diff options
Diffstat (limited to 'src/hls/HTLgenproof.v')
| -rw-r--r-- | src/hls/HTLgenproof.v | 2941 |
1 files changed, 0 insertions, 2941 deletions
diff --git a/src/hls/HTLgenproof.v b/src/hls/HTLgenproof.v deleted file mode 100644 index 1668580..0000000 --- a/src/hls/HTLgenproof.v +++ /dev/null @@ -1,2941 +0,0 @@ -(* - * Vericert: Verified high-level synthesis. - * Copyright (C) 2020 Yann Herklotz <yann@yannherklotz.com> - * 2020 James Pollard <j@mes.dev> - * - * This program is free software: you can redistribute it and/or modify - * it under the terms of the GNU General Public License as published by - * the Free Software Foundation, either version 3 of the License, or - * (at your option) any later version. - * - * This program is distributed in the hope that it will be useful, - * but WITHOUT ANY WARRANTY; without even the implied warranty of - * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - * GNU General Public License for more details. - * - * You should have received a copy of the GNU General Public License - * along with this program. If not, see <https://www.gnu.org/licenses/>. - *) - -Require compcert.backend.RTL. -Require compcert.backend.Registers. -Require compcert.common.AST. -Require Import compcert.common.Globalenvs. -Require Import compcert.common.Linking. -Require Import compcert.common.Memory. -Require Import compcert.lib.Integers. - -Require Import vericert.common.IntegerExtra. -Require Import vericert.common.Vericertlib. -Require Import vericert.common.ZExtra. -Require Import vericert.hls.Array. -Require Import vericert.hls.AssocMap. -Require vericert.hls.HTL. -Require Import vericert.hls.HTLgen. -Require Import vericert.hls.HTLgenspec. -Require Import vericert.hls.ValueInt. -Require vericert.hls.Verilog. - -Require Import Lia. - -Local Open Scope assocmap. - -#[local] Hint Resolve Smallstep.forward_simulation_plus : htlproof. -#[local] Hint Resolve AssocMap.gss : htlproof. -#[local] Hint Resolve AssocMap.gso : htlproof. - -#[local] Hint Unfold find_assocmap AssocMapExt.get_default : htlproof. - -Inductive match_assocmaps : RTL.function -> RTL.regset -> assocmap -> Prop := - match_assocmap : forall f rs am, - (forall r, Ple r (RTL.max_reg_function f) -> - val_value_lessdef (Registers.Regmap.get r rs) am#r) -> - match_assocmaps f rs am. -#[local] Hint Constructors match_assocmaps : htlproof. - -Definition state_st_wf (m : HTL.module) (s : HTL.state) := - forall st asa asr res, - s = HTL.State res m st asa asr -> - asa!(m.(HTL.mod_st)) = Some (posToValue st). -#[local] Hint Unfold state_st_wf : htlproof. - -Inductive match_arrs (m : HTL.module) (f : RTL.function) (sp : Values.val) (mem : mem) : - Verilog.assocmap_arr -> Prop := -| match_arr : forall asa stack, - asa ! (m.(HTL.mod_stk)) = Some stack /\ - stack.(arr_length) = Z.to_nat (f.(RTL.fn_stacksize) / 4) /\ - (forall ptr, - 0 <= ptr < Z.of_nat m.(HTL.mod_stk_len) -> - opt_val_value_lessdef (Mem.loadv AST.Mint32 mem - (Values.Val.offset_ptr sp (Integers.Ptrofs.repr (4 * ptr)))) - (Option.default (NToValue 0) - (Option.join (array_get_error (Z.to_nat ptr) stack)))) -> - match_arrs m f sp mem asa. - -Definition stack_based (v : Values.val) (sp : Values.block) : Prop := - match v with - | Values.Vptr sp' off => sp' = sp - | _ => True - end. - -Definition reg_stack_based_pointers (sp : Values.block) (rs : Registers.Regmap.t Values.val) : Prop := - forall r, stack_based (Registers.Regmap.get r rs) sp. - -Definition arr_stack_based_pointers (spb : Values.block) (m : mem) (stack_length : Z) - (sp : Values.val) : Prop := - forall ptr, - 0 <= ptr < (stack_length / 4) -> - stack_based (Option.default - Values.Vundef - (Mem.loadv AST.Mint32 m - (Values.Val.offset_ptr sp (Integers.Ptrofs.repr (4 * ptr))))) - spb. - -Definition stack_bounds (sp : Values.val) (hi : Z) (m : mem) : Prop := - forall ptr v, - hi <= ptr <= Integers.Ptrofs.max_unsigned -> - Z.modulo ptr 4 = 0 -> - Mem.loadv AST.Mint32 m (Values.Val.offset_ptr sp (Integers.Ptrofs.repr ptr )) = None /\ - Mem.storev AST.Mint32 m (Values.Val.offset_ptr sp (Integers.Ptrofs.repr ptr )) v = None. - -Inductive match_frames : list RTL.stackframe -> list HTL.stackframe -> Prop := -| match_frames_nil : - match_frames nil nil. - -Inductive match_constants : HTL.module -> assocmap -> Prop := - match_constant : - forall m asr, - asr!(HTL.mod_reset m) = Some (ZToValue 0) -> - asr!(HTL.mod_finish m) = Some (ZToValue 0) -> - match_constants m asr. - -Inductive match_states : RTL.state -> HTL.state -> Prop := -| match_state : forall asa asr sf f sp sp' rs mem m st res - (MASSOC : match_assocmaps f rs asr) - (TF : tr_module f m) - (WF : state_st_wf m (HTL.State res m st asr asa)) - (MF : match_frames sf res) - (MARR : match_arrs m f sp mem asa) - (SP : sp = Values.Vptr sp' (Integers.Ptrofs.repr 0)) - (RSBP : reg_stack_based_pointers sp' rs) - (ASBP : arr_stack_based_pointers sp' mem (f.(RTL.fn_stacksize)) sp) - (BOUNDS : stack_bounds sp (f.(RTL.fn_stacksize)) mem) - (CONST : match_constants m asr), - match_states (RTL.State sf f sp st rs mem) - (HTL.State res m st asr asa) -| match_returnstate : - forall - v v' stack mem res - (MF : match_frames stack res), - val_value_lessdef v v' -> - match_states (RTL.Returnstate stack v mem) (HTL.Returnstate res v') -| match_initial_call : - forall f m m0 - (TF : tr_module f m), - match_states (RTL.Callstate nil (AST.Internal f) nil m0) (HTL.Callstate nil m nil). -#[local] Hint Constructors match_states : htlproof. - -Definition match_prog (p: RTL.program) (tp: HTL.program) := - Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp /\ - main_is_internal p = true. - -#[global] Instance TransfHTLLink (tr_fun: RTL.program -> Errors.res HTL.program): - TransfLink (fun (p1: RTL.program) (p2: HTL.program) => match_prog p1 p2). -Proof. - red. intros. exfalso. destruct (link_linkorder _ _ _ H) as [LO1 LO2]. - apply link_prog_inv in H. - - unfold match_prog in *. - unfold main_is_internal in *. simplify. repeat (unfold_match H4). - repeat (unfold_match H3). simplify. - subst. rewrite H0 in *. specialize (H (AST.prog_main p2)). - exploit H. - apply Genv.find_def_symbol. exists b. split. - assumption. apply Genv.find_funct_ptr_iff. eassumption. - apply Genv.find_def_symbol. exists b0. split. - assumption. apply Genv.find_funct_ptr_iff. eassumption. - intros. inv H3. inv H5. destruct H6. inv H5. -Qed. - -Definition match_prog' (p: RTL.program) (tp: HTL.program) := - Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp. - -Lemma match_prog_matches : - forall p tp, match_prog p tp -> match_prog' p tp. -Proof. unfold match_prog. tauto. Qed. - -Lemma transf_program_match: - forall p tp, HTLgen.transl_program p = Errors.OK tp -> match_prog p tp. -Proof. - intros. unfold transl_program in H. - destruct (main_is_internal p) eqn:?; try discriminate. - unfold match_prog. split. - apply Linking.match_transform_partial_program; auto. - assumption. -Qed. - -Lemma regs_lessdef_add_greater : - forall f rs1 rs2 n v, - Plt (RTL.max_reg_function f) n -> - match_assocmaps f rs1 rs2 -> - match_assocmaps f rs1 (AssocMap.set n v rs2). -Proof. - inversion 2; subst. - intros. constructor. - intros. unfold find_assocmap. unfold AssocMapExt.get_default. - rewrite AssocMap.gso. eauto. - apply Pos.le_lt_trans with _ _ n in H2. - unfold not. intros. subst. eapply Pos.lt_irrefl. eassumption. assumption. -Qed. -#[local] Hint Resolve regs_lessdef_add_greater : htlproof. - -Lemma regs_lessdef_add_match : - forall f rs am r v v', - val_value_lessdef v v' -> - match_assocmaps f rs am -> - match_assocmaps f (Registers.Regmap.set r v rs) (AssocMap.set r v' am). -Proof. - inversion 2; subst. - constructor. intros. - destruct (peq r0 r); subst. - rewrite Registers.Regmap.gss. - unfold find_assocmap. unfold AssocMapExt.get_default. - rewrite AssocMap.gss. assumption. - - rewrite Registers.Regmap.gso; try assumption. - unfold find_assocmap. unfold AssocMapExt.get_default. - rewrite AssocMap.gso; eauto. -Qed. -#[local] Hint Resolve regs_lessdef_add_match : htlproof. - -Lemma list_combine_none : - forall n l, - length l = n -> - list_combine Verilog.merge_cell (list_repeat None n) l = l. -Proof. - induction n; intros; crush. - - symmetry. apply length_zero_iff_nil. auto. - - destruct l; crush. - rewrite list_repeat_cons. - crush. f_equal. - eauto. -Qed. - -Lemma combine_none : - forall n a, - a.(arr_length) = n -> - arr_contents (combine Verilog.merge_cell (arr_repeat None n) a) = arr_contents a. -Proof. - intros. - unfold combine. - crush. - - rewrite <- (arr_wf a) in H. - apply list_combine_none. - assumption. -Qed. - -Lemma list_combine_lookup_first : - forall l1 l2 n, - length l1 = length l2 -> - nth_error l1 n = Some None -> - nth_error (list_combine Verilog.merge_cell l1 l2) n = nth_error l2 n. -Proof. - induction l1; intros; crush. - - rewrite nth_error_nil in H0. - discriminate. - - destruct l2 eqn:EQl2. crush. - simpl in H. inv H. - destruct n; simpl in *. - inv H0. simpl. reflexivity. - eauto. -Qed. - -Lemma combine_lookup_first : - forall a1 a2 n, - a1.(arr_length) = a2.(arr_length) -> - array_get_error n a1 = Some None -> - array_get_error n (combine Verilog.merge_cell a1 a2) = array_get_error n a2. -Proof. - intros. - - unfold array_get_error in *. - apply list_combine_lookup_first; eauto. - rewrite a1.(arr_wf). rewrite a2.(arr_wf). - assumption. -Qed. - -Lemma list_combine_lookup_second : - forall l1 l2 n x, - length l1 = length l2 -> - nth_error l1 n = Some (Some x) -> - nth_error (list_combine Verilog.merge_cell l1 l2) n = Some (Some x). -Proof. - induction l1; intros; crush; auto. - - destruct l2 eqn:EQl2. crush. - simpl in H. inv H. - destruct n; simpl in *. - inv H0. simpl. reflexivity. - eauto. -Qed. - -Lemma combine_lookup_second : - forall a1 a2 n x, - a1.(arr_length) = a2.(arr_length) -> - array_get_error n a1 = Some (Some x) -> - array_get_error n (combine Verilog.merge_cell a1 a2) = Some (Some x). -Proof. - intros. - - unfold array_get_error in *. - apply list_combine_lookup_second; eauto. - rewrite a1.(arr_wf). rewrite a2.(arr_wf). - assumption. -Qed. - -Ltac inv_state := - match goal with - MSTATE : match_states _ _ |- _ => - inversion MSTATE; - match goal with - TF : tr_module _ _ |- _ => - inversion TF; - match goal with - TC : forall _ _, - Maps.PTree.get _ _ = Some _ -> tr_code _ _ _ _ _ _ _ _ _, - H : Maps.PTree.get _ _ = Some _ |- _ => - apply TC in H; inversion H; - match goal with - TI : context[tr_instr] |- _ => - inversion TI - end - end - end -end; subst. - -Ltac unfold_func H := - match type of H with - | ?f = _ => unfold f in H; repeat (unfold_match H) - | ?f _ = _ => unfold f in H; repeat (unfold_match H) - | ?f _ _ = _ => unfold f in H; repeat (unfold_match H) - | ?f _ _ _ = _ => unfold f in H; repeat (unfold_match H) - | ?f _ _ _ _ = _ => unfold f in H; repeat (unfold_match H) - end. - -Lemma init_reg_assoc_empty : - forall f l, - match_assocmaps f (RTL.init_regs nil l) (HTL.init_regs nil l). -Proof. - induction l; simpl; constructor; intros. - - rewrite Registers.Regmap.gi. unfold find_assocmap. - unfold AssocMapExt.get_default. rewrite AssocMap.gempty. - constructor. - - - rewrite Registers.Regmap.gi. unfold find_assocmap. - unfold AssocMapExt.get_default. rewrite AssocMap.gempty. - constructor. -Qed. - -Lemma arr_lookup_some: - forall (z : Z) (r0 : Registers.reg) (r : Verilog.reg) (asr : assocmap) (asa : Verilog.assocmap_arr) - (stack : Array (option value)) (H5 : asa ! r = Some stack) n, - exists x, Verilog.arr_assocmap_lookup asa r n = Some x. -Proof. - intros z r0 r asr asa stack H5 n. - eexists. - unfold Verilog.arr_assocmap_lookup. rewrite H5. reflexivity. -Qed. -#[local] Hint Resolve arr_lookup_some : htlproof. - -Section CORRECTNESS. - - Variable prog : RTL.program. - Variable tprog : HTL.program. - - Hypothesis TRANSL : match_prog prog tprog. - - Lemma TRANSL' : - Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq prog tprog. - Proof. intros; apply match_prog_matches; assumption. Qed. - - Let ge : RTL.genv := Globalenvs.Genv.globalenv prog. - Let tge : HTL.genv := Globalenvs.Genv.globalenv tprog. - - Lemma symbols_preserved: - forall (s: AST.ident), Genv.find_symbol tge s = Genv.find_symbol ge s. - Proof. intros. eapply (Genv.find_symbol_match TRANSL'). Qed. - - Lemma function_ptr_translated: - forall (b: Values.block) (f: RTL.fundef), - Genv.find_funct_ptr ge b = Some f -> - exists tf, - Genv.find_funct_ptr tge b = Some tf /\ transl_fundef f = Errors.OK tf. - Proof. - intros. exploit (Genv.find_funct_ptr_match TRANSL'); eauto. - intros (cu & tf & P & Q & R); exists tf; auto. - Qed. - - Lemma functions_translated: - forall (v: Values.val) (f: RTL.fundef), - Genv.find_funct ge v = Some f -> - exists tf, - Genv.find_funct tge v = Some tf /\ transl_fundef f = Errors.OK tf. - Proof. - intros. exploit (Genv.find_funct_match TRANSL'); eauto. - intros (cu & tf & P & Q & R); exists tf; auto. - Qed. - - Lemma senv_preserved: - Senv.equiv (Genv.to_senv ge) (Genv.to_senv tge). - Proof - (Genv.senv_transf_partial TRANSL'). - #[local] Hint Resolve senv_preserved : htlproof. - - Lemma ptrofs_inj : - forall a b, - Ptrofs.unsigned a = Ptrofs.unsigned b -> a = b. - Proof. - intros. rewrite <- Ptrofs.repr_unsigned. symmetry. rewrite <- Ptrofs.repr_unsigned. - rewrite H. auto. - Qed. - - Lemma op_stack_based : - forall F V sp v m args rs op ge pc' res0 pc f e fin rtrn st stk, - tr_instr fin rtrn st stk (RTL.Iop op args res0 pc') - (Verilog.Vnonblock (Verilog.Vvar