trans_code _) as tc. + rewrite <- is_trans_code_inv in Heqtc. + intros t s1 H. + destruct tc as [|b bl]. + { (* nil => absurd *) + inversion Heqtc. subst. + unfold dist_end_block_code; simpl. + inversion_clear H; + inversion_clear H0. + } + assert (X: Datatypes.S (dist_end_block_code c) = (size b)). + { + unfold dist_end_block_code. + subst_is_trans_code Heqtc. + lapply (size_nonzero c b bl); auto. + omega. + } + rewrite X in H; unfold size in H. + (* decomposition of starN in 3 parts: header + body + exit *) + destruct (starN_split (Mach.semantics (inv_trans_rao rao) prog) _ _ _ _ H _ _ refl_equal) as (t3&t4&s1'&H0&H3&H4). + subst t; clear X H. + destruct (starN_split (Mach.semantics (inv_trans_rao rao) prog) _ _ _ _ H0 _ _ refl_equal) as (t1&t2&s1''&H&H1&H2). + subst t3; clear H0. + exploit trans_code_decompose; eauto. clear Heqtc. + intros (c0&c1&c2&Hc0&Hc1&Hc2&Heqtc). + (* header steps *) + exploit step_simu_header; eauto. + clear H; intros [X1 X2]; subst. + (* body steps *) + exploit (star_step_simu_body_step); eauto. + clear H1; intros (rs'&m'&H0&H1&H2). subst. + autorewrite with trace_rewrite. + (* exit step *) + exploit step_simu_exit_step; eauto. + clear H3; intros (s2' & H3 & H4). + eapply ex_intro; intuition eauto. + eapply exec_bblock; eauto. + + (* Callstate *) + intros t s1' H; inversion_clear H. + eapply ex_intro; constructor 1; eauto. + inversion H1; subst; clear H1. + inversion_clear H0; simpl. + - (* function_internal*) + cutrewrite (trans_code (Mach.fn_code f0) = fn_code (transf_function f0)); eauto. + eapply exec_function_internal; eauto. + rewrite <- parent_sp_preserved; eauto. + rewrite <- parent_ra_preserved; eauto. + - (* function_external *) + autorewrite with trace_rewrite. + eapply exec_function_external; eauto. + apply find_funct_ptr_same_external; auto. + rewrite <- parent_sp_preserved; eauto. + + (* Returnstate *) + intros t s1' H; inversion_clear H. + eapply ex_intro; constructor 1; eauto. + inversion H1; subst; clear H1. + inversion_clear H0; simpl. + eapply exec_return. +Qed. + + +Lemma cfi_dist_end_block i c: +(exists cfi, trans_inst i = MB_cfi cfi) -> +dist_end_block_code (i :: c) = 0. +Proof. + unfold dist_end_block_code. + intro H. destruct H as [cfi H]. + destruct i;simpl in H;try(congruence); ( + remember (trans_code _) as bl; + rewrite <- is_trans_code_inv in Heqbl; + inversion Heqbl; subst; simpl in * |- *; try (congruence)). +Qed. + +Theorem transf_program_correct: + forward_simulation (Mach.semantics (inv_trans_rao rao) prog) (Machblock.semantics rao tprog). +Proof. + apply forward_simulation_block_trans with (dist_end_block := dist_end_block) (trans_state := trans_state). +(* simu_mid_block *) + - intros s1 t s1' H1 H2. + destruct H1; simpl in * |- *; omega || (intuition auto); + destruct H2; eapply cfi_dist_end_block; simpl; eauto. +(* public_preserved *) + - apply senv_preserved. +(* match_initial_states *) + - intros. simpl. + eapply ex_intro; constructor 1. + eapply match_states_trans_state. + destruct H. split. + apply init_mem_preserved; auto. + rewrite prog_main_preserved. rewrite <- H0. apply symbols_preserved. +(* match_final_states *) + - intros. simpl. destruct H. split with (r := r); auto. +(* final_states_end_block *) + - intros. simpl in H0. + inversion H0. + inversion H; simpl; auto. + all: try (subst; discriminate). + apply cfi_dist_end_block; exists MBreturn; eauto. +(* simu_end_block *) + - apply simu_end_block. +Qed. + +End PRESERVATION. + +(** Auxiliary lemmas used to prove existence of a Mach return adress from a Machblock return address. *) + + + +Lemma is_trans_code_monotonic i c b l: + is_trans_code c (b::l) -> + exists l' b', is_trans_code (i::c) (l' ++ (b'::l)). +Proof. + intro H; destruct c as [|i' c]. { inversion H. } + remember (trans_inst i) as ti. + destruct ti as [lbl|bi|cfi]. + - (*i=lbl *) cutrewrite (i = Mlabel lbl). 