diff options
Diffstat (limited to 'src/translation/HTLgenproof.v')
| -rw-r--r-- | src/translation/HTLgenproof.v | 4311 |
1 files changed, 2156 insertions, 2155 deletions
diff --git a/src/translation/HTLgenproof.v b/src/translation/HTLgenproof.v index 305c14f..e404c82 100644 --- a/src/translation/HTLgenproof.v +++ b/src/translation/HTLgenproof.v @@ -31,2158 +31,2159 @@ Hint Resolve AssocMap.gso : htlproof. 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. -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). -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. - - Lemma assumption_32bit : - forall v, - valueToPos (posToValue v) = v. - Proof. - Admitted. - -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), - 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). -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. - -Definition match_prog_matches : - forall p tp, - match_prog p tp -> - Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp. - Proof. intros. unfold match_prog in H. 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. -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. -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. invert H. - destruct n; simpl in *. - invert 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. invert H. - destruct n; simpl in *. - invert 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. -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'). - 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 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. - 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); simplify. - - inv Heql. - 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. - - eexists. split. constructor. constructor. auto. - - inv Heql. - 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. econstructor; eauto. constructor. trivial. - unfold Verilog.unop_run. unfold Values.Val.neg. destruct (Registers.Regmap.get r rs) eqn:?; constructor. - inv HPle. auto. - - inv Heql. - assert (HPle : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - assert (HPle0 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). - apply H in HPle. apply H in HPle0. - eexists. split. econstructor; eauto. constructor. trivial. - constructor. trivial. simplify. inv HPle. inv HPle0; constructor; auto. - + inv HPle0. constructor. unfold valueToPtr. Search Integers.Ptrofs.sub Integers.int. - pose proof Integers.Ptrofs.agree32_sub. unfold Integers.Ptrofs.agree32 in H3. - Print Integers.Ptrofs.agree32. unfold Ptrofs.of_int. simpl. - apply ptrofs_inj. assert (Archi.ptr64 = false) by auto. eapply H3 in H4. - rewrite Ptrofs.unsigned_repr. apply H4. 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. - Admitted. - - Lemma eval_cond_correct : - forall cond (args : list Registers.reg) s1 c s' i rs args m b f asr asa, - translate_condition cond args s1 = OK c s' i -> - Op.eval_condition - cond - (List.map (fun r : BinNums.positive => Registers.Regmap.get r rs) args) - m = Some b -> - Verilog.expr_runp f asr asa c (boolToValue b). - Admitted. - - (** 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'). - - Opaque combine. - - Ltac tac0 := - match goal with - | [ |- context[valueToPos (posToValue _)] ] => rewrite assumption_32bit - - | [ |- context[Verilog.merge_arrs _ _] ] => unfold Verilog.merge_arrs - | [ |- context[Verilog.merge_arr] ] => unfold Verilog.merge_arr - | [ |- context[Verilog.merge_regs _ _] ] => unfold Verilog.merge_regs; crush; unfold_merge - | [ |- context[reg_stack_based_pointers] ] => unfold reg_stack_based_pointers; intros - | [ |- context[Verilog.arr_assocmap_set _ _ _ _] ] => unfold Verilog.arr_assocmap_set - - | [ |- 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 _ _)] ] => - destruct (Pos.eq_dec s d) as [EQ|EQ]; - [> rewrite EQ | rewrite Registers.Regmap.gso; auto] - - | [ H : opt_val_value_lessdef _ _ |- _ ] => invert H - | [ H : context[Z.of_nat (Z.to_nat _)] |- _ ] => rewrite Z2Nat.id in H; [> solve crush |] - | [ H : _ ! _ = Some _ |- _] => setoid_rewrite H - end. - - Ltac small_tac := repeat (crush; try array; try ptrofs); crush; auto. - Ltac big_tac := repeat (crush; try array; try ptrofs; try tac0); crush; auto. - - 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. - apply assumption_32bit. - (* processing of state *) - econstructor. - crush. - econstructor. - econstructor. - econstructor. - - all: invert MARR; big_tac. - Unshelve. - constructor. - Qed. - 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. - exploit eval_correct; eauto. intros. inversion H1. inversion H2. - econstructor. split. - apply Smallstep.plus_one. - eapply HTL.step_module; eauto. - apply assumption_32bit. - econstructor; simpl; trivial. - constructor; trivial. - econstructor; simpl; eauto. - simpl. econstructor. econstructor. - apply H3. simplify. - - all: big_tac. - - assert (Ple res0 (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_def; eauto; simpl; auto). - - unfold Ple in H10. lia. - apply regs_lessdef_add_match. assumption. - apply regs_lessdef_add_greater. unfold Plt; lia. assumption. - assert (Ple res0 (RTL.max_reg_function f)) - by (eapply RTL.max_reg_function_def; eauto; simpl; auto). - unfold Ple in H12; lia. - unfold_merge. simpl. - rewrite AssocMap.gso. - apply AssocMap.gss. - apply st_greater_than_res. - - (*match_states*) - assert (pc' = valueToPos (posToValue 32 pc')). auto using assumption_32bit. - rewrite <- H1. - constructor; auto. - unfold_merge. - apply regs_lessdef_add_match. - constructor. - apply regs_lessdef_add_greater. - apply greater_than_max_func. - assumption. - - unfold