res0) e) - (state_goto st pc') -> - reg_stack_based_pointers sp rs -> - (RTL.fn_code f) ! pc = Some (RTL.Iop op args res0 pc') -> - @Op.eval_operation F V ge (Values.Vptr sp Ptrofs.zero) op - (map (fun r : positive => Registers.Regmap.get r rs) args) m = Some v -> - stack_based v sp. - Proof. - Ltac solve_no_ptr := - match goal with - | H: reg_stack_based_pointers ?sp ?rs |- stack_based (Registers.Regmap.get ?r ?rs) _ => - solve [apply H] - | H1: reg_stack_based_pointers ?sp ?rs, H2: Registers.Regmap.get _ _ = Values.Vptr ?b ?i - |- context[Values.Vptr ?b _] => - let H := fresh "H" in - assert (H: stack_based (Values.Vptr b i) sp) by (rewrite <- H2; apply H1); simplify; solve [auto] - | |- context[Registers.Regmap.get ?lr ?lrs] => - destruct (Registers.Regmap.get lr lrs) eqn:?; simplify; auto - | |- stack_based (?f _) _ => unfold f - | |- stack_based (?f _ _) _ => unfold f - | |- stack_based (?f _ _ _) _ => unfold f - | |- stack_based (?f _ _ _ _) _ => unfold f - | H: ?f _ _ = Some _ |- _ => - unfold f in H; repeat (unfold_match H); inv H - | H: ?f _ _ _ _ _ _ = Some _ |- _ => - unfold f in H; repeat (unfold_match H); inv H - | H: map (fun r : positive => Registers.Regmap.get r _) ?args = _ |- _ => - destruct args; inv H - | |- context[if ?c then _ else _] => destruct c; try discriminate - | H: match _ with _ => _ end = Some _ |- _ => repeat (unfold_match H) - | H: match _ with _ => _ end = OK _ _ _ |- _ => repeat (unfold_match H) - | |- context[match ?g with _ => _ end] => destruct g; try discriminate - | |- _ => simplify; solve [auto] - end. - intros F V sp v m args rs op g pc' res0 pc f e fin rtrn st stk INSTR RSBP SEL EVAL. - inv INSTR. unfold translate_instr in H5. - unfold_match H5; repeat (unfold_match H5); repeat (simplify; solve_no_ptr). - Qed. - - Lemma int_inj : - forall x y, - Int.unsigned x = Int.unsigned y -> - x = y. - Proof. - intros. rewrite <- Int.repr_unsigned at 1. rewrite <- Int.repr_unsigned. - rewrite <- H. trivial. - Qed. - - Ltac eval_correct_tac := - match goal with - | |- context[valueToPtr] => unfold valueToPtr - | |- context[valueToInt] => unfold valueToInt - | |- context[bop] => unfold bop - | H : context[bop] |- _ => unfold bop in H - | |- context[boplit] => unfold boplit - | H : context[boplit] |- _ => unfold boplit in H - | |- context[boplitz] => unfold boplitz - | H : context[boplitz] |- _ => unfold boplitz in H - | |- val_value_lessdef Values.Vundef _ => solve [constructor] - | H : val_value_lessdef _ _ |- val_value_lessdef (Values.Vint _) _ => constructor; inv H - | |- val_value_lessdef (Values.Vint _) _ => constructor; auto - | H : ret _ _ = OK _ _ _ |- _ => inv H - | H : context[RTL.max_reg_function ?f] - |- context[_ (Registers.Regmap.get ?r ?rs) (Registers.Regmap.get ?r0 ?rs)] => - let HPle1 := fresh "HPle" in - let HPle2 := fresh "HPle" in - assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - assert (HPle2 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H in HPle1; apply H in HPle2; eexists; split; - [econstructor; eauto; constructor; trivial | inv HPle1; inv HPle2; try (constructor; auto)] - | H : context[RTL.max_reg_function ?f] - |- context[_ (Registers.Regmap.get ?r ?rs) _] => - let HPle1 := fresh "HPle" in - assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H in HPle1; eexists; split; - [econstructor; eauto; constructor; trivial | inv HPle1; try (constructor; auto)] - | H : _ :: _ = _ :: _ |- _ => inv H - | |- context[match ?d with _ => _ end] => destruct d eqn:?; try discriminate - | H : match ?d with _ => _ end = _ |- _ => repeat unfold_match H - | H : match ?d with _ => _ end _ = _ |- _ => repeat unfold_match H - | |- Verilog.expr_runp _ _ _ ?f _ => - match f with - | Verilog.Vternary _ _ _ => idtac - | _ => econstructor - end - | |- val_value_lessdef (?f _ _) _ => unfold f - | |- val_value_lessdef (?f _) _ => unfold f - | H : ?f (Registers.Regmap.get _ _) _ = Some _ |- _ => - unfold f in H; repeat (unfold_match H) - | H1 : Registers.Regmap.get ?r ?rs = Values.Vint _, H2 : val_value_lessdef (Registers.Regmap.get ?r ?rs) _ - |- _ => rewrite H1 in H2; inv H2 - | |- _ => eexists; split; try constructor; solve [eauto] - | H : context[RTL.max_reg_function ?f] |- context[_ (Verilog.Vvar ?r) (Verilog.Vvar ?r0)] => - let HPle1 := fresh "H" in - let HPle2 := fresh "H" in - assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - assert (HPle2 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H in HPle1; apply H in HPle2; eexists; split; try constructor; eauto - | H : context[RTL.max_reg_function ?f] |- context[Verilog.Vvar ?r] => - let HPle := fresh "H" in - assert (HPle : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H in HPle; eexists; split; try constructor; eauto - | |- context[if ?c then _ else _] => destruct c eqn:?; try discriminate - | H : ?b = _ |- _ = boolToValue ?b => rewrite H - end. - Ltac inv_lessdef := lazymatch goal with - | H2 : context[RTL.max_reg_function ?f], - H : Registers.Regmap.get ?r ?rs = _, - H1 : Registers.Regmap.get ?r0 ?rs = _ |- _ => - let HPle1 := fresh "HPle" in - assert (HPle1 : Ple r (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H2 in HPle1; inv HPle1; - let HPle2 := fresh "HPle" in - assert (HPle2 : Ple r0 (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H2 in HPle2; inv HPle2 - | H2 : context[RTL.max_reg_function ?f], H : Registers.Regmap.get ?r ?rs = _ |- _ => - let HPle1 := fresh "HPle" in - assert (HPle1 : Ple r (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_use; eauto; simpl; auto); - apply H2 in HPle1; inv HPle1 - end. - Ltac solve_cond := - match goal with - | H : context[match _ with _ => _ end] |- _ => repeat (unfold_merge H) - | H : ?f = _ |- context[boolToValue ?f] => rewrite H; solve [auto] - | H : Values.Vptr _ _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vint _ |- _ => - rewrite H2 in H; discriminate - | H : Values.Vundef = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vint _ |- _ => - rewrite H2 in H; discriminate - | H : Values.Vint _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vundef |- _ => - rewrite H2 in H; discriminate - | H : Values.Vint _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vtrue |- _ => - rewrite H2 in H; discriminate - | H : Values.Vint _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vfalse |- _ => - rewrite H2 in H; discriminate - | H : Values.Vint _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vptr _ _ |- _ => - rewrite H2 in H; discriminate - | H : Values.Vundef = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vptr _ _ |- _ => - rewrite H2 in H; discriminate - | H : Values.Vundef = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vtrue |- _ => - rewrite H2 in H; discriminate - | H : Values.Vundef = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vfalse |- _ => - rewrite H2 in H; discriminate - | H : Values.Vptr _ _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vundef |- _ => - rewrite H2 in H; discriminate - | H : Values.Vptr _ _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vtrue |- _ => - rewrite H2 in H; discriminate - | H : Values.Vptr _ _ = Registers.Regmap.get ?r ?rs, - H2 : Registers.Regmap.get ?r ?rs = Values.Vfalse |- _ => - rewrite H2 in H; discriminate - | |- context[val_value_lessdef Values.Vundef _] => - econstructor; split; econstructor; econstructor; auto; solve [constructor] - | H1 : Registers.Regmap.get ?r ?rs = Values.Vint _, - H2 : Values.Vint _ = Registers.Regmap.get ?r ?rs, - H3 : Registers.Regmap.get ?r0 ?rs = Values.Vint _, - H4 : Values.Vint _ = Registers.Regmap.get ?r0 ?rs|- _ => - rewrite H1 in H2; rewrite H3 in H4; inv H2; inv H4; unfold valueToInt in *; constructor - | H1 : Registers.Regmap.get ?r ?rs = Values.Vptr _ _, - H2 : Values.Vptr _ _ = Registers.Regmap.get ?r ?rs, - H3 : Registers.Regmap.get ?r0 ?rs = Values.Vptr _ _, - H4 : Values.Vptr _ _ = Registers.Regmap.get ?r0 ?rs|- _ => - rewrite H1 in H2; rewrite H3 in H4; inv H2; inv H4; unfold valueToInt in *; constructor; - unfold Ptrofs.ltu, Int.ltu in *; unfold Ptrofs.of_int in *; - repeat (rewrite Ptrofs.unsigned_repr in *; auto using Int.unsigned_range_2) - | H : _ :: _ = _ :: _ |- _ => inv H - | H : ret _ _ = OK _ _ _ |- _ => inv H - | |- _ => - eexists; split; [ econstructor; econstructor; auto - | simplify; inv_lessdef; repeat (unfold valueToPtr, valueToInt in *; solve_cond); - unfold valueToPtr in * - ] - end. - - Lemma eval_cond_correct : - forall stk f sp pc rs m res ml st asr asa e b f' s s' args i cond, - match_states (RTL.State stk f sp pc rs m) (HTL.State res ml st asr asa) -> - (forall v, In v args -> Ple v (RTL.max_reg_function f)) -> - Op.eval_condition cond (map (fun r : positive => Registers.Regmap.get r rs) args) m = Some b -> - translate_condition cond args s = OK e s' i -> - Verilog.expr_runp f' asr asa e (boolToValue b). - Proof. - intros stk f sp pc rs m res ml st asr asa e b f' s s' args i cond MSTATE MAX_FUN EVAL TR_INSTR. - pose proof MSTATE as MSTATE_2. inv MSTATE. - inv MASSOC. unfold translate_condition, translate_comparison, - translate_comparisonu, translate_comparison_imm, - translate_comparison_immu in TR_INSTR; - repeat unfold_match TR_INSTR; try inv TR_INSTR; simplify_val; - unfold Values.Val.cmp_bool, Values.Val.of_optbool, bop, Values.Val.cmpu_bool, - Int.cmpu in *; - repeat unfold_match EVAL. - - repeat econstructor. repeat unfold_match Heqo. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo; simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; solve_cond. - - repeat econstructor. repeat unfold_match Heqo; simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; try solve_cond. simplify_val. - rewrite Heqv0 in H3. rewrite Heqv in H2. inv H2. inv H3. - unfold Ptrofs.ltu. unfold Int.ltu. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. auto. - - repeat econstructor. unfold Verilog.binop_run. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; simplify_val; solve_cond. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; try solve_cond. simplify_val. - rewrite Heqv0 in H3. rewrite Heqv in H2. inv H2. inv H3. - unfold Ptrofs.ltu. unfold Int.ltu. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. auto. - - repeat econstructor. unfold Verilog.binop_run. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; simplify_val; solve_cond. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; try solve_cond. simplify_val. - rewrite Heqv0 in H3. rewrite Heqv in H2. inv H2. inv H3. - unfold Ptrofs.ltu. unfold Int.ltu. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. auto. - - repeat econstructor. unfold Verilog.binop_run. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; simplify_val; solve_cond. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - pose proof (MAX_FUN p0) as MAX_FUN_P0. apply H in MAX_FUN_P0; auto. - inv MAX_FUN_P; inv MAX_FUN_P0; try solve_cond. simplify_val. - rewrite Heqv0 in H3. rewrite Heqv in H2. inv H2. inv H3. - unfold Ptrofs.ltu. unfold Int.ltu. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr by apply Int.unsigned_range_2. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - - repeat econstructor. simplify_val. - pose proof (MAX_FUN p) as MAX_FUN_P. apply H in MAX_FUN_P; auto. - inv MAX_FUN_P; simplify_val; try solve_cond. - rewrite Heqv in H0. inv H0. auto. - Qed. - - Lemma eval_cond_correct' : - forall e stk f sp pc rs m res ml st asr asa v f' s s' args i cond, - match_states (RTL.State stk f sp pc rs m) (HTL.State res ml st asr asa) -> - (forall v, In v args -> Ple v (RTL.max_reg_function f)) -> - Values.Val.of_optbool None = v -> - translate_condition cond args s = OK e s' i -> - exists v', Verilog.expr_runp f' asr asa e v' /\ val_value_lessdef v v'. - intros e stk f sp pc rs m res ml st asr asa v f' s s' args i cond MSTATE MAX_FUN EVAL TR_INSTR. - unfold translate_condition, translate_comparison, translate_comparisonu, - translate_comparison_imm, translate_comparison_immu, bop, boplit in *. - repeat unfold_match TR_INSTR; inv TR_INSTR; repeat econstructor. - Qed. - - Lemma eval_correct_Oshrximm : - forall s sp rs m v e asr asa f f' stk s' i pc res0 pc' args res ml st n, - match_states (RTL.State stk f sp pc rs m) (HTL.State res ml st asr asa) -> - (RTL.fn_code f) ! pc = Some (RTL.Iop (Op.Oshrximm n) args res0 pc') -> - Op.eval_operation ge sp (Op.Oshrximm n) - (List.map (fun r : BinNums.positive => - Registers.Regmap.get r rs) args) m = Some v -> - translate_instr (Op.Oshrximm n) args s = OK e s' i -> - exists v', Verilog.expr_runp f' asr asa e v' /\ val_value_lessdef v v'. - Proof. - intros s sp rs m v e asr asa f f' stk s' i pc pc' res0 args res ml st n MSTATE INSTR EVAL TR_INSTR. - pose proof MSTATE as MSTATE_2. inv MSTATE. - inv MASSOC. unfold translate_instr in TR_INSTR; repeat (unfold_match TR_INSTR); inv TR_INSTR; - unfold Op.eval_operation in EVAL; repeat (unfold_match EVAL); inv EVAL. - (*repeat (simplify; eval_correct_tac; unfold valueToInt in * ). - destruct (Z_lt_ge_dec (Int.signed i0) 0). - econstructor.