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| author | Nadesh Ramanathan <nadeshramanathan88@gmail.com> | 2020-07-07 15:33:32 +0100 |
|---|---|---|
| committer | Nadesh Ramanathan <nadeshramanathan88@gmail.com> | 2020-07-07 15:33:32 +0100 |
| commit | 5ca10a134176352b25a0b434d5508852c7314670 (patch) | |
| tree | cb24d9e17782c3ad08079b9090463008e70cb019 /src/translation/HTLgenproof.v | |
| parent | 65ac86da554770ba0e3a24d187037c6a72a8725b (diff) | |
| parent | e0fe2958baaa0c61159b2a0e330c323f9f2f646c (diff) | |
| download | vericert-kvx-5ca10a134176352b25a0b434d5508852c7314670.tar.gz vericert-kvx-5ca10a134176352b25a0b434d5508852c7314670.zip | |
Merge branch 'dev-nadesh-proven' of https://github.com/ymherklotz/coqup into dev-nadesh-proven
Diffstat (limited to 'src/translation/HTLgenproof.v')
| -rw-r--r-- | src/translation/HTLgenproof.v | 3830 |
1 files changed, 1972 insertions, 1858 deletions
diff --git a/src/translation/HTLgenproof.v b/src/translation/HTLgenproof.v index 6e470d5..813a94b 100644 --- a/src/translation/HTLgenproof.v +++ b/src/translation/HTLgenproof.v @@ -16,9 +16,9 @@ * along with this program. If not, see <https://www.gnu.org/licenses/>. *) -From compcert Require RTL Registers AST Integers. -From compcert Require Import Globalenvs Memory. -From coqup Require Import Coquplib HTLgenspec HTLgen Value AssocMap Array IntegerExtra ZExtra. +From compcert Require RTL Registers AST. +From compcert Require Import Integers Globalenvs Memory Linking. +From coqup Require Import Coquplib HTLgenspec HTLgen ValueInt AssocMap Array IntegerExtra ZExtra. From coqup Require HTL Verilog. Require Import Lia. @@ -41,7 +41,7 @@ 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 32 st). + 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) : @@ -53,7 +53,7 @@ Inductive match_arrs (m : HTL.module) (f : RTL.function) (sp : Values.val) (mem 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 32 0) + (Option.default (NToValue 0) (Option.join (array_get_error (Z.to_nat ptr) stack)))) -> match_arrs m f sp mem asa. @@ -86,18 +86,13 @@ Definition stack_bounds (sp : Values.val) (hi : Z) (m : mem) : Prop := Inductive match_frames : list RTL.stackframe -> list HTL.stackframe -> Prop := | match_frames_nil : match_frames nil nil. -(* | match_frames_cons : *) -(* forall cs lr r f sp sp' pc rs m asr asa *) -(* (TF : tr_module f m) *) -(* (ST: match_frames mem cs lr) *) -(* (MA: match_assocmaps f rs asr) *) -(* (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_frames mem (RTL.Stackframe r f sp pc rs :: cs) *) -(* (HTL.Stackframe r m pc asr asa :: lr). *) + +Inductive match_constants : HTL.module -> assocmap -> Prop := + match_constant : + forall m asr, + asr!(HTL.mod_reset m) = Some (ZToValue 0) -> + asr!(HTL.mod_finish m) = Some (ZToValue 0) -> + match_constants m asr. Inductive match_states : RTL.state -> HTL.state -> Prop := | match_state : forall asa asr sf f sp sp' rs mem m st res @@ -109,7 +104,8 @@ Inductive match_states : RTL.state -> HTL.state -> Prop := (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), + (BOUNDS : stack_bounds sp (f.(RTL.fn_stacksize)) mem) + (CONST : match_constants m asr), match_states (RTL.State sf f sp st rs mem) (HTL.State res m st asr asa) | match_returnstate : @@ -128,11 +124,30 @@ 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. +Instance TransfHTLLink (tr_fun: RTL.program -> Errors.res HTL.program): + TransfLink (fun (p1: RTL.program) (p2: HTL.program) => match_prog p1 p2). +Proof. + red. intros. exfalso. destruct (link_linkorder _ _ _ H) as [LO1 LO2]. + apply link_prog_inv in H. + + unfold match_prog in *. + unfold main_is_internal in *. simplify. repeat (unfold_match H4). + repeat (unfold_match H3). simplify. + subst. rewrite H0 in *. specialize (H (AST.prog_main p2)). + exploit H. + apply