state_st_wf. intros. inversion H2. subst. - unfold_merge. - rewrite AssocMap.gso. - apply AssocMap.gss. - apply st_greater_than_res. - - + econstructor. split. - apply Smallstep.plus_one. - eapply HTL.step_module; eauto. - econstructor; simpl; trivial. - constructor; trivial. - econstructor; simpl; eauto. - eapply eval_correct; eauto. - constructor. rewrite valueToInt_intToValue. trivial. - unfold_merge. simpl. - rewrite AssocMap.gso. - apply AssocMap.gss. - apply st_greater_than_res. - - match_states - assert (pc' = valueToPos (posToValue 32 pc')). auto using assumption_32bit. - rewrite <- H1. - constructor. - unfold_merge. - apply regs_lessdef_add_match. - constructor. - symmetry. apply valueToInt_intToValue. - apply regs_lessdef_add_greater. - apply greater_than_max_func. - assumption. assumption. - - unfold state_st_wf. intros. inversion H2. subst. - unfold_merge. - rewrite AssocMap.gso. - apply AssocMap.gss. - apply st_greater_than_res. - assumption. - Admitted. - 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 _ _ = _ |- _ ] => invert 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 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 *) - invert MARR. 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'. - invert 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 H6 in HPler0. - invert HPler0; try congruence. - rewrite EQr0 in H8. - invert H8. - clear H0. clear H6. - - 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. - { rewrite HeqOFFSET. - apply PtrofsExtra.add_mod; crush. - rewrite Integers.Ptrofs.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. - apply PtrofsExtra.of_int_mod. - rewrite Integers.Int.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. } - - (** 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 (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; 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. - apply assumption_32bit. - econstructor. econstructor. econstructor. crush. - econstructor. econstructor. econstructor. crush. - econstructor. econstructor. - 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 *) - match goal with - | [ |- context [valueToNat ?x] ] => - assert (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4))) - = - valueToNat x) - as EXPR_OK by admit - end. - rewrite <- EXPR_OK. - - specialize (H7 (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4)))). - exploit H7; 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. - - + (** Preamble *) - invert MARR. 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'. - invert 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 H6 in HPler0. - apply H8 in HPler1. - invert HPler0; invert HPler1; try congruence. - rewrite EQr0 in H9. - rewrite EQr1 in H11. - invert H9. invert H11. - clear H0. clear H6. clear H8. - - 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. - { rewrite HeqOFFSET. - apply PtrofsExtra.add_mod; crush; try lia. - rewrite Integers.Ptrofs.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. - apply PtrofsExtra.of_int_mod. - apply IntExtra.add_mod; crush. - apply IntExtra.mul_mod2; crush. - rewrite Integers.Int.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. - rewrite Integers.Int.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. } - - (** 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 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. - apply assumption_32bit. - econstructor. econstructor. econstructor. crush. - econstructor. econstructor. econstructor. crush. - econstructor. econstructor. econstructor. - econstructor. econstructor. econstructor. econstructor. - econstructor. - eapply Verilog.erun_Vbinop with (EQ := ?[EQ6]). - econstructor. econstructor. 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 *) - match goal with - | [ |- context [valueToNat ?x] ] => - assert (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4))) - = - valueToNat x) - as EXPR_OK by admit - end. - rewrite <- EXPR_OK. - - specialize (H7 (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4)))). - exploit H7; 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. - - + invert MARR. 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 *) - rename H0 into MOD_PRESERVE. - - (** 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. - apply assumption_32bit. - 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 *) - match goal with (* Prevents issues with evars *) - | [ |- context [valueToNat ?x] ] => - assert (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4))) - = - valueToNat x) - as EXPR_OK by admit - end. - rewrite <- EXPR_OK. - - specialize (H7 (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4)))). - exploit H7; 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. - Admitted. - 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 *) - invert MARR. 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'. - invert 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 H6 in HPler0. - invert HPler0; try congruence. - rewrite EQr0 in H8. - invert H8. - clear H0. clear H6. - - 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. - { rewrite HeqOFFSET. - apply PtrofsExtra.add_mod; crush; try lia. - rewrite Integers.Ptrofs.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. - apply PtrofsExtra.of_int_mod. - rewrite Integers.Int.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. } - - (** 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. - apply assumption_32bit. - econstructor. econstructor. econstructor. - eapply Verilog.stmnt_runp_Vnonblock_arr. crush. - econstructor. - eapply Verilog.erun_Vbinop with (EQ := ?