2: ( destruct i; simpl in * |- *; try congruence ). + exists nil; simpl; eexists. eapply Tr_add_label; eauto. + - (*i=basic*) + destruct i'. + 10: { exists (add_to_new_bblock (MB_basic bi)::nil). exists b. + cutrewrite ((add_to_new_bblock (MB_basic bi) :: nil) ++ (b::l)=(add_to_new_bblock (MB_basic bi) :: (b::l)));eauto. + rewrite Heqti. + eapply Tr_end_block; eauto. + rewrite <-Heqti. + eapply End_basic. inversion H; try(simpl; congruence). + simpl in H5; congruence. } + all: try(exists nil; simpl; eexists; eapply Tr_add_basic; eauto; inversion H; try(eauto || congruence)). + - (*i=cfi*) + destruct i; try(simpl in Heqti; congruence). + all: exists (add_to_new_bblock (MB_cfi cfi)::nil); exists b; + cutrewrite ((add_to_new_bblock (MB_cfi cfi) :: nil) ++ (b::l)=(add_to_new_bblock (MB_cfi cfi) :: (b::l)));eauto; + rewrite Heqti; + eapply Tr_end_block; eauto; + rewrite <-Heqti; + eapply End_cfi; congruence. +Qed. + +Lemma trans_code_monotonic i c b l: + (b::l) = trans_code c -> + exists l' b', trans_code (i::c) = (l' ++ (b'::l)). +Proof. + intro H; rewrite <- is_trans_code_inv in H. + destruct (is_trans_code_monotonic i c b l H) as (l' & b' & H0). + subst_is_trans_code H0. + eauto. +Qed. + +(* FIXME: these two lemma should go into [Coqlib.v] *) +Lemma is_tail_app A (l1: list A): forall l2, is_tail l2 (l1 ++ l2). +Proof. + induction l1; simpl; auto with coqlib. +Qed. +Hint Resolve is_tail_app: coqlib. + +Lemma is_tail_app_inv A (l1: list A): forall l2 l3, is_tail (l1 ++ l2) l3 -> is_tail l2 l3. +Proof. + induction l1; simpl; auto with coqlib. + intros l2 l3 H; inversion H; eauto with coqlib. +Qed. +Hint Resolve is_tail_app_inv: coqlib. + + +Lemma Mach_Machblock_tail sg ros c c1 c2: c1=(Mcall sg ros :: c) -> is_tail c1 c2 -> + exists b, is_tail (b :: trans_code c) (trans_code c2). +Proof. + intros H; induction 1. + - intros; subst. + remember (trans_code (Mcall _ _::c)) as tc2. + rewrite <- is_trans_code_inv in Heqtc2. + inversion Heqtc2; simpl in * |- *; subst; try congruence. + subst_is_trans_code H1. + eapply ex_intro; eauto with coqlib. + - intros; exploit IHis_tail; eauto. clear IHis_tail. + intros (b & Hb). inversion Hb; clear Hb. + * exploit (trans_code_monotonic i c2); eauto. + intros (l' & b' & Hl'); rewrite Hl'. + exists b'; simpl; eauto with coqlib. + * exploit (trans_code_monotonic i c2); eauto. + intros (l' & b' & Hl'); rewrite Hl'. + simpl; eapply ex_intro. + eapply is_tail_trans; eauto with coqlib. +Qed. + +Section Mach_Return_Address. + +Variable return_address_offset: function -> code -> ptrofs -> Prop. + +Hypothesis ra_exists: forall (b: bblock) (f: function) (c : list bblock), + is_tail (b :: c) (fn_code f) -> exists ra : ptrofs, return_address_offset f c ra. + +Definition Mach_return_address_offset (f: Mach.function) (c: Mach.code) (ofs: ptrofs) : Prop := + return_address_offset (transf_function f) (trans_code c) ofs. + +Lemma Mach_return_address_exists: + forall f sg ros c, is_tail (Mcall sg ros :: c) f.(Mach.fn_code) -> + exists ra, Mach_return_address_offset f c ra. +Proof. + intros. + exploit Mach_Machblock_tail; eauto. + destruct 1. + eapply ra_exists; eauto. +Qed. + +End Mach_Return_Address.
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