Genv.find_def_symbol. exists b. split. + assumption. apply Genv.find_funct_ptr_iff. eassumption. + apply Genv.find_def_symbol. exists b0. split. + assumption. apply Genv.find_funct_ptr_iff. eassumption. + intros. inv H3. inv H5. destruct H6. inv H5. +Qed. + +Definition match_prog' (p: RTL.program) (tp: HTL.program) := + Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp. + +Lemma match_prog_matches : + forall p tp, match_prog p tp -> match_prog' p tp. +Proof. unfold match_prog. tauto. Qed. Lemma transf_program_match: forall p tp, HTLgen.transl_program p = Errors.OK tp -> match_prog p tp. @@ -266,32 +281,6 @@ Proof. assumption. Qed. -(* Need to eventually move posToValue 32 to posToValueAuto, as that has this proof. *) -Lemma assumption_32bit : - forall v, - valueToPos (posToValue 32 v) = v. -Admitted. - -Lemma st_greater_than_res : - forall m res : positive, - m <> res. -Admitted. - -Lemma finish_not_return : - forall r f : positive, - r <> f. -Admitted. - -Lemma finish_not_res : - forall f r : positive, - f <> r. -Admitted. - -Lemma greater_than_max_func : - forall f st, - Plt (RTL.max_reg_function f) st. -Proof. Admitted. - Ltac inv_state := match goal with MSTATE : match_states _ _ |- _ => @@ -353,1818 +342,1943 @@ Section CORRECTNESS. 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 eval_correct : - forall sp op rs args m v v' e asr asa f s s' i, - 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 -> - val_value_lessdef v v' -> - Verilog.expr_runp f asr asa e v'. - 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 32 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 32 _)] ] => 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 32 _) ] => - apply regs_lessdef_add_greater; [> apply greater_than_max_func | 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. - - (* Iop *) - (* destruct v eqn:?; *) - (* try ( *) - (* destruct op eqn:?; inversion H21; simpl in H0; repeat (unfold_match H0); *) - (* inversion H0; subst; simpl in *; try (unfold_func H4); try (unfold_func H5); *) - (* try (unfold_func H6); *) - (* try (unfold Op.eval_addressing32 in H6; repeat (unfold_match H6); inversion H6; *) - (* unfold_func H3); *) - - (* inversion Heql; inversion MASSOC; subst; *) - (* assert (HPle : Ple r (RTL.max_reg_function f)) *) - (* by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) - (* apply H1 in HPle; inversion HPle; *) - (* rewrite H2 in *; discriminate *) - (* ). *) - - (* + econstructor. split. *) - (* apply Smallstep.plus_one. *) - (* eapply HTL.step_module; eauto. *) - (* econstructor; simpl; trivial. *) - (* constructor; trivial. *) - (* econstructor; simpl; eauto. *) - (* eapply eval_correct; eauto. constructor. *) - (* 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 (valueToZ 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. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - rewrite Integers.Ptrofs.signed_repr; try assumption. - admit. (* FIXME: Register bounds. *) - apply PtrofsExtra.of_int_mod. - rewrite Integers.Int.signed_repr; 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: { apply st_greater_than_res. } - 2: { apply st_greater_than_res. } - - (** 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 H10. - rewrite EQr1 in H12. - invert H10. invert H12. - 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 (valueToZ 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. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - rewrite Integers.Ptrofs.signed_repr; try assumption. - admit. (* FIXME: Register bounds. *) - apply PtrofsExtra.of_int_mod. - apply IntExtra.add_mod; crush; try lia. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - apply IntExtra.mul_mod; crush; try lia. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - admit. (* FIXME: Register bounds. *) - rewrite Integers.Int.signed_repr; crush. - rewrite Integers.Int.signed_repr; 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: { apply st_greater_than_res. } - 2: { apply st_greater_than_res. } - - (** 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: { apply st_greater_than_res. } - 2: { apply st_greater_than_res. } - - (** 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. - 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 (valueToZ 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. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - rewrite Integers.Ptrofs.signed_repr; try assumption. - admit. (* FIXME: Register bounds. *) - apply PtrofsExtra.of_int_mod. - rewrite Integers.Int.signed_repr; 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. apply greater_than_max_func. - 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 (valueToZ 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 H19; try lia. - apply Zmult_lt_compat_r with (p := 4) in H19; try lia. - rewrite ZLib.div_mul_undo in H19; 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 H10. - rewrite EQr1 in H12. - invert H10. invert H12. - 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 (valueToZ 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. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - rewrite Integers.Ptrofs.signed_repr; try assumption. - admit. (* FIXME: Register bounds. *) - apply PtrofsExtra.of_int_mod. - apply IntExtra.add_mod; crush; try lia. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - apply IntExtra.mul_mod; crush; try lia. - exists 1073741824. reflexivity. (* FIXME: This is sadness inducing. *) - admit. (* FIXME: Register bounds. *) - rewrite Integers.Int.signed_repr; crush; try split; try assumption. - rewrite Integers.Int.signed_repr; crush; try split; try assumption. - } - - (** Write bounds proof *) - assert (Integers.Ptrofs.unsigned OFFSET < f.(RTL.fn_stacksize)) as WRITE_BOUND_HIGH. - { destruct (Integers.Ptrofs.unsigned OFFSET <? f.(RTL.fn_stacksize)) eqn:EQ; crush; auto. - unfold stack_bounds in BOUNDS. - exploit (BOUNDS (Integers.Ptrofs.unsigned OFFSET) (Registers.Regmap.get src rs)); auto. - split; try lia; apply Integers.Ptrofs.unsigned_range_2. - small_tac. } - - (** Start of proof proper *) - eexists. split. - eapply Smallstep.plus_one. - eapply HTL.step_module; eauto. - 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. apply greater_than_max_func. - 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 (valueToZ 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 H20. - destruct (Registers.Regmap.get src rs) eqn:EQ_SRC; constructor; invert H20; 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 H20. - rewrite Z_div_mult in H20; try lia. - replace 4 with (Integers.Ptrofs.unsigned (Integers.Ptrofs.repr 4)) in H20 by reflexivity. - rewrite <- PtrofsExtra.divu_unsigned in H20; unfold_constants; try lia. - rewrite H20. 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 H22; try lia. - apply Zmult_lt_compat_r with (p := 4) in H22; try lia. - rewrite ZLib.div_mul_undo in H22; 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 H20; 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 H21; try lia. - rewrite ZLib.div_mul_undo in H21; 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. apply greater_than_max_func. - 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 H12; try lia. - apply Zmult_lt_compat_r with (p := 4) in H12; try lia. - rewrite ZLib.div_mul_undo in H12; 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 H10; try lia. - rewrite ZLib.div_mul_undo in H10; 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 st_greater_than_res. - apply st_greater_than_res. - - apply HTL.step_finish. - unfold Verilog.merge_regs. - unfold_merge; simpl. - rewrite AssocMap.gso. - apply