[EQ7]). - eapply Verilog.erun_Vbinop with (EQ := ?[EQ8]). - econstructor. - econstructor. - econstructor. econstructor. econstructor. econstructor. - econstructor. econstructor. econstructor. econstructor. - - all: crush. - - (** State Lookup *) - unfold Verilog.merge_regs. - crush. - unfold_merge. - apply AssocMap.gss. - - (** Match states *) - rewrite assumption_32bit. - 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 *) - match goal with - | [ |- context [valueToNat ?x] ] => - assert (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4))) - = - valueToNat x) - as EXPR_OK by admit - end. - - 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. - 2: { reflexivity. } - unfold Verilog.arr_assocmap_set. - rewrite AssocMap.gss. - unfold Verilog.merge_arr. - rewrite AssocMap.gss. - setoid_rewrite H5. - reflexivity. - - rewrite combine_length. - rewrite <- array_set_len. - unfold arr_repeat. crush. - apply list_repeat_len. - - rewrite <- array_set_len. - unfold arr_repeat. crush. - rewrite list_repeat_len. - rewrite H4. reflexivity. - - 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. - 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 H13. - destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; invert H13; 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 H13. - rewrite Z_div_mult in H13; try lia. - replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H13 by reflexivity. - rewrite <- PtrofsExtra.divu_unsigned in H13; unfold_constants; try lia. - rewrite H13. rewrite EXPR_OK. - 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 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 <- 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 H7; 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; 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'. - invert H0. - 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: { 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. - invert 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. invert H0. congruence. - - + (** Preamble *) - invert MARR. 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'. - invert 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 H6 in HPler0. - apply H8 in HPler1. - invert HPler0; invert HPler1; try congruence. - rewrite EQr0 in H9. - rewrite EQr1 in H11. - invert H9. invert H11. - clear H0. clear H6. clear H8. - - 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. - { rewrite HeqOFFSET. - apply PtrofsExtra.add_mod; crush; try lia. - rewrite Integers.Ptrofs.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. - apply PtrofsExtra.of_int_mod. - apply IntExtra.add_mod; crush. - apply IntExtra.mul_mod2; crush. - rewrite Integers.Int.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. - rewrite Integers.Int.unsigned_repr_eq. - rewrite <- Zmod_div_mod; crush. } - - (** 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. - apply assumption_32bit. - econstructor. econstructor. econstructor. - eapply Verilog.stmnt_runp_Vnonblock_arr. crush. - econstructor. - eapply Verilog.erun_Vbinop with (EQ := ?[EQ9]). - eapply Verilog.erun_Vbinop with (EQ := ?[EQ10]). - eapply Verilog.erun_Vbinop with (EQ := ?[EQ11]). - econstructor. econstructor. econstructor. econstructor. - econstructor. - eapply Verilog.erun_Vbinop with (EQ := ?[EQ12]). - econstructor. econstructor. econstructor. econstructor. - econstructor. econstructor. econstructor. econstructor. - econstructor. econstructor. econstructor. econstructor. - - all: crush. - - (** State Lookup *) - unfold Verilog.merge_regs. - crush. - unfold_merge. - apply AssocMap.gss. - - (** Match states *) - rewrite assumption_32bit. - 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 (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4))) - = - valueToNat (vdiv - (vplus (vplus asr # r0 (ZToValue 32 z0) ?EQ11) (vmul asr # r1 (ZToValue 32 z) ?EQ12) - ?EQ10) (ZToValue 32 4) ?EQ9)) - as EXPR_OK by admit. - - 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. - 2: { reflexivity. } - unfold Verilog.arr_assocmap_set. - rewrite AssocMap.gss. - unfold Verilog.merge_arr. - rewrite AssocMap.gss. - setoid_rewrite H5. - reflexivity. - - rewrite combine_length. - rewrite <- array_set_len. - unfold arr_repeat. crush. - apply list_repeat_len. - - rewrite <- array_set_len. - unfold arr_repeat. crush. - rewrite list_repeat_len. - rewrite H4. reflexivity. - - 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. - 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 H16. - destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; invert H16; 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 H16. - rewrite Z_div_mult in H16; try lia. - replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H16 by reflexivity. - rewrite <- PtrofsExtra.divu_unsigned in H16; unfold_constants; try lia. - rewrite H16. rewrite EXPR_OK. - 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 H18; try lia. - apply Zmult_lt_compat_r with (p := 4) in H18; try lia. - rewrite ZLib.div_mul_undo in H18; try lia. - split; try lia. - apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. - } - - rewrite <- 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 H7; 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 H16; 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'. - invert H0. - 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. - } - 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. - invert 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. invert H0. congruence. - - + invert MARR. 