*) - unfold Values.Val.shrx in *. - destruct v0; try discriminate. - destruct (Int.ltu n (Int.repr 31)) eqn:?; try discriminate. - inversion H1. clear H1. - assert (Int.unsigned n <= 30). - { unfold Int.ltu in *. destruct (zlt (Int.unsigned n) (Int.unsigned (Int.repr 31))); try discriminate. - rewrite Int.unsigned_repr in l by (simplify; lia). - replace 31 with (Z.succ 30) in l by auto. - apply Zlt_succ_le in l. - auto. - } - rewrite IntExtra.shrx_shrx_alt_equiv in H2 by auto. - unfold IntExtra.shrx_alt in *. - destruct (zlt (Int.signed i0) 0). - - repeat econstructor; unfold valueToBool, boolToValue, uvalueToZ, natToValue; - repeat (simplify; eval_correct_tac). - inv_lessdef. unfold valueToInt in *. rewrite H3 in H1. - inv H1. - unfold Int.lt in *. rewrite zlt_true in Heqb0. simplify. - rewrite Int.unsigned_repr in Heqb0. discriminate. - simplify; lia. - unfold ZToValue. rewrite Int.signed_repr by (simplify; lia). - auto. - rewrite H3 in H1; discriminate. - rewrite H3 in H2; discriminate. - rewrite IntExtra.shrx_shrx_alt_equiv; auto. unfold IntExtra.shrx_alt. rewrite zlt_true; try lia. - simplify. inv_lessdef. unfold valueToInt in *. - rewrite H3 in H1. auto. inv H1. auto. - rewrite H3 in H1. discriminate. - rewrite H3 in H2. discriminate. - - econstructor; econstructor; [eapply Verilog.erun_Vternary_false|idtac]; repeat econstructor; unfold valueToBool, boolToValue, uvalueToZ, natToValue; - repeat (simplify; eval_correct_tac). - inv_lessdef. unfold valueToInt in *. rewrite H3 in H1. - inv H1. - unfold Int.lt in *. rewrite zlt_false in Heqb0. simplify. - rewrite Int.unsigned_repr in Heqb0. lia. - simplify; lia. - unfold ZToValue. rewrite Int.signed_repr by (simplify; lia). - auto. - rewrite H3 in H1; discriminate. - rewrite H3 in H2; discriminate. - rewrite IntExtra.shrx_shrx_alt_equiv; auto. unfold IntExtra.shrx_alt. rewrite zlt_false; try lia. - simplify. inv_lessdef. unfold valueToInt in *. - rewrite H3 in H1. auto. inv H1. auto. - rewrite H3 in H1. discriminate. - rewrite H3 in H2. discriminate. - Qed. - - Lemma eval_correct : - forall s sp op rs m v e asr asa f f' stk s' i pc res0 pc' args res ml st, - match_states (RTL.State stk f sp pc rs m) (HTL.State res ml st asr asa) -> - (RTL.fn_code f) ! pc = Some (RTL.Iop op args res0 pc') -> - Op.eval_operation ge sp op - (List.map (fun r : BinNums.positive => Registers.Regmap.get r rs) args) m = Some v -> - translate_instr op args s = OK e s' i -> - exists v', Verilog.expr_runp f' asr asa e v' /\ val_value_lessdef v v'. - Proof. - intros s sp op rs m v e asr asa f f' stk s' i pc pc' res0 args res ml st MSTATE INSTR EVAL TR_INSTR. - pose proof MSTATE as MSTATE_2. inv MSTATE. - inv MASSOC. unfold translate_instr in TR_INSTR; repeat (unfold_match TR_INSTR); inv TR_INSTR; - unfold Op.eval_operation in EVAL; repeat (unfold_match EVAL); inv EVAL; - repeat (simplify; eval_correct_tac; unfold valueToInt in *). - - pose proof Integers.Ptrofs.agree32_sub as H2; unfold Integers.Ptrofs.agree32 in H2. - unfold Ptrofs.of_int. simpl. - apply ptrofs_inj. assert (Archi.ptr64 = false) by auto. eapply H2 in H3. - rewrite Ptrofs.unsigned_repr. apply H3. replace Ptrofs.max_unsigned with Int.max_unsigned; auto. - apply Int.unsigned_range_2. - auto. rewrite Ptrofs.unsigned_repr. replace Ptrofs.max_unsigned with Int.max_unsigned; auto. - apply Int.unsigned_range_2. rewrite Ptrofs.unsigned_repr. auto. - replace Ptrofs.max_unsigned with Int.max_unsigned; auto. - apply Int.unsigned_range_2. - - pose proof Integers.Ptrofs.agree32_sub as AGR; unfold Integers.Ptrofs.agree32 in AGR. - assert (ARCH: Archi.ptr64 = false) by auto. eapply AGR in ARCH. - apply int_inj. unfold Ptrofs.to_int. rewrite Int.unsigned_repr. - apply ARCH. pose proof Ptrofs.unsigned_range_2. - replace Ptrofs.max_unsigned with Int.max_unsigned; auto. - pose proof Ptrofs.agree32_of_int. unfold Ptrofs.agree32 in H2. - eapply H2 in ARCH. apply ARCH. - pose proof Ptrofs.agree32_of_int. unfold Ptrofs.agree32 in H2. - eapply H2 in ARCH. apply ARCH. - - rewrite H0 in Heqb. rewrite H1 in Heqb. discriminate. - - rewrite Heqb in Heqb0. discriminate. - - rewrite H0 in Heqb. rewrite H1 in Heqb. discriminate. - - rewrite Heqb in Heqb0. discriminate. - (*- unfold Int.ror. unfold Int.or. unfold Int.shru, Int.shl, Int.sub. unfold intToValue. unfold Int.modu, - repeat (rewrite Int.unsigned_repr). auto.*) - - assert (Int.unsigned n <= 30). - { unfold Int.ltu in *. destruct (zlt (Int.unsigned n) (Int.unsigned (Int.repr 31))); try discriminate. - rewrite Int.unsigned_repr in l by (simplify; lia). - replace 31 with (Z.succ 30) in l by auto. - apply Zlt_succ_le in l. - auto. - } - destruct (zlt (Int.signed i0) 0). - + repeat econstructor; unfold valueToBool, boolToValue, uvalueToZ, natToValue; - repeat (simplify; eval_correct_tac). - rewrite IntExtra.shrx_shrx_alt_equiv; auto. unfold IntExtra.shrx_alt. rewrite zlt_true; try lia. - simplify. inv_lessdef. unfold valueToInt in *. - rewrite Heqv0 in H0. auto. inv H0. auto. - rewrite Heqv0 in H2. discriminate. - unfold valueToInt in l. auto. - inv_lessdef. unfold valueToInt in *. rewrite Heqv0 in H0. - inv H0. - unfold Int.lt in *. rewrite zlt_true in Heqb0. simplify. - rewrite Int.unsigned_repr in Heqb0. discriminate. - simplify; lia. - unfold ZToValue. rewrite Int.signed_repr by (simplify; lia). - auto. - rewrite Heqv0 in H0; discriminate. - rewrite Heqv0 in H2; discriminate. - + eapply Verilog.erun_Vternary_false; repeat econstructor; unfold valueToBool, boolToValue, uvalueToZ, natToValue; - repeat (simplify; eval_correct_tac). - rewrite IntExtra.shrx_shrx_alt_equiv; auto. unfold IntExtra.shrx_alt. rewrite zlt_false; try lia. - simplify. inv_lessdef. unfold valueToInt in *. - rewrite Heqv0 in H0. auto. inv H0. auto. - rewrite Heqv0 in H2. discriminate. - unfold valueToInt in *. auto. - inv_lessdef. unfold valueToInt in *. - rewrite Heqv0 in H0. - inv H0. - unfold Int.lt in *. rewrite zlt_false in Heqb0. simplify. - rewrite Int.unsigned_repr in Heqb0. lia. - simplify; lia. - unfold ZToValue. rewrite Int.signed_repr by (simplify; lia). - auto. - rewrite Heqv0 in H0; discriminate. - rewrite Heqv0 in H2; discriminate. - - unfold Op.eval_addressing32 in *. repeat (unfold_match H2); inv H2. - + unfold translate_eff_addressing in *. repeat (unfold_match H1). - destruct v0; inv Heql; rewrite H2; inv H1; repeat eval_correct_tac. - pose proof Integers.Ptrofs.agree32_add as AGR; unfold Integers.Ptrofs.agree32 in AGR. unfold ZToValue. - apply ptrofs_inj. unfold Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. - apply AGR. auto. rewrite H2 in H0. inv H0. unfold valueToPtr. unfold Ptrofs.of_int. - rewrite Ptrofs.unsigned_repr. auto. replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr. auto. replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - + unfold translate_eff_addressing in *. repeat (unfold_match H1). inv H1. - inv Heql. unfold boplitz. repeat (simplify; eval_correct_tac). - all: repeat (unfold_match Heqv). - * inv Heqv. unfold valueToInt in *. inv H2. inv H0. unfold valueToInt in *. trivial. - * constructor. unfold valueToPtr, ZToValue in *. - pose proof Integers.Ptrofs.agree32_add as AGR; unfold Integers.Ptrofs.agree32 in AGR. unfold ZToValue. - apply ptrofs_inj. unfold Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. - apply AGR. auto. inv Heqv. rewrite Int.add_commut. - apply AGR. auto. inv H1. inv H0. unfold valueToPtr. unfold Ptrofs.of_int. - rewrite Ptrofs.unsigned_repr. auto. replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - unfold Ptrofs.of_int. - rewrite Ptrofs.unsigned_repr. inv H0. auto. replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr. auto. replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - apply Int.unsigned_range_2. - * constructor. unfold valueToPtr, ZToValue in *. - pose proof Integers.Ptrofs.agree32_add as AGR; unfold Integers.Ptrofs.agree32 in AGR. unfold ZToValue. - apply ptrofs_inj. unfold Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. - apply AGR. auto. inv Heqv. - apply AGR. auto. inv H0. unfold valueToPtr, Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. auto. - replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - inv H1. unfold valueToPtr, Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. auto. - replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. - rewrite Ptrofs.unsigned_repr. auto. - replace Ptrofs.max_unsigned with Int.max_unsigned by auto. - apply Int.unsigned_range_2. apply Int.unsigned_range_2. - + unfold translate_eff_addressing in *. repeat (unfold_match H1). inv H1. - inv Heql. unfold boplitz. repeat (simplify; eval_correct_tac). - all: repeat (unfold_match Heqv). - * unfold Values.Val.mul in Heqv. repeat (unfold_match Heqv). inv Heqv. inv H3. - unfold valueToInt, ZToValue. auto. - * unfold Values.Val.mul in Heqv. repeat (unfold_match Heqv). - * unfold Values.Val.mul in Heqv. repeat (unfold_match Heqv). - * constructor. unfold valueToPtr, ZToValue. unfold Values.Val.mul in Heqv. repeat (unfold_match Heqv). - + unfold translate_eff_addressing in *. repeat (unfold_match H1). inv H1. - inv Heql. unfold boplitz. repeat (simplify; eval_correct_tac). - all: repeat (unfold_match Heqv). - unfold valueToPtr, ZToValue. - repeat unfold_match Heqv0. unfold Values.Val.mul in Heqv1. repeat unfold_match Heqv1. - inv Heqv1. inv Heqv0. unfold valueToInt in *. - assert (HPle1 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle1. inv HPle1. unfold valueToInt in *. rewrite Heqv2 in H2. inv H2. auto. - rewrite Heqv2 in H2. inv H2. - rewrite Heqv2 in H3. discriminate. - repeat unfold_match Heqv0. unfold Values.Val.mul in Heqv1. repeat unfold_match Heqv1. - repeat unfold_match Heqv0. unfold Values.Val.mul in Heqv1. repeat unfold_match Heqv1. - constructor. unfold valueToPtr, ZToValue. inv Heqv0. inv Heqv1. - assert (HPle1 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle1. inv HPle1. unfold valueToInt in *. rewrite Heqv2 in H3. inv H3. - - pose proof Integers.Ptrofs.agree32_add as AGR; unfold Integers.Ptrofs.agree32 in AGR. unfold ZToValue. - apply ptrofs_inj. unfold Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. - apply AGR. auto. inv H2. unfold valueToPtr, Ptrofs.of_int. rewrite Ptrofs.unsigned_repr. auto. - replace Ptrofs.max_unsigned with Int.max_unsigned by auto. apply Int.unsigned_range_2. - apply Ptrofs.unsigned_repr. apply Int.unsigned_range_2. apply Int.unsigned_range_2. - - rewrite Heqv2 in H3. inv H3. - - rewrite Heqv2 in H4. inv H4. - + unfold translate_eff_addressing in *. repeat (unfold_match H1). inv H1. - inv Heql. unfold boplitz. repeat (simplify; eval_correct_tac). - all: repeat (unfold_match Heqv). - eexists. split. constructor. - constructor. unfold valueToPtr, ZToValue. rewrite Ptrofs.add_zero_l. unfold Ptrofs.of_int. - rewrite Int.unsigned_repr. symmetry. apply Ptrofs.repr_unsigned. - unfold check_address_parameter_unsigned in *. apply Ptrofs.unsigned_range_2. - - destruct (Op.eval_condition cond (map (fun r : positive => Registers.Regmap.get r rs) args) m) eqn:EQ. - + exploit eval_cond_correct; eauto. intros. eapply RTL.max_reg_function_use. apply INSTR. auto. - intros. econstructor. econstructor. eassumption. unfold boolToValue, Values.Val.of_optbool. - destruct b; constructor; auto. - + eapply eval_cond_correct'; eauto. intros. eapply RTL.max_reg_function_use. apply INSTR. auto. - - monadInv H1. - destruct (Op.eval_condition c (map (fun r1 : positive => Registers.Regmap.get r1 rs) l0) m) eqn:EQN; - simplify. destruct b eqn:B. - + exploit eval_cond_correct; eauto. intros. eapply RTL.max_reg_function_use. apply INSTR. - simplify; tauto. intros. - econstructor. econstructor. eapply Verilog.erun_Vternary_true. eassumption. econstructor. auto. - auto. unfold Values.Val.normalize. - destruct (Registers.Regmap.get r rs) eqn:EQN2; constructor. - * assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle1. inv HPle1. unfold valueToInt in H1. rewrite EQN2 in H1. inv H1. auto. - rewrite EQN2 in H1. discriminate. rewrite EQN2 in H2. discriminate. - * assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle1. inv HPle1. rewrite EQN2 in H1. inv H1. rewrite EQN2 in H1. inv H1. auto. - rewrite EQN2 in H2. discriminate. - + exploit eval_cond_correct; eauto. intros. eapply RTL.max_reg_function_use. apply INSTR. - simplify; tauto. intros. - econstructor. econstructor. eapply Verilog.erun_Vternary_false. eassumption. econstructor. auto. - auto. unfold Values.Val.normalize. - destruct (Registers.Regmap.get r0 rs) eqn:EQN2; constructor. - * assert (HPle1 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle1. inv HPle1. unfold valueToInt in H1. rewrite EQN2 in H1. inv H1. auto. - rewrite EQN2 in H1. discriminate. rewrite EQN2 in H2. discriminate. - * assert (HPle1 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle1. inv HPle1. rewrite EQN2 in H1. inv H1. rewrite EQN2 in H1. inv H1. auto. - rewrite EQN2 in H2. discriminate. - + exploit eval_cond_correct'; eauto. intros. eapply RTL.max_reg_function_use. apply INSTR. - simplify; tauto. intros. inv H0. inv H1. destruct (Int.eq_dec x0 Int.zero). - econstructor. econstructor. eapply Verilog.erun_Vternary_false. - eassumption. econstructor. auto. subst. auto. constructor. - econstructor. econstructor. eapply Verilog.erun_Vternary_true. - eassumption. econstructor. auto. unfold valueToBool. pose proof n. apply Int.eq_false in n. - unfold uvalueToZ. unfold Int.eq in n. unfold zeq in *. - destruct (Int.unsigned x0 ==Z Int.unsigned Int.zero); try discriminate. - rewrite <- Z.eqb_neq in n0. rewrite Int.unsigned_zero in n0. rewrite n0. auto. - constructor. - Qed. - - (** The proof of semantic preservation for the translation of instructions - is a simulation argument based on diagrams of the following form: -<< - match_states - code st rs ---------------- State m st assoc - || | - || | - || | - \/ v - code st rs' --------------- State m st assoc' - match_states ->> - where [tr_code c data control fin rtrn st] is assumed to hold. - - The precondition and postcondition is that that should hold is [match_assocmaps rs assoc]. - *) - - Definition transl_instr_prop (instr : RTL.instruction) : Prop := - forall m asr asa fin rtrn st stmt trans res, - tr_instr fin rtrn st (m.