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 *) - rename H0 into MOD_PRESERVE. - - (** 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. - apply assumption_32bit. - econstructor. econstructor. econstructor. - eapply Verilog.stmnt_runp_Vnonblock_arr. crush. - econstructor. econstructor. econstructor. econstructor. - - all: crush. - - (** State Lookup *) - unfold Verilog.merge_regs. - crush. - unfold_merge. - apply AssocMap.gss. - - (** Match states *) - rewrite assumption_32bit. - 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 (Z.to_nat - (Integers.Ptrofs.unsigned - (Integers.Ptrofs.divu - OFFSET - (Integers.Ptrofs.repr 4))) - = - valueToNat (ZToValue 32 (Integers.Ptrofs.unsigned OFFSET / 4))) - as EXPR_OK by admit. - - 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. - 2: { reflexivity. } - unfold Verilog.arr_assocmap_set. - rewrite AssocMap.gss. - unfold Verilog.merge_arr. - rewrite AssocMap.gss. - setoid_rewrite H5. - reflexivity. - - rewrite combine_length. - rewrite <- array_set_len. - unfold arr_repeat. crush. - apply list_repeat_len. - - rewrite <- array_set_len. - unfold arr_repeat. crush. - rewrite list_repeat_len. - rewrite H4. reflexivity. - - 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; invert 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 EXPR_OK. - 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 H11; try lia. - 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. - } - - rewrite <- 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 H7; 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'. - invert H0. - apply Zmult_lt_compat_r with (p := 4) in H9; try lia. - rewrite ZLib.div_mul_undo in H9; 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. - invert 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. invert H0. congruence. - Admitted. - 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. - - eexists. split. apply Smallstep.plus_one. - eapply HTL.step_module; eauto. - apply assumption_32bit. - eapply Verilog.stmnt_runp_Vnonblock_reg with - (rhsval := if b then posToValue 32 ifso else posToValue 32 ifnot). - constructor. - - simpl. - destruct b. - eapply Verilog.erun_Vternary_true. - eapply eval_cond_correct; eauto. - constructor. - apply boolToValue_ValueToBool. - eapply Verilog.erun_Vternary_false. - eapply eval_cond_correct; eauto. - constructor. - apply boolToValue_ValueToBool. - constructor. - - big_tac. - - invert MARR. - destruct b; rewrite assumption_32bit; big_tac. - - Unshelve. - constructor. - Qed. - 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. - Admitted. - 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. - apply assumption_32bit. - constructor. - econstructor; simpl; trivial. - econstructor; simpl; trivial. - constructor. - econstructor; simpl; trivial. - constructor. - - constructor. constructor. - - unfold state_st_wf in WF; big_tac; eauto. - - 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. - apply assumption_32bit. - constructor. - econstructor; simpl; trivial. - econstructor; simpl; trivial. - constructor. constructor. constructor. - constructor. constructor. constructor. - - unfold state_st_wf in WF; big_tac; eauto. - - apply HTL.step_finish. - unfold Verilog.merge_regs. - 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. - 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 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. - invert H. - crush. - unfold Mem.load. - intros. - match goal with | |- context[if ?x then _ else _] => destruct x end; try congruence. - invert 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. - invert H. - crush. - unfold Mem.store. - intros. - match goal with | |- context[if ?x then _ else _] => destruct x end; try congruence. - invert 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. - Opaque Mem.alloc. - Opaque Mem.load. - Opaque Mem.store. - Qed. - 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. - 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. - 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. invert MF. constructor. reflexivity. - Qed. - 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. - 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. +(* 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. *) +(* 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). *) +(* 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. *) + +(* Lemma assumption_32bit : *) +(* forall v, *) +(* valueToPos (posToValue v) = v. *) +(* Proof. *) +(* Admitted. *) + +(* 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), *) +(* 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). *) +(* 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. *) + +(* Definition match_prog_matches : *) +(* forall p tp, *) +(* match_prog p tp -> *) +(* Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp. *) +(* Proof. intros. unfold match_prog in H. 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. *) +(* 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. *) +(* 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. invert H. *) +(* destruct n; simpl in *. *) +(* invert 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. invert H. *) +(* destruct n; simpl in *. *) +(* invert 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. *) +(* 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'). *) +(* 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 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. *) +(* 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); simplify. *) +(* - inv Heql. *) +(* 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. *) +(* - eexists. split. constructor. constructor. auto. *) +(* - inv Heql. *) +(* 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. econstructor; eauto. constructor. trivial. *) +(* unfold Verilog.unop_run. unfold Values.Val.neg. destruct (Registers.Regmap.get r rs) eqn:?; constructor. *) +(* inv HPle. auto. *) +(* - inv Heql. *) +(* assert (HPle : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). *) +(* assert (HPle0 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto). *) +(* apply H in HPle. apply H in HPle0. *) +(* eexists. split. econstructor; eauto. constructor. trivial. *) +(* constructor. trivial. simplify. inv HPle. inv HPle0; constructor; auto. *) +(* + inv HPle0. constructor. unfold valueToPtr. Search Integers.Ptrofs.sub Integers.int. *) +(* pose proof Integers.Ptrofs.agree32_sub. unfold Integers.Ptrofs.agree32 in H3. *) +(* Print Integers.Ptrofs.agree32. unfold Ptrofs.of_int. simpl. *) +(* apply ptrofs_inj. assert (Archi.ptr64 = false) by auto. eapply H3 in H4. *) +(* rewrite Ptrofs.unsigned_repr. apply H4. 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. *) +(* Admitted. *) + +(* Lemma eval_cond_correct : *) +(* forall cond (args : list Registers.reg) s1 c s' i rs args m b f asr asa, *) +(* translate_condition cond args s1 = OK c s' i -> *) +(* Op.eval_condition *) +(* cond *) +(* (List.map (fun r : BinNums.positive => Registers.Regmap.get r rs) args) *) +(* m = Some b -> *) +(* Verilog.expr_runp f asr asa c (boolToValue b). *) +(* Admitted. *) + +(* (** 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'). *) + +(* Opaque combine. *) + +(* Ltac tac0 := *) +(* match goal with *) +(* | [ |- context[valueToPos (posToValue _)] ] => rewrite assumption_32bit *) + +(* | [ |- context[Verilog.merge_arrs _ _] ] => unfold Verilog.merge_arrs *) +(* | [ |- context[Verilog.merge_arr] ] => unfold Verilog.merge_arr *) +(* | [ |- context[Verilog.merge_regs _ _] ] => unfold Verilog.merge_regs; crush; unfold_merge *) +(* | [ |- context[reg_stack_based_pointers] ] => unfold reg_stack_based_pointers; intros *) +(* | [ |- context[Verilog.arr_assocmap_set _ _ _ _] ] => unfold Verilog.arr_assocmap_set *) + +(* | [ |- 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 _ _)] ] => *) +(* destruct (Pos.eq_dec s d) as [EQ|EQ]; *) +(* [> rewrite EQ | rewrite Registers.Regmap.gso; auto] *) + +(* | [ H : opt_val_value_lessdef _ _ |- _ ] => invert H *) +(* | [ H : context[Z.of_nat (Z.to_nat _)] |- _ ] => rewrite Z2Nat.id in H; [> solve crush |] *) +(* | [ H : _ ! _ = Some _ |- _] => setoid_rewrite H *) +(* end. *) + +(* Ltac small_tac := repeat (crush; try array; try ptrofs); crush; auto. *) +(* Ltac big_tac := repeat (crush; try array; try ptrofs; try tac0); crush; auto. *) + + (* 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. *) + (* apply assumption_32bit. *) + (* (* processing of state *) *) + (* econstructor. *) + (* crush. *) + (* econstructor. *) + (* econstructor. *) + (* econstructor. *) + + (* all: invert MARR; big_tac. *) + (* Unshelve. *) + (* constructor. *) + (* Qed. *) + (* 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. *) + (* exploit eval_correct; eauto. intros. inversion H1. inversion H2. *) + (* econstructor. split. *) + (* apply Smallstep.plus_one. *) + (* eapply HTL.step_module; eauto. *) + (* apply assumption_32bit. *) + (* econstructor; simpl; trivial. *) + (* constructor; trivial. *) + (* econstructor; simpl; eauto. *) + (* simpl. econstructor. econstructor. *) + (* apply H3. simplify. *) + + (* all: big_tac. *) + + (* assert (Ple res0 (RTL.max_reg_function f)) *) + (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) + + (* unfold Ple in H10. lia. *) + (* apply regs_lessdef_add_match. assumption. *) + (* apply regs_lessdef_add_greater. unfold Plt; lia. assumption. *) + (* assert (Ple res0 (RTL.max_reg_function f)) *) + (* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) + (* unfold Ple in H12; lia. *) + (* unfold_merge. simpl. *) + (* rewrite AssocMap.gso. *) + (* apply AssocMap.gss. *) + (* apply st_greater_than_res. *) + + (* (*match_states*) *) + (* assert (pc' = valueToPos (posToValue 32 pc')). auto using assumption_32bit. *) + (* rewrite <- H1. *) + (* constructor; auto. *) + (* unfold_merge. *) + (* apply regs_lessdef_add_match. *) + (* constructor. *) + (* apply regs_lessdef_add_greater. *) + (* apply greater_than_max_func. *) + (* assumption. *) + + (* unfold state_st_wf. intros. inversion H2. subst. *) + (* unfold_merge. *) + (* rewrite AssocMap.gso. *) + (* apply AssocMap.gss. *) + (* apply st_greater_than_res. *) + + (* + econstructor. split. *) + (* apply Smallstep.plus_one. *) + (* eapply HTL.step_module; eauto. *) + (* econstructor; simpl; trivial. *) + (* constructor; trivial. *) + (* econstructor; simpl; eauto. *) + (* eapply eval_correct; eauto. *) + (* constructor. rewrite valueToInt_intToValue. trivial. *) + (* unfold_merge. simpl. *) + (* rewrite AssocMap.gso. *) + (* apply AssocMap.gss. *) + (* apply st_greater_than_res. *) + + (* match_states *) + (* assert (pc' = valueToPos (posToValue 32 pc')). auto using assumption_32bit. *) + (* rewrite <- H1. *) + (* constructor. *) + (* unfold_merge. *) + (* apply regs_lessdef_add_match. *) + (* constructor. *) + (* symmetry. apply valueToInt_intToValue. *) + (* apply regs_lessdef_add_greater. *) + (* apply greater_than_max_func. *) + (* assumption. assumption. *) + + (* unfold state_st_wf. intros. inversion H2. subst. *) + (* unfold_merge. *) + (* rewrite AssocMap.gso. *) + (* apply AssocMap.gss. *) + (* apply st_greater_than_res. *) + (* assumption. *) + (* Admitted. *) + (* 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 _ _ = _ |- _ ] => invert 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 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 *) *) + (* invert MARR. 