(HTL.mod_stk)) instr stmt trans -> - exists asr' asa', - HTL.step tge (HTL.State res m st asr asa) Events.E0 (HTL.State res m st asr' asa'). - -Ltac name_goal name := refine ?[name]. - -Ltac unfold_merge := - unfold merge_assocmap; repeat (rewrite AssocMapExt.merge_add_assoc); - try (rewrite AssocMapExt.merge_base_1). - - Ltac tac0 := - match goal with - | [ |- HTL.exec_ram _ _ _ _ _ ] => constructor - | [ |- context[Verilog.merge_regs _ _] ] => unfold Verilog.merge_regs; cbn; unfold_merge - | [ |- context[reg_stack_based_pointers] ] => unfold reg_stack_based_pointers; intros - - | [ |- context[HTL.empty_stack] ] => unfold HTL.empty_stack - - | [ |- context[_ # ?d <- _ ! ?d] ] => rewrite AssocMap.gss - | [ |- context[_ # ?d <- _ ! ?s] ] => rewrite AssocMap.gso - | [ |- context[(AssocMap.empty _) ! _] ] => rewrite AssocMap.gempty - - | [ |- context[array_get_error _ (combine Verilog.merge_cell (arr_repeat None _) _)] ] => - rewrite combine_lookup_first - - | [ |- state_st_wf _ _ ] => unfold state_st_wf; inversion 1 - | [ |- context[match_states _ _] ] => econstructor; auto - | [ |- match_arrs _ _ _ _ _ ] => econstructor; auto - | [ |- match_assocmaps _ _ _ # _ <- (posToValue _) ] => - apply regs_lessdef_add_greater; [> unfold Plt; lia | assumption] - - | [ H : ?asa ! ?r = Some _ |- Verilog.arr_assocmap_lookup ?asa ?r _ = Some _ ] => - unfold Verilog.arr_assocmap_lookup; setoid_rewrite H; f_equal - | [ |- context[(AssocMap.combine _ _ _) ! _] ] => - try (rewrite AssocMap.gcombine; [> | reflexivity]) - - | [ |- context[Registers.Regmap.get ?d (Registers.Regmap.set ?d _ _)] ] => - rewrite Registers.Regmap.gss - | [ |- context[Registers.Regmap.get ?s (Registers.Regmap.set ?d _ _)] ] => - let EQ := fresh "EQ" in - destruct (Pos.eq_dec s d) as [EQ|EQ]; - [> rewrite EQ | rewrite Registers.Regmap.gso; auto] - - | [ H : opt_val_value_lessdef _ _ |- _ ] => inv H - | [ H : context[Z.of_nat (Z.to_nat _)] |- _ ] => rewrite Z2Nat.id in H; [> solve crush |] - | [ H : _ ! _ = Some _ |- _] => setoid_rewrite H - | [ |- context[AssocMapExt.merge]] => progress unfold_merge - end. - - Ltac simplify_local := intros; unfold_constants; cbn in *; - repeat (nicify_hypotheses; nicify_goals; kill_bools; substpp); - cbn in *. - - Ltac simplify_val := repeat (simplify_local; unfold uvalueToZ, valueToPtr, Ptrofs.of_int, valueToInt, intToValue, - ptrToValue in *). - - Ltac crush_val := simplify_val; try discriminate; try congruence; try lia; liapp; try assumption. - - Ltac small_tac := repeat (crush_val; try array; try ptrofs); crush_val; auto. - Ltac big_tac := repeat (crush_val; try array; try ptrofs; try tac0); crush_val; auto. - - Lemma match_assocmaps_merge_empty: - forall f rs ars, - match_assocmaps f rs ars -> - match_assocmaps f rs (AssocMapExt.merge value empty_assocmap ars). - Proof. - inversion 1; subst; clear H. - constructor; intros. - rewrite merge_get_default2; auto. - Qed. - - Opaque AssocMap.get. - Opaque AssocMap.set. - Opaque AssocMapExt.merge. - Opaque Verilog.merge_arr. - - Lemma match_assocmaps_ext : - forall f rs ars1 ars2, - (forall x, Ple x (RTL.max_reg_function f) -> ars1 ! x = ars2 ! x) -> - match_assocmaps f rs ars1 -> - match_assocmaps f rs ars2. - Proof. - intros * YFRL YMATCH. - inv YMATCH. constructor; intros x' YPLE. - unfold "#", AssocMapExt.get_default in *. - rewrite <- YFRL by auto. - eauto. - Qed. - - Lemma transl_inop_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (sp : Values.val) (pc : positive) - (rs : RTL.regset) (m : mem) (pc' : RTL.node), - (RTL.fn_code f) ! pc = Some (RTL.Inop pc') -> - forall R1 : HTL.state, - match_states (RTL.State s f sp pc rs m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ match_states (RTL.State s f sp pc' rs m) R2. - Proof. - intros s f sp pc rs m pc' H R1 MSTATE. - inv_state. - - unfold match_prog in TRANSL. - econstructor. - split. - apply Smallstep.plus_one. - eapply HTL.step_module; eauto. - inv CONST; assumption. - inv CONST; assumption. - (* processing of state *) - econstructor. - crush. - econstructor. - econstructor. - econstructor. - big_tac. - cbn. - - solve [inv MARR; big_tac]. - - (* inv MARR; big_tac. *) - inv MARR; big_tac; auto. - - - eapply match_assocmaps_ext; [|eauto]; intros. - repeat unfold_merge. rewrite AssocMap.gso by (unfold Ple in *; lia). - rewrite AssocMapExt.merge_base_1; auto. - - rewrite <- H1. unfold Verilog.merge_arrs. - rewrite !AssocMap.gcombine by auto. rewrite !AssocMap.gss. - setoid_rewrite H1. - repeat erewrite Verilog.merge_arr_empty2; eauto. - - inv CONST; cbn in *. constructor; cbn in *. - + repeat unfold_merge. rewrite AssocMap.gso by lia. - unfold_merge; auto. - + repeat unfold_merge. rewrite AssocMap.gso by lia. - unfold_merge; auto. - - Unshelve. exact tt. - Qed. - #[local] Hint Resolve transl_inop_correct : htlproof. - - Lemma transl_iop_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (sp : Values.val) (pc : positive) - (rs : Registers.Regmap.t Values.val) (m : mem) (op : Op.operation) (args : list Registers.reg) - (res0 : Registers.reg) (pc' : RTL.node) (v : Values.val), - (RTL.fn_code f) ! pc = Some (RTL.Iop op args res0 pc') -> - Op.eval_operation ge sp op (map (fun r : positive => Registers.Regmap.get r rs) args) m = Some v -> - forall R1 : HTL.state, - match_states (RTL.State s f sp pc rs m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ - match_states (RTL.State s f sp pc' (Registers.Regmap.set res0 v rs) m) R2. - Proof. - intros s f sp pc rs m op args res0 pc' v H H0 R1 MSTATE. - inv_state. inv MARR. - exploit eval_correct; eauto. intros. inversion H1. inversion H2. - econstructor. split. - apply Smallstep.plus_one. - eapply HTL.step_module; eauto. - inv CONST. assumption. - inv CONST. assumption. - econstructor; simpl; trivial. - constructor; trivial. - econstructor; simpl; eauto. - simpl. econstructor. econstructor. - apply H5. simplify. - - all: big_tac. - - - assert (HPle: Ple res0 (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_def; eauto; simpl; auto). - unfold Ple in HPle. lia. - - eapply match_assocmaps_merge_empty. eapply match_assocmaps_ext; intros. - unfold Ple in *. instantiate (1 := asr # res0 <- x). - destruct (peq res0 x1); subst. - + rewrite merge_get_default4 with (x := x); - apply AssocMap.gss. - + rewrite merge_get_default3; [now rewrite AssocMap.gso by auto|]. - rewrite AssocMap.gso by auto. - now rewrite AssocMap.gso by lia. - + now apply regs_lessdef_add_match. - - assert (HPle: Ple res0 (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_def; eauto; simpl; auto). - unfold Ple in HPle. lia. - - unfold Verilog.merge_arrs. - rewrite ! AssocMap.gcombine by auto. rewrite ! AssocMap.gss. - erewrite ! Verilog.merge_arr_empty2; eauto. - erewrite ! Verilog.merge_arr_empty2; eauto. - - assumption. - - eapply op_stack_based; eauto. - - assert (HPle: Ple res0 (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_def; eauto; simpl; auto). - unfold Ple in *. - inv CONST. constructor; cbn. - repeat rewrite merge_get_default3 by solve [auto | lia]. - rewrite merge_get_default3; [eauto | ]. - repeat rewrite AssocMap.gso by solve [auto | lia]. auto. - repeat rewrite merge_get_default3 by solve [auto | lia]. - rewrite merge_get_default3; [eauto | ]. - repeat rewrite AssocMap.gso by solve [auto | lia]. auto. - Unshelve. apply tt. - Qed. - #[local] Hint Resolve transl_iop_correct : htlproof. - - Ltac tac := - repeat match goal with - | [ _ : error _ _ = OK _ _ _ |- _ ] => discriminate - | [ _ : context[if (?x && ?y) then _ else _] |- _ ] => - let EQ1 := fresh "EQ" in - let EQ2 := fresh "EQ" in - destruct x eqn:EQ1; destruct y eqn:EQ2; simpl in * - | [ _ : context[if ?x then _ else _] |- _ ] => - let EQ := fresh "EQ" in - destruct x eqn:EQ; simpl in * - | [ H : ret _ _ = _ |- _ ] => inv H - | [ _ : context[match ?x with | _ => _ end] |- _ ] => destruct x - end. - - Ltac inv_arr_access := - match goal with - | [ _ : translate_arr_access ?chunk ?addr ?args _ _ = OK ?c _ _ |- _] => - destruct c, chunk, addr, args; crush; tac; crush - end. - - Lemma offset_expr_ok : - forall v z, (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ v)) - (Integers.Ptrofs.of_int (Integers.Int.repr z))) - (Integers.Ptrofs.repr 4))) - = valueToNat (Int.divu (Int.add v (ZToValue z)) (ZToValue 4))). - Proof. - simplify_val. unfold valueToNat. unfold Int.divu, Ptrofs.divu. - pose proof Integers.Ptrofs.agree32_add as AGR. - unfold Integers.Ptrofs.agree32 in AGR. - assert (Ptrofs.unsigned (Ptrofs.add (Ptrofs.repr (Int.unsigned v)) - (Ptrofs.repr (Int.unsigned (Int.repr z)))) = - Int.unsigned (Int.add v (ZToValue z))). - apply AGR; auto. - apply Ptrofs.unsigned_repr. apply Int.unsigned_range_2. - apply Ptrofs.unsigned_repr. apply Int.unsigned_range_2. - rewrite H. replace (Ptrofs.unsigned (Ptrofs.repr 4)) with 4. - replace (Int.unsigned (ZToValue 4)) with 4. - pose proof Ptrofs.agree32_repr. unfold Ptrofs.agree32 in *. - rewrite H0. trivial. auto. - unfold ZToValue. symmetry. apply Int.unsigned_repr. - unfold_constants. lia. - unfold ZToValue. symmetry. apply Int.unsigned_repr. - unfold_constants. lia. - Qed. - - Lemma offset_expr_ok_2 : - forall v0 v1 z0 z1, - (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ v0)) - (Integers.Ptrofs.of_int - (Integers.Int.add - (Integers.Int.mul (valueToInt v1) (Integers.Int.repr z1)) - (Integers.Int.repr z0)))) (Ptrofs.repr 4)))) - = valueToNat (Int.divu (Int.add (Int.add v0 (ZToValue z0)) - (Int.mul v1 (ZToValue z1))) (ZToValue 4)). - intros. unfold ZToValue, valueToNat, valueToInt, Ptrofs.divu, Int.divu, Ptrofs.of_int. - - assert (H : (Ptrofs.unsigned - (Ptrofs.add (Ptrofs.repr (uvalueToZ v0)) - (Ptrofs.of_int (Int.add (Int.mul (valueToInt v1) (Int.repr z1)) (Int.repr z0)))) / - Ptrofs.unsigned (Ptrofs.repr 4)) - = (Int.unsigned (Int.add (Int.add v0 (Int.repr z0)) (Int.mul v1 (Int.repr z1))) / - Int.unsigned (Int.repr 4))). - { unfold ZToValue, valueToNat, valueToInt, Ptrofs.divu, Int.divu, Ptrofs.of_int. - rewrite Ptrofs.unsigned_repr by (unfold_constants; lia). - rewrite Int.unsigned_repr by (unfold_constants; lia). - - unfold Ptrofs.of_int. rewrite Int.add_commut. - pose proof Integers.Ptrofs.agree32_add as AGR. unfold Ptrofs.agree32 in *. - erewrite AGR. - 3: { unfold uvalueToZ. rewrite Ptrofs.unsigned_repr. trivial. apply Int.unsigned_range_2. } - 3: { rewrite Ptrofs.unsigned_repr. trivial. apply Int.unsigned_range_2. } - rewrite Int.add_assoc. trivial. auto. - } - - rewrite <- H. auto. - - Qed. - - Lemma offset_expr_ok_3 : - forall OFFSET, - Z.to_nat (Ptrofs.unsigned (Ptrofs.divu OFFSET (Ptrofs.repr 4))) - = valueToNat (ZToValue (Ptrofs.unsigned OFFSET / 4)). - Proof. auto. Qed. - - Lemma transl_iload_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (sp : Values.val) (pc : positive) - (rs : Registers.Regmap.t Values.val) (m : mem) (chunk : AST.memory_chunk) - (addr : Op.addressing) (args : list Registers.reg) (dst : Registers.reg) - (pc' : RTL.node) (a v : Values.val), - (RTL.fn_code f) ! pc = Some (RTL.Iload chunk addr args dst pc') -> - Op.eval_addressing ge sp addr (map (fun r : positive => Registers.Regmap.get r rs) args) = Some a -> - Mem.loadv chunk m a = Some v -> - forall R1 : HTL.state, - match_states (RTL.State s f sp pc rs m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ - match_states (RTL.State s f sp pc' (Registers.Regmap.set dst v rs) m) R2. - Proof. - (* intros s f sp pc rs m chunk addr args dst pc' a v H H0 H1 R1 MSTATE. *) - (* inv_state. inv_arr_access. *) - - (* + (** Preamble *) *) - (* inv MARR. inv CONST. crush. *) - - (* unfold Op.eval_addressing in H0. *) - (* destruct (Archi.ptr64) eqn:ARCHI; crush. *) - - (* unfold reg_stack_based_pointers in RSBP. *) - (* pose proof (RSBP r0) as RSBPr0. *) - - (* destruct (Registers.Regmap.get r0 rs) eqn:EQr0; crush. *) - - (* rewrite ARCHI in H1. crush. *) - (* subst. *) - - (* pose proof MASSOC as MASSOC'. *) - (* inv MASSOC'. *) - (* pose proof (H0 r0). *) - (* assert (HPler0 : Ple r0 (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; crush; eauto). *) - (* apply H0 in HPler0. *) - (* inv HPler0; try congruence. *) - (* rewrite EQr0 in H11. *) - (* inv H11. *) - - (* unfold check_address_parameter_signed in *; *) - (* unfold check_address_parameter_unsigned in *; crush. *) - - (* remember (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ asr # r0)) *) - (* (Integers.Ptrofs.of_int (Integers.Int.repr z))) as OFFSET. *) - - (* (** Modular preservation proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET mod 4 = 0) as MOD_PRESERVE. *) - (* { apply Mem.load_valid_access in H1. unfold Mem.valid_access in *. simplify. *) - (* apply Zdivide_mod. assumption. } *) - - (* (** Read bounds proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as READ_BOUND_HIGH. *) - (* { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:EQ; crush; auto. *) - (* unfold stack_bounds in BOUNDS. *) - (* exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET)); auto. *) - (* split; try lia; apply Integers.Ptrofs.unsigned_range_2. *) - (* small_tac. } *) - - (* (** Normalisation proof *) *) - (* assert (Integers.Ptrofs.repr *) - (* (4 * Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4))) = OFFSET) *) - (* as