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'. *) + (* invert 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 H6 in HPler0. *) + (* invert HPler0; try congruence. *) + (* rewrite EQr0 in H8. *) + (* invert H8. *) + (* clear H0. clear H6. *) + + (* 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. *) + (* { rewrite HeqOFFSET. *) + (* apply PtrofsExtra.add_mod; crush. *) + (* rewrite Integers.Ptrofs.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. *) + (* apply PtrofsExtra.of_int_mod. *) + (* rewrite Integers.Int.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. } *) + + (* (** 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 (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; 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. *) + (* apply assumption_32bit. *) + (* econstructor. econstructor. econstructor. crush. *) + (* econstructor. econstructor. econstructor. crush. *) + (* econstructor. econstructor. *) + (* 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 *) *) + (* match goal with *) + (* | [ |- context [valueToNat ?x] ] => *) + (* assert (Z.to_nat *) + (* (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4))) *) + (* = *) + (* valueToNat x) *) + (* as EXPR_OK by admit *) + (* end. *) + (* rewrite <- EXPR_OK. *) + + (* specialize (H7 (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4)))). *) + (* exploit H7; 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. *) + + (* + (** Preamble *) *) + (* invert MARR. 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'. *) + (* invert 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 H6 in HPler0. *) + (* apply H8 in HPler1. *) + (* invert HPler0; invert HPler1; try congruence. *) + (* rewrite EQr0 in H9. *) + (* rewrite EQr1 in H11. *) + (* invert H9. invert H11. *) + (* clear H0. clear H6. clear H8. *) + + (* 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. *) + (* { rewrite HeqOFFSET. *) + (* apply PtrofsExtra.add_mod; crush; try lia. *) + (* rewrite Integers.Ptrofs.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. *) + (* apply PtrofsExtra.of_int_mod. *) + (* apply IntExtra.add_mod; crush. *) + (* apply IntExtra.mul_mod2; crush. *) + (* rewrite Integers.Int.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. *) + (* rewrite Integers.Int.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. } *) + + (* (** 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 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. *) + (* apply assumption_32bit. *) + (* econstructor. econstructor. econstructor. crush. *) + (* econstructor. econstructor. econstructor. crush. *) + (* econstructor. econstructor. econstructor. *) + (* econstructor. econstructor. econstructor. econstructor. *) + (* econstructor. *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ6]). *) + (* econstructor. econstructor. 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 *) *) + (* match goal with *) + (* | [ |- context [valueToNat ?x] ] => *) + (* assert (Z.to_nat *) + (* (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4))) *) + (* = *) + (* valueToNat x) *) + (* as EXPR_OK by admit *) + (* end. *) + (* rewrite <- EXPR_OK. *) + + (* specialize (H7 (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4)))). *) + (* exploit H7; 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. *) + + (* + invert MARR. 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 *) *) + (* rename H0 into MOD_PRESERVE. *) + + (* (** 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. *) + (* apply assumption_32bit. *) + (* 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 *) *) + (* match goal with (* Prevents issues with evars *) *) + (* | [ |- context [valueToNat ?x] ] => *) + (* assert (Z.to_nat *) + (* (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4))) *) + (* = *) + (* valueToNat x) *) + (* as EXPR_OK by admit *) + (* end. *) + (* rewrite <- EXPR_OK. *) + + (* specialize (H7 (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4)))). *) + (* exploit H7; 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. *) + (* Admitted. *) + (* 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 *) *) + (* invert MARR. 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'. *) + (* invert 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 H6 in HPler0. *) + (* invert HPler0; try congruence. *) + (* rewrite EQr0 in H8. *) + (* invert H8. *) + (* clear H0. clear H6. *) + + (* 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. *) + (* { rewrite HeqOFFSET. *) + (* apply PtrofsExtra.add_mod; crush; try lia. *) + (* rewrite Integers.Ptrofs.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. *) + (* apply PtrofsExtra.of_int_mod. *) + (* rewrite Integers.Int.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. } *) + + (* (** 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. *) + (* apply assumption_32bit. *) + (* econstructor. econstructor. econstructor. *) + (* eapply Verilog.stmnt_runp_Vnonblock_arr. crush. *) + (* econstructor. *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ7]). *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ8]). *) + (* econstructor. *) + (* econstructor. *) + (* econstructor. econstructor. econstructor. econstructor. *) + (* econstructor. econstructor. econstructor. econstructor. *) + + (* all: crush. *) + + (* (** State Lookup *) *) + (* unfold Verilog.merge_regs. *) + (* crush. *) + (* unfold_merge. *) + (* apply AssocMap.gss. *) + + (* (** Match states *) *) + (* rewrite assumption_32bit. *) + (* 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 *) *) + (* match goal with *) + (* | [ |- context [valueToNat ?x] ] => *) + (* assert (Z.to_nat *) + (* (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4))) *) + (* = *) + (* valueToNat x) *) + (* as EXPR_OK by admit *) + (* end. *) + + (* 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. *) + (* 2: { reflexivity. } *) + (* unfold Verilog.arr_assocmap_set. *) + (* rewrite AssocMap.gss. *) + (* unfold Verilog.merge_arr. *) + (* rewrite AssocMap.gss. *) + (* setoid_rewrite H5. *) + (* reflexivity. *) + + (* rewrite combine_length. *) + (* rewrite <- array_set_len. *) + (* unfold arr_repeat. crush. *) + (* apply list_repeat_len. *) + + (* rewrite <- array_set_len. *) + (* unfold arr_repeat. crush. *) + (* rewrite list_repeat_len. *) + (* rewrite H4. reflexivity. *) + + (* 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. *) + (* 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 H13. *) + (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; invert H13; 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 H13. *) + (* rewrite Z_div_mult in H13; try lia. *) + (* replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H13 by reflexivity. *) + (* rewrite <- PtrofsExtra.divu_unsigned in H13; unfold_constants; try lia. *) + (* rewrite H13. rewrite EXPR_OK. *) + (* 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 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 <- 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 H7; 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; 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'. *) + (* invert H0. *) + (* 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: { 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. *) + (* invert 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. invert H0. congruence. *) + + (* + (** Preamble *) *) + (* invert MARR. 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'. *) + (* invert 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 H6 in HPler0. *) + (* apply H8 in HPler1. *) + (* invert HPler0; invert HPler1; try congruence. *) + (* rewrite EQr0 in H9. *) + (* rewrite EQr1 in H11. *) + (* invert H9. invert H11. *) + (* clear H0. clear H6. clear H8. *) + + (* 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. *) + (* { rewrite HeqOFFSET. *) + (* apply PtrofsExtra.add_mod; crush; try lia. *) + (* rewrite Integers.Ptrofs.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. *) + (* apply PtrofsExtra.of_int_mod. *) + (* apply IntExtra.add_mod; crush. *) + (* apply IntExtra.mul_mod2; crush. *) + (* rewrite Integers.Int.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. *) + (* rewrite Integers.Int.unsigned_repr_eq. *) + (* rewrite <- Zmod_div_mod; crush. } *) + + (* (** 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. *) + (* apply assumption_32bit. *) + (* econstructor. econstructor. econstructor. *) + (* eapply Verilog.stmnt_runp_Vnonblock_arr. crush. *) + (* econstructor. *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ9]). *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ10]). *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ11]). *) + (* econstructor. econstructor. econstructor. econstructor. *) + (* econstructor. *) + (* eapply Verilog.erun_Vbinop with (EQ := ?[EQ12]). *) + (* econstructor. econstructor. econstructor. econstructor. *) + (* econstructor. econstructor. econstructor. econstructor. *) + (* econstructor. econstructor. econstructor. econstructor. *) + + (* all: crush. *) + + (* (** State Lookup *) *) + (* unfold Verilog.merge_regs. *) + (* crush. *) + (* unfold_merge. *) + (* apply AssocMap.gss. *) + + (* (** Match states *) *) + (* rewrite assumption_32bit. *) + (* 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 (Z.to_nat *) + (* (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4))) *) + (* = *) + (* valueToNat (vdiv *) + (* (vplus (vplus asr # r0 (ZToValue 32 z0) ?EQ11) (vmul asr # r1 (ZToValue 32 z) ?EQ12) *) + (* ?EQ10) (ZToValue 32 4) ?EQ9)) *) + (* as EXPR_OK by admit. *) + + (* 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. *) + (* 2: { reflexivity. } *) + (* unfold Verilog.arr_assocmap_set. *) + (* rewrite AssocMap.gss. *) + (* unfold Verilog.merge_arr. *) + (* rewrite AssocMap.gss. *) + (* setoid_rewrite H5. *) + (* reflexivity. *) + + (* rewrite combine_length. *) + (* rewrite <- array_set_len. *) + (* unfold arr_repeat. crush. *) + (* apply list_repeat_len. *) + + (* rewrite <- array_set_len. *) + (* unfold arr_repeat. crush. *) + (* rewrite list_repeat_len. *) + (* rewrite H4. reflexivity. *) + + (* 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. *) + (* 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 H16. *) + (* destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; invert H16; 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 H16. *) + (* rewrite Z_div_mult in H16; try lia. *) + (* replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H16 by reflexivity. *) + (* rewrite <- PtrofsExtra.divu_unsigned in H16; unfold_constants; try lia. *) + (* rewrite H16. rewrite EXPR_OK. *) + (* 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 H18; try lia. *) + (* apply Zmult_lt_compat_r with (p := 4) in H18; try lia. *) + (* rewrite ZLib.div_mul_undo in H18; try lia. *) + (* split; try lia. *) + (* apply Z.le_trans with (m := RTL.fn_stacksize f); crush; lia. *) + (* } *) + + (* rewrite <- 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 H7; 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 H16; 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'. *) + (* invert H0. *) + (* 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. *) + (* } *) + (* 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. *) + (* invert 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. invert H0. congruence. *) + + (* + invert MARR. 