NORMALISE. *) - (* { replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) at 1 by reflexivity. *) - (* rewrite <- PtrofsExtra.mul_unsigned. *) - (* apply PtrofsExtra.mul_divu; crush; auto. } *) - - (* (** Normalised bounds proof *) *) - (* assert (0 <= *) - (* Integers.Ptrofs.unsigned (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4)) *) - (* < (RTL.fn_stacksize f / 4)) *) - (* as NORMALISE_BOUND. *) - (* { split. *) - (* apply Integers.Ptrofs.unsigned_range_2. *) - (* assert (HDIV: forall x y, Integers.Ptrofs.divu x y = Integers.Ptrofs.divu x y ) by reflexivity. *) - (* unfold Integers.Ptrofs.divu at 2 in HDIV. *) - (* rewrite HDIV. clear HDIV. *) - (* rewrite Integers.Ptrofs.unsigned_repr; crush. *) - (* apply Zmult_lt_reg_r with (p := 4); try lia. *) - (* repeat rewrite ZLib.div_mul_undo; try lia. *) - (* apply Z.div_pos; small_tac. *) - (* apply Z.div_le_upper_bound; small_tac. } *) - - (* inversion NORMALISE_BOUND as [ NORMALISE_BOUND_LOW NORMALISE_BOUND_HIGH ]; *) - (* clear NORMALISE_BOUND. *) - - (* (** Start of proof proper *) *) - (* eexists. split. *) - (* eapply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor. econstructor. econstructor. crush. *) - (* econstructor. econstructor. econstructor. crush. *) - (* econstructor. econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* econstructor. econstructor. *) - - (* all: big_tac. *) - - (* 1: { *) - (* assert (HPle : Ple dst (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_def. eassumption. auto. *) - (* apply ZExtra.Pge_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. *) - (* } *) - - (* 2: { *) - (* assert (HPle : Ple dst (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_def. eassumption. auto. *) - (* apply ZExtra.Pge_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. *) - (* } *) - - (* (** Match assocmaps *) *) - (* apply regs_lessdef_add_match; big_tac. *) - - (* (** Equality proof *) *) - (* rewrite <- offset_expr_ok. *) - - (* specialize (H9 (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu *) - (* OFFSET *) - (* (Integers.Ptrofs.repr 4)))). *) - (* exploit H9; big_tac. *) - - (* (** RSBP preservation *) *) - (* unfold arr_stack_based_pointers in ASBP. *) - (* specialize (ASBP (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4)))). *) - (* exploit ASBP; big_tac. *) - (* rewrite NORMALISE in H14. rewrite HeqOFFSET in H14. rewrite H1 in H14. assumption. *) - (* constructor; simplify. rewrite AssocMap.gso. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* assert (HPle: Ple dst (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) - (* unfold Ple in HPle. lia. *) - (* rewrite AssocMap.gso. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* assert (HPle: Ple dst (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) - (* unfold Ple in HPle. lia. *) - (* + (** Preamble *) *) - (* inv MARR. inv CONST. crush. *) - - (* unfold Op.eval_addressing in H0. *) - (* destruct (Archi.ptr64) eqn:ARCHI; crush. *) - - (* unfold reg_stack_based_pointers in RSBP. *) - (* pose proof (RSBP r0) as RSBPr0. *) - (* pose proof (RSBP r1) as RSBPr1. *) - - (* destruct (Registers.Regmap.get r0 rs) eqn:EQr0; *) - (* destruct (Registers.Regmap.get r1 rs) eqn:EQr1; crush. *) - - (* rewrite ARCHI in H1. crush. *) - (* subst. *) - (* clear RSBPr1. *) - - (* pose proof MASSOC as MASSOC'. *) - (* inv MASSOC'. *) - (* pose proof (H0 r0). *) - (* pose proof (H0 r1). *) - (* assert (HPler0 : Ple r0 (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; crush; eauto). *) - (* assert (HPler1 : Ple r1 (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; simpl; auto). *) - (* apply H8 in HPler0. *) - (* apply H11 in HPler1. *) - (* inv HPler0; inv HPler1; try congruence. *) - (* rewrite EQr0 in H13. *) - (* rewrite EQr1 in H14. *) - (* inv H13. inv H14. *) - (* clear H0. clear H8. clear H11. *) - - (* unfold check_address_parameter_signed in *; *) - (* unfold check_address_parameter_unsigned in *; crush. *) - - (* remember (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ asr # r0)) *) - (* (Integers.Ptrofs.of_int *) - (* (Integers.Int.add (Integers.Int.mul (valueToInt asr # r1) (Integers.Int.repr z)) *) - (* (Integers.Int.repr z0)))) as OFFSET. *) - - (* (** Modular preservation proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET mod 4 = 0) as MOD_PRESERVE. *) - (* { apply Mem.load_valid_access in H1. unfold Mem.valid_access in *. simplify. *) - (* apply Zdivide_mod. assumption. } *) - - (* (** Read bounds proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as READ_BOUND_HIGH. *) - (* { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:EQ; crush; auto. *) - (* unfold stack_bounds in BOUNDS. *) - (* exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET)); auto. *) - (* split; try lia; apply Integers.Ptrofs.unsigned_range_2. *) - (* small_tac. } *) - - (* (** Normalisation proof *) *) - (* assert (Integers.Ptrofs.repr *) - (* (4 * Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4))) = OFFSET) *) - (* as NORMALISE. *) - (* { replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) at 1 by reflexivity. *) - (* rewrite <- PtrofsExtra.mul_unsigned. *) - (* apply PtrofsExtra.mul_divu; crush. } *) - - (* (** Normalised bounds proof *) *) - (* assert (0 <= *) - (* Integers.Ptrofs.unsigned (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4)) *) - (* < (RTL.fn_stacksize f / 4)) *) - (* as NORMALISE_BOUND. *) - (* { split. *) - (* apply Integers.Ptrofs.unsigned_range_2. *) - (* assert (forall x y, Integers.Ptrofs.divu x y = Integers.Ptrofs.divu x y ) by reflexivity. *) - (* unfold Integers.Ptrofs.divu at 2 in H14. *) - (* rewrite H14. clear H14. *) - (* rewrite Integers.Ptrofs.unsigned_repr; crush. *) - (* apply Zmult_lt_reg_r with (p := 4); try lia. *) - (* repeat rewrite ZLib.div_mul_undo; try lia. *) - (* apply Z.div_pos; small_tac. *) - (* apply Z.div_le_upper_bound; lia. } *) - - (* inversion NORMALISE_BOUND as [ NORMALISE_BOUND_LOW NORMALISE_BOUND_HIGH ]; *) - (* clear NORMALISE_BOUND. *) - - (* (** Start of proof proper *) *) - (* eexists. split. *) - (* eapply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor. econstructor. econstructor. crush. *) - (* econstructor. econstructor. econstructor. crush. *) - (* econstructor. econstructor. econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* econstructor. econstructor. auto. econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* all: big_tac. *) - - (* 1: { assert (HPle : Ple dst (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_def. eassumption. auto. *) - (* apply ZExtra.Pge_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. } *) - - (* 2: { assert (HPle : Ple dst (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_def. eassumption. auto. *) - (* apply ZExtra.Pge_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. } *) - - (* (** Match assocmaps *) *) - (* apply regs_lessdef_add_match; big_tac. *) - - (* (** Equality proof *) *) - (* rewrite <- offset_expr_ok_2. *) - - (* specialize (H9 (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu *) - (* OFFSET *) - (* (Integers.Ptrofs.repr 4)))). *) - (* exploit H9; big_tac. *) - - (* (** RSBP preservation *) *) - (* unfold arr_stack_based_pointers in ASBP. *) - (* specialize (ASBP (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4)))). *) - (* exploit ASBP; big_tac. *) - (* rewrite NORMALISE in H14. rewrite HeqOFFSET in H14. rewrite H1 in H14. assumption. *) - - (* constructor; simplify. rewrite AssocMap.gso. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* assert (HPle: Ple dst (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) - (* unfold Ple in HPle. lia. *) - (* rewrite AssocMap.gso. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* assert (HPle: Ple dst (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) - (* unfold Ple in HPle. lia. *) - - (* + inv MARR. inv CONST. crush. *) - - (* unfold Op.eval_addressing in H0. *) - (* destruct (Archi.ptr64) eqn:ARCHI; crush. *) - (* rewrite ARCHI in H0. crush. *) - - (* unfold check_address_parameter_unsigned in *; *) - (* unfold check_address_parameter_signed in *; crush. *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* rewrite ZERO in H1. clear ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l in H1. *) - - (* remember i0 as OFFSET. *) - - (* (** Modular preservation proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET mod 4 = 0) as MOD_PRESERVE. *) - (* { apply Mem.load_valid_access in H1. unfold Mem.valid_access in *. simplify. *) - (* apply Zdivide_mod. assumption. } *) - - (* (** Read bounds proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as READ_BOUND_HIGH. *) - (* { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:?EQ; crush; auto. *) - (* unfold stack_bounds in BOUNDS. *) - (* exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET)); big_tac. } *) - - (* (** Normalisation proof *) *) - (* assert (Integers.Ptrofs.repr *) - (* (4 * Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4))) = OFFSET) *) - (* as NORMALISE. *) - (* { replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) at 1 by reflexivity. *) - (* rewrite <- PtrofsExtra.mul_unsigned. *) - (* apply PtrofsExtra.mul_divu; crush. } *) - - (* (** Normalised bounds proof *) *) - (* assert (0 <= *) - (* Integers.Ptrofs.unsigned (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4)) *) - (* < (RTL.fn_stacksize f / 4)) *) - (* as NORMALISE_BOUND. *) - (* { split. *) - (* apply Integers.Ptrofs.unsigned_range_2. *) - (* assert (forall x y, Integers.Ptrofs.divu x y = Integers.Ptrofs.divu x y ) by reflexivity. *) - (* unfold Integers.Ptrofs.divu at 2 in H0. *) - (* rewrite H0. clear H0. *) - (* rewrite Integers.Ptrofs.unsigned_repr; crush. *) - (* apply Zmult_lt_reg_r with (p := 4); try lia. *) - (* repeat rewrite ZLib.div_mul_undo; try lia. *) - (* apply Z.div_pos; small_tac. *) - (* apply Z.div_le_upper_bound; lia. } *) - - (* inversion NORMALISE_BOUND as [ NORMALISE_BOUND_LOW NORMALISE_BOUND_HIGH ]; *) - (* clear NORMALISE_BOUND. *) - - (* (** Start of proof proper *) *) - (* eexists. split. *) - (* eapply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor. econstructor. econstructor. crush. *) - (* econstructor. econstructor. econstructor. econstructor. crush. *) - - (* all: big_tac. *) - - (* 1: { assert (HPle : Ple dst (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_def. eassumption. auto. *) - (* apply ZExtra.Pge_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. } *) - - (* 2: { assert (HPle : Ple dst (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_def. eassumption. auto. *) - (* apply ZExtra.Pge_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. } *) - - (* (** Match assocmaps *) *) - (* apply regs_lessdef_add_match; big_tac. *) - - (* (** Equality proof *) *) - (* rewrite <- offset_expr_ok_3. *) - - (* specialize (H9 (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu *) - (* OFFSET *) - (* (Integers.Ptrofs.repr 4)))). *) - (* exploit H9; big_tac. *) - - (* (** RSBP preservation *) *) - (* unfold arr_stack_based_pointers in ASBP. *) - (* specialize (ASBP (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.divu OFFSET (Integers.Ptrofs.repr 4)))). *) - (* exploit ASBP; big_tac. *) - (* rewrite NORMALISE in H0. rewrite H1 in H0. assumption. *) - - (* constructor; simplify. rewrite AssocMap.gso. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* assert (HPle: Ple dst (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) - (* unfold Ple in HPle. lia. *) - (* rewrite AssocMap.gso. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* assert (HPle: Ple dst (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) - (* unfold Ple in HPle. lia. *) - - (* Unshelve. *) - (* exact (Values.Vint (Int.repr 0)). *) - (* exact tt. *) - (* exact (Values.Vint (Int.repr 0)). *) - (* exact tt. *) - (* exact (Values.Vint (Int.repr 0)). *) - (* exact tt. *) - (* Qed. *) - Admitted. - #[local] Hint Resolve transl_iload_correct : htlproof. - - Lemma transl_istore_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (sp : Values.val) (pc : positive) - (rs : Registers.Regmap.t Values.val) (m : mem) (chunk : AST.memory_chunk) - (addr : Op.addressing) (args : list Registers.reg) (src : Registers.reg) - (pc' : RTL.node) (a : Values.val) (m' : mem), - (RTL.fn_code f) ! pc = Some (RTL.Istore chunk addr args src pc') -> - Op.eval_addressing ge sp addr (map (fun r : positive => Registers.Regmap.get r rs) args) = Some a -> - Mem.storev chunk m a (Registers.Regmap.get src rs) = Some m' -> - forall R1 : HTL.state, - match_states (RTL.State s f sp pc rs m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ match_states (RTL.State s f sp pc' rs m') R2. - Proof. - (* intros s f sp pc rs m chunk addr args src pc' a m' H H0 H1 R1 MSTATES. *) - (* inv_state. inv_arr_access. *) - - (* + (** Preamble *) *) - (* inv MARR. inv CONST. crush. *) - - (* unfold Op.eval_addressing in H0. *) - (* destruct (Archi.ptr64) eqn:ARCHI; crush. *) - - (* unfold reg_stack_based_pointers in RSBP. *) - (* pose proof (RSBP r0) as RSBPr0. *) - - (* destruct (Registers.Regmap.get r0 rs) eqn:EQr0; crush. *) - - (* rewrite ARCHI in H1. crush. *) - (* subst. *) - - (* pose proof MASSOC as MASSOC'. *) - (* inv MASSOC'. *) - (* pose proof (H0 r0). *) - (* assert (HPler0 : Ple r0 (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; crush; eauto). *) - (* apply H8 in HPler0. *) - (* inv HPler0; try congruence. *) - (* rewrite EQr0 in H11. *) - (* inv H11. *) - (* clear H0. clear H8. *) - - (* unfold check_address_parameter_unsigned in *; *) - (* unfold check_address_parameter_signed in *; crush. *) - - (* remember (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ asr # r0)) *) - (* (Integers.Ptrofs.of_int (Integers.Int.repr z))) as OFFSET. *) - - (* (** Modular preservation proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET mod 4 = 0) as MOD_PRESERVE. *) - (* { apply Mem.store_valid_access_3 in H1. unfold Mem.valid_access in *. simplify. *) - (* apply Zdivide_mod. assumption. } *) - - (* (** Write bounds proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as WRITE_BOUND_HIGH. *) - (* { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:EQ; crush; auto. *) - (* unfold stack_bounds in BOUNDS. *) - (* exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET) (Registers.Regmap.get src rs)); big_tac. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - - (* (** Start of proof proper *) *) - (* eexists. split. *) - (* eapply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor. econstructor. econstructor. *) - (* eapply Verilog.stmnt_runp_Vnonblock_arr. crush. *) - (* econstructor. *) - (* econstructor. *) - (* econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - - (* all: try constructor; crush. *) - - (* (** State Lookup *) *) - (* unfold Verilog.merge_regs. *) - (* crush. *) - (* unfold_merge. *) - (* apply AssocMap.gss. *) - - (* (** Match states *) *) - (* econstructor; eauto. *) - - (* (** Match assocmaps *) *) - (* unfold Verilog.merge_regs. crush. unfold_merge. *) - (* apply regs_lessdef_add_greater. *) - (* unfold Plt; lia. *) - (* assumption. *) - - (* (** States well formed *) *) - (* unfold state_st_wf. inversion 1. crush. *) - (* unfold Verilog.merge_regs. *) - (* unfold_merge. *) - (* apply AssocMap.gss. *) - - (* (** Equality proof *) *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* inversion MASSOC; revert HeqOFFSET; subst; clear MASSOC; intros HeqOFFSET. *) - - (* econstructor. *) - (* repeat split; crush. *) - (* unfold HTL.empty_stack. *) - (* crush. *) - (* unfold Verilog.merge_arrs. *) - - (* rewrite AssocMap.gcombine by reflexivity. *) - (* rewrite AssocMap.gss. *) - (* erewrite Verilog.merge_arr_empty2. *) - (* unfold Verilog.arr_assocmap_set. *) - (* rewrite AssocMap.gcombine by reflexivity. *) - (* rewrite AssocMap.gss. *) - (* rewrite AssocMap.gss. *) - (* unfold Verilog.merge_arr. *) - (* setoid_rewrite H7. *) - (* reflexivity. *) - - (* rewrite AssocMap.gcombine by reflexivity. *) - (* unfold Verilog.merge_arr. *) - (* unfold Verilog.arr_assocmap_set. *) - (* rewrite AssocMap.gss. *) - (* rewrite AssocMap.gss. *) - (* setoid_rewrite H7. *) - (* reflexivity. *) - - (* rewrite combine_length. *) - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* symmetry. *) - (* apply list_repeat_len. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite H4. *) - (* apply list_repeat_len. *) - - (* rewrite combine_length. *) - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* apply list_repeat_len. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite H4. *) - (* apply list_repeat_len. *) - - (* remember (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ asr # r0)) *) - (* (Integers.Ptrofs.of_int (Integers.Int.repr z))) as OFFSET. *) - - (* destruct (4 * ptr ==Z Integers.Ptrofs.unsigned OFFSET). *) - - (* erewrite Mem.load_store_same. *) - (* 2: { rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite e. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* rewrite HeqOFFSET. *) - (* exact H1. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - (* constructor. *) - (* erewrite combine_lookup_second. *) - (* simplify. *) - (* assert (HPle : Ple src (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) - (* apply H11 in HPle. *) - (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; inv HPle; eauto. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite list_repeat_len. auto. *) - - (* assert (HMul : 4 * ptr / 4 = Integers.Ptrofs.unsigned OFFSET / 4) by (f_equal; assumption). *) - (* rewrite Z.mul_comm in HMul. *) - (* rewrite Z_div_mult in HMul; try lia. *) - (* replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in HMul by reflexivity. *) - (* rewrite <- PtrofsExtra.divu_unsigned in HMul; unfold_constants; try lia. *) - (* rewrite HMul. rewrite <- offset_expr_ok. *) - (* rewrite HeqOFFSET. *) - (* rewrite array_get_error_set_bound. *) - (* reflexivity. *) - (* unfold arr_length, arr_repeat. simpl. *) - (* rewrite list_repeat_len. rewrite HeqOFFSET in HMul. lia. *) - - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* simpl. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* destruct (Z_le_gt_dec (4 * ptr + 4) (Integers.Ptrofs.unsigned OFFSET)); eauto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* left; auto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* right. *) - (* apply ZExtra.mod_0_bounds; try lia. *) - (* apply ZLib.Z_mod_mult'. *) - (* rewrite Z2Nat.id in H15; try lia. *) - (* apply Zmult_lt_compat_r with (p := 4) in H15; try lia. *) - (* rewrite ZLib.div_mul_undo in H15; try lia. *) - (* split; try lia. *) - (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) - (* } *) - - (* rewrite <- offset_expr_ok. *) - (* rewrite PtrofsExtra.divu_unsigned; auto; try (unfold_constants; lia). *) - (* destruct (ptr ==Z Integers.Ptrofs.unsigned OFFSET / 4). *) - (* apply Z.mul_cancel_r with (p := 4) in e; try lia. *) - (* rewrite ZLib.div_mul_undo in e; try lia. *) - (* rewrite combine_lookup_first. *) - (* eapply H9; eauto. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite list_repeat_len. auto. *) - (* rewrite array_gso. *) - (* unfold array_get_error. *) - (* unfold arr_repeat. *) - (* crush. *) - (* apply list_repeat_lookup. *) - (* lia. *) - (* unfold_constants. *) - (* intro. *) - (* apply Z2Nat.inj_iff in H13; rewrite HeqOFFSET in n0; try lia. *) - (* apply Z.div_pos; try lia. *) - (* apply Integers.Ptrofs.unsigned_range. *) - (* apply Integers.Ptrofs.unsigned_range_2. *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO1 by reflexivity. *) - (* unfold arr_stack_based_pointers. *) - (* intros. *) - (* destruct (4 * ptr ==Z Integers.Ptrofs.unsigned OFFSET). *) - - (* crush. *) - (* erewrite Mem.load_store_same. *) - (* 2: { rewrite ZERO1. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite e. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* exact H1. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - (* crush. *) - (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; try constructor. *) - (* destruct (Archi.ptr64); try discriminate. *) - (* pose proof (RSBP src). rewrite EQ_SRC in H11. *) - (* assumption. *) - - (* simpl. *) - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. *) - (* rewrite ZERO1. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* simpl. *) - (* destruct (Z_le_gt_dec (4 * ptr + 4) (Integers.Ptrofs.unsigned OFFSET)); eauto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* left; auto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* right. *) - (* apply ZExtra.mod_0_bounds; try lia. *) - (* apply ZLib.Z_mod_mult'. *) - (* inv H11. *) - (* apply Zmult_lt_compat_r with (p := 4) in H14; try lia. *) - (* rewrite ZLib.div_mul_undo in H14; try lia. *) - (* split; try lia. *) - (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) - (* } *) - (* apply ASBP; assumption. *) - - (* unfold stack_bounds in *. intros. *) - (* simpl. *) - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { rewrite HeqOFFSET in *. simplify_val. *) - (* right. right. simpl. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr; crush; try lia. *) - (* apply ZExtra.mod_0_bounds; crush; try lia. } *) - (* crush. *) - (* exploit (BOUNDS ptr); try lia. intros. crush. *) - (* exploit (BOUNDS ptr v); try lia. intros. *) - (* inv H11. *) - (* match goal with | |- ?x = _ => destruct x eqn:EQ end; try reflexivity. *) - (* assert (Mem.valid_access m AST.Mint32 sp' *) - (* (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.add (Integers.Ptrofs.repr 0) *) - (* (Integers.Ptrofs.repr ptr))) Writable). *) - (* { pose proof H1. eapply Mem.store_valid_access_2 in H11. *) - (* exact H11. eapply Mem.store_valid_access_3. eassumption. } *) - (* pose proof (Mem.valid_access_store m AST.Mint32 sp' *) - (* (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.add (Integers.Ptrofs.repr 0) *) - (* (Integers.Ptrofs.repr ptr))) v). *) - (* apply X in H11. inv H11. congruence. *) - - (* constructor; simplify. unfold Verilog.merge_regs. unfold_merge. *) - (* rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* unfold Verilog.merge_regs. unfold_merge. *) - (* rewrite AssocMap.gso. *) - (* assumption. lia. *) - - (* + (** Preamble *) *) - (* inv MARR. inv CONST. crush. *) - - (* unfold Op.eval_addressing in H0. *) - (* destruct (Archi.ptr64) eqn:ARCHI; crush. *) - - (* unfold reg_stack_based_pointers in RSBP. *) - (* pose proof (RSBP r0) as RSBPr0. *) - (* pose proof (RSBP r1) as RSBPr1. *) - - (* destruct (Registers.Regmap.get r0 rs) eqn:EQr0; *) - (* destruct (Registers.Regmap.get r1 rs) eqn:EQr1; crush. *) - - (* rewrite ARCHI in H1. crush. *) - (* subst. *) - (* clear RSBPr1. *) - - (* pose proof MASSOC as MASSOC'. *) - (* inv MASSOC'. *) - (* pose proof (H0 r0). *) - (* pose proof (H0 r1). *) - (* assert (HPler0 : Ple r0 (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; crush; eauto). *) - (* assert (HPler1 : Ple r1 (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; simpl; auto). *) - (* apply H8 in HPler0. *) - (* apply H11 in HPler1. *) - (* inv HPler0; inv HPler1; try congruence. *) - (* rewrite EQr0 in H13. *) - (* rewrite EQr1 in H14. *) - (* inv H13. inv H14. *) - (* clear H0. clear H8. clear H11. *) - - (* unfold check_address_parameter_signed in *; *) - (* unfold check_address_parameter_unsigned in *; crush. *) - - (* remember (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ asr # r0)) *) - (* (Integers.Ptrofs.of_int *) - (* (Integers.Int.add (Integers.Int.mul (valueToInt asr # r1) (Integers.Int.repr z)) *) - (* (Integers.Int.repr z0)))) as OFFSET. *) - - (* (** Modular preservation proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET mod 4 = 0) as MOD_PRESERVE. *) - (* { apply Mem.store_valid_access_3 in H1. unfold Mem.valid_access in *. simplify. *) - (* apply Zdivide_mod. assumption. } *) - - (* (** Write bounds proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as WRITE_BOUND_HIGH. *) - (* { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:EQ; crush; auto. *) - (* unfold stack_bounds in BOUNDS. *) - (* exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET) (Registers.Regmap.get src rs)); auto. *) - (* split; try lia; apply Integers.Ptrofs.unsigned_range_2. *) - (* small_tac. } *) - - (* (** Start of proof proper *) *) - (* eexists. split. *) - (* eapply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor. econstructor. econstructor. *) - (* eapply Verilog.stmnt_runp_Vnonblock_arr. crush. *) - (* econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - (* econstructor. econstructor. econstructor. econstructor. *) - - (* all: try constructor; crush. *) - - (* (** State Lookup *) *) - (* unfold Verilog.merge_regs. *) - (* crush. *) - (* unfold_merge. *) - (* apply AssocMap.gss. *) - - (* (** Match states *) *) - (* econstructor; eauto. *) - - (* (** Match assocmaps *) *) - (* unfold Verilog.merge_regs. crush. unfold_merge. *) - (* apply regs_lessdef_add_greater. *) - (* unfold Plt; lia. *) - (* assumption. *) - - (* (** States well formed *) *) - (* unfold state_st_wf. inversion 1. crush. *) - (* unfold Verilog.merge_regs. *) - (* unfold_merge. *) - (* apply AssocMap.gss. *) - - (* (** Equality proof *) *) - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* inversion MASSOC; revert HeqOFFSET; subst; clear MASSOC; intros HeqOFFSET. *) - - (* econstructor. *) - (* repeat split; crush. *) - (* unfold HTL.empty_stack. *) - (* crush. *) - (* unfold Verilog.merge_arrs. *) - - (* rewrite AssocMap.gcombine by reflexivity. *) - (* rewrite AssocMap.gss. *) - (* erewrite Verilog.merge_arr_empty2. *) - (* unfold Verilog.arr_assocmap_set. *) - (* rewrite AssocMap.gcombine by reflexivity. *) - (* rewrite AssocMap.gss. *) - (* rewrite AssocMap.gss. *) - (* unfold Verilog.merge_arr. *) - (* setoid_rewrite H7. *) - (* reflexivity. *) - - (* rewrite AssocMap.gcombine by reflexivity. *) - (* unfold Verilog.merge_arr. *) - (* unfold Verilog.arr_assocmap_set. *) - (* rewrite AssocMap.gss. *) - (* rewrite AssocMap.gss. *) - (* setoid_rewrite H7. *) - (* reflexivity. *) - - (* rewrite combine_length. *) - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* symmetry. *) - (* apply list_repeat_len. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite H4. *) - (* apply list_repeat_len. *) - - (* rewrite combine_length. *) - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* apply list_repeat_len. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite H4. *) - (* apply list_repeat_len. *) - - (* remember (Integers.Ptrofs.add (Integers.Ptrofs.repr (uvalueToZ asr # r0)) *) - (* (Integers.Ptrofs.of_int *) - (* (Integers.Int.add (Integers.Int.mul (valueToInt asr # r1) (Integers.Int.repr z)) *) - (* (Integers.Int.repr z0)))) as OFFSET. *) - (* destruct (4 * ptr ==Z Integers.Ptrofs.unsigned OFFSET). *) - - (* erewrite Mem.load_store_same. *) - (* 2: { rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite e. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* rewrite HeqOFFSET. *) - (* exact H1. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - (* constructor. *) - (* erewrite combine_lookup_second. *) - (* simpl. *) - (* assert (Ple src (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) - (* apply H14 in H15. *) - (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; inv H15; eauto. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite list_repeat_len. auto. *) - - (* assert (4 * ptr / 4 = Integers.Ptrofs.unsigned OFFSET / 4) by (f_equal; assumption). *) - (* rewrite Z.mul_comm in H15. *) - (* rewrite Z_div_mult in H15; try lia. *) - (* replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H15 by reflexivity. *) - (* rewrite <- PtrofsExtra.divu_unsigned in H15; unfold_constants; try lia. *) - (* rewrite H15. rewrite <- offset_expr_ok_2. *) - (* rewrite HeqOFFSET in *. *) - (* rewrite array_get_error_set_bound. *) - (* reflexivity. *) - (* unfold arr_length, arr_repeat. simpl. *) - (* rewrite list_repeat_len. lia. *) - - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* simpl. *) - (* destruct (Z_le_gt_dec (4 * ptr + 4) (Integers.Ptrofs.unsigned OFFSET)); eauto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* left; auto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* right. *) - (* apply ZExtra.mod_0_bounds; try lia. *) - (* apply ZLib.Z_mod_mult'. *) - (* rewrite Z2Nat.id in H17; try lia. *) - (* apply Zmult_lt_compat_r with (p := 4) in H17; try lia. *) - (* rewrite ZLib.div_mul_undo in H17; try lia. *) - (* split; try lia. *) - (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) - (* } *) - - (* rewrite <- offset_expr_ok_2. *) - (* rewrite PtrofsExtra.divu_unsigned; auto; try (unfold_constants; lia). *) - (* destruct (ptr ==Z Integers.Ptrofs.unsigned OFFSET / 4). *) - (* apply Z.mul_cancel_r with (p := 4) in e; try lia. *) - (* rewrite ZLib.div_mul_undo in e; try lia. *) - (* rewrite combine_lookup_first. *) - (* eapply H9; eauto. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite list_repeat_len. auto. *) - (* rewrite array_gso. *) - (* unfold array_get_error. *) - (* unfold arr_repeat. *) - (* crush. *) - (* apply list_repeat_lookup. *) - (* lia. *) - (* unfold_constants. *) - (* intro. *) - (* rewrite HeqOFFSET in *. *) - (* apply Z2Nat.inj_iff in H15; try lia. *) - (* apply Z.div_pos; try lia. *) - (* apply Integers.Ptrofs.unsigned_range. *) - (* apply Integers.Ptrofs.unsigned_range_2. *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO1 by reflexivity. *) - (* unfold arr_stack_based_pointers. *) - (* intros. *) - (* destruct (4 * ptr ==Z Integers.Ptrofs.unsigned OFFSET). *) - - (* crush. *) - (* erewrite Mem.load_store_same. *) - (* 2: { rewrite ZERO1. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite e. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* exact H1. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - (* crush. *) - (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; try constructor. *) - (* destruct (Archi.ptr64); try discriminate. *) - (* pose proof (RSBP src). rewrite EQ_SRC in H14. *) - (* assumption. *) - - (* simpl. *) - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. *) - (* rewrite ZERO1. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* simpl. *) - (* destruct (Z_le_gt_dec (4 * ptr + 4) (Integers.Ptrofs.unsigned OFFSET)); eauto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* left; auto. *) - (* rewrite HeqOFFSET in *. simplify_val. *) - (* right. *) - (* apply ZExtra.mod_0_bounds; try lia. *) - (* apply ZLib.Z_mod_mult'. *) - (* inv H14. *) - (* apply Zmult_lt_compat_r with (p := 4) in H16; try lia. *) - (* rewrite ZLib.div_mul_undo in H16; try lia. *) - (* split; try lia. *) - (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) - (* } *) - (* apply ASBP; assumption. *) - - (* unfold stack_bounds in *. intros. *) - (* simpl. *) - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { rewrite HeqOFFSET in *. simplify_val. *) - (* right. right. simpl. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr; crush; try lia. *) - (* apply ZExtra.mod_0_bounds; crush; try lia. } *) - (* crush. *) - (* exploit (BOUNDS ptr); try lia. intros. crush. *) - (* exploit (BOUNDS ptr v); try lia. intros. *) - (* simplify. *) - (* match goal with | |- ?x = _ => destruct x eqn:EQ end; try reflexivity. *) - (* assert (Mem.valid_access m AST.Mint32 sp' *) - (* (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.add (Integers.Ptrofs.repr 0) *) - (* (Integers.Ptrofs.repr ptr))) Writable). *) - (* { pose proof H1. eapply Mem.store_valid_access_2 in H14. *) - (* exact H14. eapply Mem.store_valid_access_3. eassumption. } *) - (* pose proof (Mem.valid_access_store m AST.Mint32 sp' *) - (* (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.add (Integers.Ptrofs.repr 0) *) - (* (Integers.Ptrofs.repr ptr))) v). *) - (* apply X in H14. inv H14. congruence. *) - - (* constructor; simplify. unfold Verilog.merge_regs. unfold_merge. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* unfold Verilog.merge_regs. unfold_merge. rewrite AssocMap.gso. *) - (* assumption. lia. *) - - (* + inv MARR. inv CONST. crush. *) - - (* unfold Op.eval_addressing in H0. *) - (* destruct (Archi.ptr64) eqn:ARCHI; crush. *) - (* rewrite ARCHI in H0. crush. *) - - (* unfold check_address_parameter_unsigned in *; *) - (* unfold check_address_parameter_signed in *; crush. *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* rewrite ZERO in H1. clear ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l in H1. *) - - (* remember i0 as OFFSET. *) - - (* (** Modular preservation proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET mod 4 = 0) as MOD_PRESERVE. *) - (* { apply Mem.store_valid_access_3 in H1. unfold Mem.valid_access in *. simplify. *) - (* apply Zdivide_mod. assumption. } *) - - (* (** Write bounds proof *) *) - (* assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as WRITE_BOUND_HIGH. *) - (* { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:?EQ; crush; auto. *) - (* unfold stack_bounds in BOUNDS. *) - (* exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET) (Registers.Regmap.get src rs)); auto. *) - (* crush. *) - (* replace (Integers.Ptrofs.repr 0) with (Integers.Ptrofs.zero) by reflexivity. *) - (* small_tac. } *) - - (* (** Start of proof proper *) *) - (* eexists. split. *) - (* eapply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor. econstructor. econstructor. *) - (* eapply Verilog.stmnt_runp_Vnonblock_arr. crush. *) - (* econstructor. econstructor. econstructor. econstructor. *) - - (* all: try constructor; crush. *) - - (* (** State Lookup *) *) - (* unfold Verilog.merge_regs. *) - (* crush. *) - (* unfold_merge. *) - (* apply AssocMap.gss. *) - - (* (** Match states *) *) - (* econstructor; eauto. *) - - (* (** Match assocmaps *) *) - (* unfold Verilog.merge_regs. crush. unfold_merge. *) - (* apply regs_lessdef_add_greater. *) - (* unfold Plt; lia. *) - (* assumption. *) - - (* (** States well formed *) *) - (* unfold state_st_wf. inversion 1. crush. *) - (* unfold Verilog.merge_regs. *) - (* unfold_merge. *) - (* apply AssocMap.gss. *) - - (* (** Equality proof *) *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* inversion MASSOC; revert HeqOFFSET; subst; clear MASSOC; intros HeqOFFSET. *) - - (* econstructor. *) - (* repeat split; crush. *) - (* unfold HTL.empty_stack. *) - (* crush. *) - (* unfold Verilog.merge_arrs. *) - - (* rewrite AssocMap.gcombine by reflexivity. *) - (* rewrite AssocMap.gss. *) - (* erewrite Verilog.merge_arr_empty2. *) - (* unfold Verilog.arr_assocmap_set. *) - (* rewrite AssocMap.gcombine by reflexivity. *) - (* rewrite AssocMap.gss. *) - (* rewrite AssocMap.gss. *) - (* unfold Verilog.merge_arr. *) - (* setoid_rewrite H7. *) - (* reflexivity. *) - - (* rewrite AssocMap.gcombine by reflexivity. *) - (* unfold Verilog.merge_arr. *) - (* unfold Verilog.arr_assocmap_set. *) - (* rewrite AssocMap.gss. *) - (* rewrite AssocMap.gss. *) - (* setoid_rewrite H7. *) - (* reflexivity. *) - - (* rewrite combine_length. *) - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* symmetry. *) - (* apply list_repeat_len. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite H4. *) - (* apply list_repeat_len. *) - - (* rewrite combine_length. *) - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* apply list_repeat_len. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite H4. *) - (* apply list_repeat_len. *) - - (* remember i0 as OFFSET. *) - (* destruct (4 * ptr ==Z Integers.Ptrofs.unsigned OFFSET). *) - - (* erewrite Mem.load_store_same. *) - (* 2: { rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite e. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* exact H1. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - (* constructor. *) - (* erewrite combine_lookup_second. *) - (* simpl. *) - (* assert (Ple src (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) - (* apply H0 in H8. *) - (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; inv H8; eauto. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite list_repeat_len. auto. *) - - (* assert (4 * ptr / 4 = Integers.Ptrofs.unsigned OFFSET / 4) by (f_equal; assumption). *) - (* rewrite Z.mul_comm in H8. *) - (* rewrite Z_div_mult in H8; try lia. *) - (* replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H8 by reflexivity. *) - (* rewrite <- PtrofsExtra.divu_unsigned in H8; unfold_constants; try lia. *) - (* rewrite H8. rewrite <- offset_expr_ok_3. *) - (* rewrite array_get_error_set_bound. *) - (* reflexivity. *) - (* unfold arr_length, arr_repeat. simpl. *) - (* rewrite list_repeat_len. lia. *) - - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* simpl. *) - (* destruct (Z_le_gt_dec (4 * ptr + 4) (Integers.Ptrofs.unsigned OFFSET)); eauto. *) - (* right. *) - (* apply ZExtra.mod_0_bounds; try lia. *) - (* apply ZLib.Z_mod_mult'. *) - (* rewrite Z2Nat.id in H13; try lia. *) - (* apply Zmult_lt_compat_r with (p := 4) in H13; try lia. *) - (* rewrite ZLib.div_mul_undo in H13; try lia. *) - (* split; try lia. *) - (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) - (* } *) - - (* rewrite <- offset_expr_ok_3. *) - (* rewrite PtrofsExtra.divu_unsigned; auto; try (unfold_constants; lia). *) - (* destruct (ptr ==Z Integers.Ptrofs.unsigned OFFSET / 4). *) - (* apply Z.mul_cancel_r with (p := 4) in e; try lia. *) - (* rewrite ZLib.div_mul_undo in e; try lia. *) - (* rewrite combine_lookup_first. *) - (* eapply H9; eauto. *) - - (* rewrite <- array_set_len. *) - (* unfold arr_repeat. crush. *) - (* rewrite list_repeat_len. auto. *) - (* rewrite array_gso. *) - (* unfold array_get_error. *) - (* unfold arr_repeat. *) - (* crush. *) - (* apply list_repeat_lookup. *) - (* lia. *) - (* unfold_constants. *) - (* intro. *) - (* apply Z2Nat.inj_iff in H8; try lia. *) - (* apply Z.div_pos; try lia. *) - (* apply Integers.Ptrofs.unsigned_range. *) - - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* unfold arr_stack_based_pointers. *) - (* intros. *) - (* destruct (4 * ptr ==Z Integers.Ptrofs.unsigned OFFSET). *) - - (* crush. *) - (* erewrite Mem.load_store_same. *) - (* 2: { rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite e. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* exact H1. *) - (* apply Integers.Ptrofs.unsigned_range_2. } *) - (* crush. *) - (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; try constructor. *) - (* destruct (Archi.ptr64); try discriminate. *) - (* pose proof (RSBP src). rewrite EQ_SRC in H0. *) - (* assumption. *) - - (* simpl. *) - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr. *) - (* simpl. *) - (* destruct (Z_le_gt_dec (4 * ptr + 4) (Integers.Ptrofs.unsigned OFFSET)); eauto. *) - (* right. *) - (* apply ZExtra.mod_0_bounds; try lia. *) - (* apply ZLib.Z_mod_mult'. *) - (* inv H0. *) - (* apply Zmult_lt_compat_r with (p := 4) in H11; try lia. *) - (* rewrite ZLib.div_mul_undo in H11; try lia. *) - (* split; try lia. *) - (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) - (* } *) - (* apply ASBP; assumption. *) - - (* unfold stack_bounds in *. intros. *) - (* simpl. *) - (* assert (Integers.Ptrofs.repr 0 = Integers.Ptrofs.zero) as ZERO by reflexivity. *) - (* erewrite Mem.load_store_other with (m1 := m). *) - (* 2: { exact H1. } *) - (* 2: { right. right. simpl. *) - (* rewrite ZERO. *) - (* rewrite Integers.Ptrofs.add_zero_l. *) - (* rewrite Integers.Ptrofs.unsigned_repr; crush; try lia. *) - (* apply ZExtra.mod_0_bounds; crush; try lia. } *) - (* crush. *) - (* exploit (BOUNDS ptr); try lia. intros. crush. *) - (* exploit (BOUNDS ptr v); try lia. intros. *) - (* inv H0. *) - (* match goal with | |- ?x = _ => destruct x eqn:?EQ end; try reflexivity. *) - (* assert (Mem.valid_access m AST.Mint32 sp' *) - (* (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.add (Integers.Ptrofs.repr 0) *) - (* (Integers.Ptrofs.repr ptr))) Writable). *) - (* { pose proof H1. eapply Mem.store_valid_access_2 in H0. *) - (* exact H0. eapply Mem.store_valid_access_3. eassumption. } *) - (* pose proof (Mem.valid_access_store m AST.Mint32 sp' *) - (* (Integers.Ptrofs.unsigned *) - (* (Integers.Ptrofs.add (Integers.Ptrofs.repr 0) *) - (* (Integers.Ptrofs.repr ptr))) v). *) - (* apply X in H0. inv H0. congruence. *) - - (* constructor; simplify. unfold Verilog.merge_regs. unfold_merge. rewrite AssocMap.gso. *) - (* assumption. lia. *) - (* unfold Verilog.merge_regs. unfold_merge. rewrite AssocMap.gso. *) - (* assumption. lia. *) - - (* Unshelve. *) - (* exact tt. *) - (* exact (Values.Vint (Int.repr 0)). *) - (* exact tt. *) - (* exact (Values.Vint (Int.repr 0)). *) - (* exact tt. *) - (* exact (Values.Vint (Int.repr 0)). *) - (* Qed. *) Admitted. - #[local] Hint Resolve transl_istore_correct : htlproof. - - Lemma transl_icond_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (sp : Values.val) (pc : positive) - (rs : Registers.Regmap.t Values.val) (m : mem) (cond : Op.condition) (args : list Registers.reg) - (ifso ifnot : RTL.node) (b : bool) (pc' : RTL.node), - (RTL.fn_code f) ! pc = Some (RTL.Icond cond args ifso ifnot) -> - Op.eval_condition cond (map (fun r : positive => Registers.Regmap.get r rs) args) m = Some b -> - pc' = (if b then ifso else ifnot) -> - forall R1 : HTL.state, - match_states (RTL.State s f sp pc rs m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ match_states (RTL.State s f sp pc' rs m) R2. - Proof. - intros s f sp pc rs m cond args ifso ifnot b pc' H H0 H1 R1 MSTATE. - inv_state. - destruct b. - - eexists. split. apply Smallstep.plus_one. - clear H33. - eapply HTL.step_module; eauto. - inv CONST; assumption. - inv CONST; assumption. - econstructor; simpl; trivial. - constructor; trivial. - eapply Verilog.erun_Vternary_true; simpl; eauto. - eapply eval_cond_correct; eauto. intros. - intros. eapply RTL.max_reg_function_use. apply H22. auto. - econstructor. auto. - simpl. econstructor. constructor. unfold Verilog.merge_regs. unfold_merge. simpl. - unfold_merge. apply AssocMap.gss. - - inv MARR. inv CONST. - big_tac. - (* constructor; rewrite AssocMap.gso; simplify; try assumption; lia. *) - (* - eexists. split. apply Smallstep.plus_one. *) - (* clear H32. *) - (* eapply HTL.step_module; eauto. *) - (* inv CONST; assumption. *) - (* inv CONST; assumption. *) - (* econstructor; simpl; trivial. *) - (* constructor; trivial. *) - (* eapply Verilog.erun_Vternary_false; simpl; eauto. *) - (* eapply eval_cond_correct; eauto. intros. *) - (* intros. eapply RTL.max_reg_function_use. apply H22. auto. *) - (* econstructor. auto. *) - (* simpl. econstructor. constructor. unfold Verilog.merge_regs. unfold_merge. simpl. *) - (* apply AssocMap.gss. *) - - (* inv MARR. inv CONST. *) - (* big_tac. *) - (* constructor; rewrite AssocMap.gso; simplify; try assumption; lia. *) - - (* Unshelve. all: exact tt. *) - (* Qed. *) Admitted. - #[local] Hint Resolve transl_icond_correct : htlproof. - - (*Lemma transl_ijumptable_correct: *) - (* forall (s : list RTL.stackframe) (f : RTL.function) (sp : Values.val) (pc : positive) *) - (* (rs : Registers.Regmap.t Values.val) (m : mem) (arg : Registers.reg) (tbl : list RTL.node) *) - (* (n : Integers.Int.int) (pc' : RTL.node), *) - (* (RTL.fn_code f) ! pc = Some (RTL.Ijumptable arg tbl) -> *) - (* Registers.Regmap.get arg rs = Values.Vint n -> *) - (* list_nth_z tbl (Integers.Int.unsigned n) = Some pc' -> *) - (* forall R1 : HTL.state, *) - (* match_states (RTL.State s f sp pc rs m) R1 -> *) - (* exists R2 : HTL.state, *) - (* Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ match_states (RTL.State s f sp pc' rs m) R2. *) - (* Proof. *) - (* intros s f sp pc rs m arg tbl n pc' H H0 H1 R1 MSTATE. *) - - (* #[local] Hint Resolve transl_ijumptable_correct : htlproof.*) - - Lemma transl_ireturn_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (stk : Values.block) - (pc : positive) (rs : RTL.regset) (m : mem) (or : option Registers.reg) - (m' : mem), - (RTL.fn_code f) ! pc = Some (RTL.Ireturn or) -> - Mem.free m stk 0 (RTL.fn_stacksize f) = Some m' -> - forall R1 : HTL.state, - match_states (RTL.State s f (Values.Vptr stk Integers.Ptrofs.zero) pc rs m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ - match_states (RTL.Returnstate s (Registers.regmap_optget or Values.Vundef rs) m') R2. - Proof. - intros s f stk pc rs m or m' H H0 R1 MSTATE. - inv_state. - - - econstructor. split. - eapply Smallstep.plus_two. - - eapply HTL.step_module; eauto. - inv CONST; assumption. - inv CONST; assumption. - constructor. - econstructor; simpl; trivial. - econstructor; simpl; trivial. - constructor. - econstructor; simpl; trivial. - constructor. - - constructor. constructor. constructor. - - unfold state_st_wf in WF; big_tac; eauto. - destruct wf1 as [HCTRL HDATA]. apply HCTRL. - apply AssocMapExt.elements_iff. eexists. - match goal with H: control ! pc = Some _ |- _ => apply H end. - - apply HTL.step_finish. - unfold Verilog.merge_regs. - unfold_merge; simpl. - (* rewrite AssocMap.gso. *) - (* apply AssocMap.gss. lia. *) - (* apply AssocMap.gss. *) - (* rewrite Events.E0_left. reflexivity. *) - - (* constructor; auto. *) - (* constructor. *) - - (* (* FIXME: Duplication *) *) - (* - econstructor. split. *) - (* eapply Smallstep.plus_two. *) - (* eapply HTL.step_module; eauto. *) - (* inv CONST; assumption. *) - (* inv CONST; assumption. *) - (* constructor. *) - (* econstructor; simpl; trivial. *) - (* econstructor; simpl; trivial. *) - (* constructor. constructor. constructor. *) - (* constructor. constructor. constructor. *) - (* constructor. *) - - (* unfold state_st_wf in WF; big_tac; eauto. *) - - (* destruct wf1 as [HCTRL HDATA]. apply HCTRL. *) - (* apply AssocMapExt.elements_iff. eexists. *) - (* match goal with H: control ! pc = Some _ |- _ => apply H end. *) - - (* apply HTL.step_finish. *) - (* unfold Verilog.merge_regs. *) - (* unfold_merge. *) - (* unfold_merge. *) - (* rewrite AssocMap.gso. *) - (* apply AssocMap.gss. simpl; lia. *) - (* apply AssocMap.gss. *) - (* rewrite Events.E0_left. trivial. *) - - (* constructor; auto. *) - - (* simpl. inversion MASSOC. subst. *) - (* unfold find_assocmap, AssocMapExt.get_default. rewrite AssocMap.gso. *) - (* apply H1. eapply RTL.max_reg_function_use. eauto. simpl; tauto. *) - (* assert (HPle : Ple r (RTL.max_reg_function f)). *) - (* eapply RTL.max_reg_function_use. eassumption. simpl; auto. *) - (* apply ZExtra.Ple_not_eq. apply ZExtra.Ple_Plt_Succ. assumption. *) - - (* Unshelve. *) - (* all: constructor. *) - (* Qed. *) Admitted. - #[local] Hint Resolve transl_ireturn_correct : htlproof. - - Lemma transl_callstate_correct: - forall (s : list RTL.stackframe) (f : RTL.function) (args : list Values.val) - (m : mem) (m' : Mem.mem') (stk : Values.block), - Mem.alloc m 0 (RTL.fn_stacksize f) = (m', stk) -> - forall R1 : HTL.state, - match_states (RTL.Callstate s (AST.Internal f) args m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ - match_states - (RTL.State s f (Values.Vptr stk Integers.Ptrofs.zero) (RTL.fn_entrypoint f) - (RTL.init_regs args (RTL.fn_params f)) m') R2. - Proof. - intros s f args m m' stk H R1 MSTATE. - - inversion MSTATE; subst. inversion TF; subst. - econstructor. split. apply Smallstep.plus_one. - eapply HTL.step_call. crush. - - apply match_state with (sp' := stk); eauto. - - all: big_tac. - - apply regs_lessdef_add_greater. unfold Plt; lia. - apply regs_lessdef_add_greater. unfold Plt; lia. - apply regs_lessdef_add_greater. unfold Plt; lia. - apply init_reg_assoc_empty. - - constructor. - - destruct (Mem.load AST.Mint32 m' stk - (Integers.Ptrofs.unsigned (Integers.Ptrofs.add - Integers.Ptrofs.zero - (Integers.Ptrofs.repr (4 * ptr))))) eqn:LOAD. - pose proof Mem.load_alloc_same as LOAD_ALLOC. - pose proof H as ALLOC. - eapply LOAD_ALLOC in ALLOC. - 2: { exact LOAD. } - ptrofs. rewrite LOAD. - rewrite ALLOC. - repeat constructor. - - ptrofs. rewrite LOAD. - repeat constructor. - - unfold reg_stack_based_pointers. intros. - unfold RTL.init_regs; crush. - destruct (RTL.fn_params f); - rewrite Registers.Regmap.gi; constructor. - - unfold arr_stack_based_pointers. intros. - crush. - destruct (Mem.load AST.Mint32 m' stk - (Integers.Ptrofs.unsigned (Integers.Ptrofs.add - Integers.Ptrofs.zero - (Integers.Ptrofs.repr (4 * ptr))))) eqn:LOAD. - pose proof Mem.load_alloc_same as LOAD_ALLOC. - pose proof H as ALLOC. - eapply LOAD_ALLOC in ALLOC. - 2: { exact LOAD. } - rewrite ALLOC. - repeat constructor. - constructor. - - Transparent Mem.alloc. (* TODO: Since there are opaque there's probably a lemma. *) - Transparent Mem.load. - Transparent Mem.store. - unfold stack_bounds. - split. - - unfold Mem.alloc in H. - inv H. - crush. - unfold Mem.load. - intros. - match goal with | |- context[if ?x then _ else _] => destruct x end; try congruence. - inv v0. unfold Mem.range_perm in H4. - unfold Mem.perm in H4. crush. - unfold Mem.perm_order' in H4. - small_tac. - exploit (H4 ptr). rewrite Integers.Ptrofs.unsigned_repr; small_tac. intros. - rewrite Maps.PMap.gss in H8. - match goal with | H8 : context[if ?x then _ else _] |- _ => destruct x eqn:EQ end; try contradiction. - crush. - apply proj_sumbool_true in H10. lia. - - unfold Mem.alloc in H. - inv H. - crush. - unfold Mem.store. - intros. - match goal with | |- context[if ?x then _ else _] => destruct x end; try congruence. - inv v0. unfold Mem.range_perm in H4. - unfold Mem.perm in H4. crush. - unfold Mem.perm_order' in H4. - small_tac. - exploit (H4 ptr). rewrite Integers.Ptrofs.unsigned_repr; small_tac. intros. - rewrite Maps.PMap.gss in H8. - match goal with | H8 : context[if ?x then _ else _] |- _ => destruct x eqn:EQ end; try contradiction. - crush. - apply proj_sumbool_true in H10. lia. - constructor. simplify. rewrite AssocMap.gss. - simplify. rewrite AssocMap.gso. apply AssocMap.gss. simplify. lia. - Opaque Mem.alloc. - Opaque Mem.load. - Opaque Mem.store. - Qed. - #[local] Hint Resolve transl_callstate_correct : htlproof. - - Lemma transl_returnstate_correct: - forall (res0 : Registers.reg) (f : RTL.function) (sp : Values.val) (pc : RTL.node) - (rs : RTL.regset) (s : list RTL.stackframe) (vres : Values.val) (m : mem) - (R1 : HTL.state), - match_states (RTL.Returnstate (RTL.Stackframe res0 f sp pc rs :: s) vres m) R1 -> - exists R2 : HTL.state, - Smallstep.plus HTL.step tge R1 Events.E0 R2 /\ - match_states (RTL.State s f sp pc (Registers.Regmap.set res0 vres rs) m) R2. - Proof. - intros res0 f sp pc rs s vres m R1 MSTATE. - inversion MSTATE. inversion MF. - Qed. - #[local] Hint Resolve transl_returnstate_correct : htlproof. - - Lemma option_inv : - forall A x y, - @Some A x = Some y -> x = y. - Proof. intros. inversion H. trivial. Qed. - - Lemma main_tprog_internal : - forall b, - Globalenvs.Genv.find_symbol tge tprog.(AST.prog_main) = Some b -> - exists f, Genv.find_funct_ptr (Genv.globalenv tprog) b = Some (AST.Internal f). - Proof. - intros. - destruct TRANSL. unfold main_is_internal in H1. - repeat (unfold_match H1). replace b with b0. - exploit function_ptr_translated; eauto. intros [tf [A B]]. - unfold transl_fundef, AST.transf_partial_fundef, Errors.bind in B. - unfold_match B. inv B. econstructor. apply A. - - apply option_inv. rewrite <- Heqo. rewrite <- H. - rewrite symbols_preserved. replace (AST.prog_main tprog) with (AST.prog_main prog). - trivial. symmetry; eapply Linking.match_program_main; eauto. - Qed. - - Lemma transl_initial_states : - forall s1 : Smallstep.state (RTL.semantics prog), - Smallstep.initial_state (RTL.semantics prog) s1 -> - exists s2 : Smallstep.state (HTL.semantics tprog), - Smallstep.initial_state (HTL.semantics tprog) s2 /\ match_states s1 s2. - Proof. - induction 1. - destruct TRANSL. unfold main_is_internal in H4. - repeat (unfold_match H4). - assert (f = AST.Internal f1). apply option_inv. - rewrite <- Heqo0. rewrite <- H1. replace b with b0. - auto. apply option_inv. rewrite <- H0. rewrite <- Heqo. - trivial. - exploit function_ptr_translated; eauto. - intros [tf [A B]]. - unfold transl_fundef, Errors.bind in B. - unfold AST.transf_partial_fundef, Errors.bind in B. - repeat (unfold_match B). inversion B. subst. - exploit main_tprog_internal; eauto; intros. - rewrite symbols_preserved. replace (AST.prog_main tprog) with (AST.prog_main prog). - apply Heqo. symmetry; eapply Linking.match_program_main; eauto. - inversion H5. - econstructor; split. econstructor. - apply (Genv.init_mem_transf_partial TRANSL'); eauto. - replace (AST.prog_main tprog) with (AST.prog_main prog). - rewrite symbols_preserved; eauto. - symmetry; eapply Linking.match_program_main; eauto. - apply H6. - - constructor. - apply transl_module_correct. - assert (Some (AST.Internal x) = Some (AST.Internal m)). - replace (AST.fundef HTL.module) with (HTL.fundef). - rewrite <- H6. setoid_rewrite <- A. trivial. - trivial. inv H7. assumption. - Qed. - #[local] Hint Resolve transl_initial_states : htlproof. - - Lemma transl_final_states : - forall (s1 : Smallstep.state (RTL.semantics prog)) - (s2 : Smallstep.state (HTL.semantics tprog)) - (r : Integers.Int.int), - match_states s1 s2 -> - Smallstep.final_state (RTL.semantics prog) s1 r -> - Smallstep.final_state (HTL.semantics tprog) s2 r. - Proof. - intros. inv H0. inv H. inv H4. inv MF. constructor. reflexivity. - Qed. - #[local] Hint Resolve transl_final_states : htlproof. - - Theorem transl_step_correct: - forall (S1 : RTL.state) t S2, - RTL.step ge S1 t S2 -> - forall (R1 : HTL.state), - match_states S1 R1 -> - exists R2, Smallstep.plus HTL.step tge R1 t R2 /\ match_states S2 R2. - Proof. - induction 1; eauto with htlproof; (intros; inv_state). - Qed. - #[local] Hint Resolve transl_step_correct : htlproof. - - Theorem transf_program_correct: - Smallstep.forward_simulation (RTL.semantics prog) (HTL.semantics tprog). - Proof. - eapply Smallstep.forward_simulation_plus; eauto with htlproof. - apply senv_preserved. - Qed. - -End CORRECTNESS. |