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 *) *) + (* rename H0 into MOD_PRESERVE. *) + + (* (** 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. *) + (* apply assumption_32bit. *) + (* econstructor. econstructor. econstructor. *) + (* eapply Verilog.stmnt_runp_Vnonblock_arr. crush. *) + (* econstructor. econstructor. econstructor. econstructor. *) + + (* all: crush. *) + + (* (** State Lookup *) *) + (* unfold Verilog.merge_regs. *) + (* crush. *) + (* unfold_merge. *) + (* apply AssocMap.gss. *) + + (* (** Match states *) *) + (* rewrite assumption_32bit. *) + (* 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 (Z.to_nat *) + (* (Integers.Ptrofs.unsigned *) + (* (Integers.Ptrofs.divu *) + (* OFFSET *) + (* (Integers.Ptrofs.repr 4))) *) + (* = *) + (* valueToNat (ZToValue 32 (Integers.Ptrofs.unsigned OFFSET / 4))) *) + (* as EXPR_OK by admit. *) + + (* 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. *) + (* 2: { reflexivity. } *) + (* unfold Verilog.arr_assocmap_set. *) + (* rewrite AssocMap.gss. *) + (* unfold Verilog.merge_arr. *) + (* rewrite AssocMap.gss. *) + (* setoid_rewrite H5. *) + (* reflexivity. *) + + (* rewrite combine_length. *) + (* rewrite <- array_set_len. *) + (* unfold arr_repeat. crush. *) + (* apply list_repeat_len. *) + + (* rewrite <- array_set_len. *) + (* unfold arr_repeat. crush. *) + (* rewrite list_repeat_len. *) + (* rewrite H4. reflexivity. *) + + (* 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; invert 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 EXPR_OK. *) + (* 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 H11; try lia. *) + (* 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. *) + (* } *) + + (* rewrite <- 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 H7; 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'. *) + (* invert H0. *) + (* apply Zmult_lt_compat_r with (p := 4) in H9; try lia. *) + (* rewrite ZLib.div_mul_undo in H9; 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. *) + (* invert 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. invert H0. congruence. *) + (* Admitted. *) + (* 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. *) + + (* eexists. split. apply Smallstep.plus_one. *) + (* eapply HTL.step_module; eauto. *) + (* apply assumption_32bit. *) + (* eapply Verilog.stmnt_runp_Vnonblock_reg with *) + (* (rhsval := if b then posToValue 32 ifso else posToValue 32 ifnot). *) + (* constructor. *) + + (* simpl. *) + (* destruct b. *) + (* eapply Verilog.erun_Vternary_true. *) + (* eapply eval_cond_correct; eauto. *) + (* constructor. *) + (* apply boolToValue_ValueToBool. *) + (* eapply Verilog.erun_Vternary_false. *) + (* eapply eval_cond_correct; eauto. *) + (* constructor. *) + (* apply boolToValue_ValueToBool. *) + (* constructor. *) + + (* big_tac. *) + + (* invert MARR. *) + (* destruct b; rewrite assumption_32bit; big_tac. *) + + (* Unshelve. *) + (* constructor. *) + (* Qed. *) + (* 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. *) + (* Admitted. *) + (* 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. *) + (* apply assumption_32bit. *) + (* constructor. *) + (* econstructor; simpl; trivial. *) + (* econstructor; simpl; trivial. *) + (* constructor. *) + (* econstructor; simpl; trivial. *) + (* constructor. *) + + (* constructor. constructor. *) + + (* unfold state_st_wf in WF; big_tac; eauto. *) + + (* 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. *) + (* apply assumption_32bit. *) + (* constructor. *) + (* econstructor; simpl; trivial. *) + (* econstructor; simpl; trivial. *) + (* constructor. constructor. constructor. *) + (* constructor. constructor. constructor. *) + + (* unfold state_st_wf in WF; big_tac; eauto. *) + + (* apply HTL.step_finish. *) + (* unfold Verilog.merge_regs. *) + (* 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. *) + (* 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 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. *) + (* invert H. *) + (* crush. *) + (* unfold Mem.load. *) + (* intros. *) + (* match goal with | |- context[if ?x then _ else _] => destruct x end; try congruence. *) + (* invert 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. *) + (* invert H. *) + (* crush. *) + (* unfold Mem.store. *) + (* intros. *) + (* match goal with | |- context[if ?x then _ else _] => destruct x end; try congruence. *) + (* invert 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. *) + (* Opaque Mem.alloc. *) + (* Opaque Mem.load. *) + (* Opaque Mem.store. *) + (* Qed. *) + (* 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. *) + (* 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. *) + (* 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. invert MF. constructor. reflexivity. *) + (* Qed. *) + (* 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. *) + (* Hint Resolve transl_step_correct : htlproof. *) + +(* Theorem transf_program_correct: *) +(* Smallstep.forward_simulation (RTL.semantics prog) (HTL.semantics tprog). *) +(* Proof. *) +(* Admitted. *) +(* (* eapply Smallstep.forward_simulation_plus; eauto with htlproof. *) *) +(* (* apply senv_preserved. *) *) +(* (* Qed. *) *) + +(* End CORRECTNESS. *) |