AssocMap.gss. - apply finish_not_return. - 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 st_greater_than_res. - apply st_greater_than_res. - - apply HTL.step_finish. - unfold Verilog.merge_regs. - unfold_merge. - rewrite AssocMap.gso. - apply AssocMap.gss. - apply finish_not_return. - 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. - apply st_greater_than_res. - - 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. - apply greater_than_max_func. - 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 H3. - unfold Mem.perm in H3. crush. - unfold Mem.perm_order' in H3. - small_tac. - exploit (H3 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 H3. - unfold Mem.perm in H3. crush. - unfold Mem.perm_order' in H3. - small_tac. - exploit (H3 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. - - (* Had to admit proof because currently there is no way to force main to be Internal. *) - 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. +(* 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 op_stack_based : *) +(* forall F V sp v m args rs op ge pc' res0 pc f e fin rtrn st stk, *) +(* tr_instr fin rtrn st stk (RTL.Iop op args res0 pc') *) +(* (Verilog.Vnonblock (Verilog.Vvar res0) e) *) +(* (state_goto st pc') -> *) +(* reg_stack_based_pointers sp rs -> *) +(* (RTL.fn_code f) ! pc = Some (RTL.Iop op args res0 pc') -> *) +(* @Op.eval_operation F V ge (Values.Vptr sp Ptrofs.zero) op *) +(* (map (fun r : positive => Registers.Regmap.get r rs) args) m = Some v -> *) +(* stack_based v sp. *) +(* Proof. *) +(* Ltac solve_no_ptr := *) +(* match goal with *) +(* | H: reg_stack_based_pointers ?sp ?rs |- stack_based (Registers.Regmap.get ?r ?rs) _ => *) +(* solve [apply H] *) +(* | H1: reg_stack_based_pointers ?sp ?rs, H2: Registers.Regmap.get _ _ = Values.Vptr ?b ?i *) +(* |- context[Values.Vptr ?b _] => *) +(* let H := fresh "H" in *) +(* assert (H: stack_based (Values.Vptr b i) sp) by (rewrite <- H2; apply H1); simplify; solve [auto] *) +(* | |- context[Registers.Regmap.get ?lr ?lrs] => *) +(* destruct (Registers.Regmap.get lr lrs) eqn:?; simplify; auto *) +(* | |- stack_based (?f _) _ => unfold f *) +(* | |- stack_based (?f _ _) _ => unfold f *) +(* | |- stack_based (?f _ _ _) _ => unfold f *) +(* | |- stack_based (?f _ _ _ _) _ => unfold f *) +(* | H: ?f _ _ = Some _ |- _ => *) +(* unfold f in H; repeat (unfold_match H); inv H *) +(* | H: ?f _ _ _ _ _ _ = Some _ |- _ => *) +(* unfold f in H; repeat (unfold_match H); inv H *) +(* | H: map (fun r : positive => Registers.Regmap.get r _) ?args = _ |- _ => *) +(* destruct args; inv H *) +(* | |- context[if ?c then _ else _] => destruct c; try discriminate *) +(* | H: match _ with _ => _ end = Some _ |- _ => repeat (unfold_match H) *) +(* | |- context[match ?g with _ => _ end] => destruct g; try discriminate *) +(* | |- _ => simplify; solve [auto] *) +(* end. *) +(* intros F V sp v m args rs op g pc' res0 pc f e fin rtrn st stk INSTR RSBP SEL EVAL. *) +(* inv INSTR. unfold translate_instr in H5. *) +(* unfold_match H5; repeat (unfold_match H5); repeat (simplify; solve_no_ptr). *) +(* Qed. *) + +(* Lemma int_inj : *) +(* forall x y, *) +(* Int.unsigned x = Int.unsigned y -> *) +(* x = y. *) +(* Proof. *) +(* intros. rewrite <- Int.repr_unsigned at 1. rewrite <- Int.repr_unsigned. *) +(* rewrite <- H. trivial. *) +(* Qed. *) + +(* 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. *) +(* Ltac eval_correct_tac := *) +(* match goal with *) +(* | |- context[valueToPtr] => unfold valueToPtr *) +(* | |- context[valueToInt] => unfold valueToInt *) +(* | |- context[bop] => unfold bop *) +(* | |- context[boplit] => unfold boplit *) +(* | |- val_value_lessdef Values.Vundef _ => solve [constructor] *) +(* | H : val_value_lessdef _ _ |- val_value_lessdef (Values.Vint _) _ => constructor; inv H *) +(* | |- val_value_lessdef (Values.Vint _) _ => constructor; auto *) +(* | H : context[RTL.max_reg_function ?f] *) +(* |- context[_ (Registers.Regmap.get ?r ?rs) (Registers.Regmap.get ?r0 ?rs)] => *) +(* let HPle1 := fresh "HPle" in *) +(* let HPle2 := fresh "HPle" in *) +(* assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) +(* assert (HPle2 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) +(* apply H in HPle1; apply H in HPle2; eexists; split; *) +(* [econstructor; eauto; constructor; trivial | inv HPle1; inv HPle2; try (constructor; auto)] *) +(* | H : context[RTL.max_reg_function ?f] *) +(* |- context[_ (Registers.Regmap.get ?r ?rs) _] => *) +(* let HPle1 := fresh "HPle" in *) +(* assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) +(* apply H in HPle1; eexists; split; *) +(* [econstructor; eauto; constructor; trivial | inv HPle1; try (constructor; auto)] *) +(* | H : _ :: _ = _ :: _ |- _ => inv H *) +(* | |- context[match ?d with _ => _ end] => destruct d eqn:?; try discriminate *) +(* | |- Verilog.expr_runp _ _ _ _ _ => econstructor *) +(* | |- val_value_lessdef (?f _ _) _ => unfold f *) +(* | |- val_value_lessdef (?f _) _ => unfold f *) +(* | H : ?f (Registers.Regmap.get _ _) _ = Some _ |- _ => *) +(* unfold f in H; repeat (unfold_match H) *) +(* | H1 : Registers.Regmap.get ?r ?rs = Values.Vint _, H2 : val_value_lessdef (Registers.Regmap.get ?r ?rs) _ *) +(* |- _ => rewrite H1 in H2; inv H2 *) +(* | |- _ => eexists; split; try constructor; solve [eauto] *) +(* | H : context[RTL.max_reg_function ?f] |- context[_ (Verilog.Vvar ?r) (Verilog.Vvar ?r0)] => *) +(* let HPle1 := fresh "H" in *) +(* let HPle2 := fresh "H" in *) +(* assert (HPle1 : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) +(* assert (HPle2 : Ple r0 (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) +(* apply H in HPle1; apply H in HPle2; eexists; split; try constructor; eauto *) +(* | H : context[RTL.max_reg_function ?f] |- context[Verilog.Vvar ?r] => *) +(* let HPle := fresh "H" in *) +(* assert (HPle : Ple r (RTL.max_reg_function f)) by (eapply RTL.max_reg_function_use; eauto; simpl; auto); *) +(* apply H in HPle; eexists; split; try constructor; eauto *) +(* | |- context[if ?c then _ else _] => destruct c eqn:?; try discriminate *) +(* end. *) +(* 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); inv EVAL; *) +(* repeat (simplify; eval_correct_tac; unfold valueToInt in *) +(* - pose proof Integers.Ptrofs.agree32_sub as H2; unfold Integers.Ptrofs.agree32 in H2. *) +(* unfold Ptrofs.of_int. simpl. *) +(* apply ptrofs_inj. assert (Archi.ptr64 = false) by auto. eapply H2 in H3. *) +(* rewrite Ptrofs.unsigned_repr. apply H3. replace Ptrofs.max_unsigned with Int.max_unsigned; auto. *) +(* apply Int.unsigned_range_2. *) +(* auto. rewrite Ptrofs.unsigned_repr. replace Ptrofs.max_unsigned with Int.max_unsigned; auto. *) +(* apply Int.unsigned_range_2. rewrite Ptrofs.unsigned_repr. auto. *) +(* replace Ptrofs.max_unsigned with Int.max_unsigned; auto. *) +(* apply Int.unsigned_range_2. *) +(* - pose proof Integers.Ptrofs.agree32_sub as AGR; unfold Integers.Ptrofs.agree32 in AGR. *) +(* assert (ARCH: Archi.ptr64 = false) by auto. eapply AGR in ARCH. *) +(* apply int_inj. unfold Ptrofs.to_int. rewrite Int.unsigned_repr. *) +(* apply ARCH. Search Ptrofs.unsigned. pose proof Ptrofs.unsigned_range_2. *) +(* replace Ptrofs.max_unsigned with Int.max_unsigned; auto. *) +(* pose proof Ptrofs.agree32_of_int. unfold Ptrofs.agree32 in H2. *) +(* eapply H2 in ARCH. apply ARCH. *) +(* pose proof Ptrofs.agree32_of_int. unfold Ptrofs.agree32 in H2. *) +(* eapply H2 in ARCH. apply ARCH. *) +(* - admit. (* mulhs *) *) +(* - admit. (* mulhu *) *) +(* - rewrite H0 in Heqb. rewrite H1 in Heqb. discriminate. *) +(* - rewrite Heqb in Heqb0. discriminate. *) +(* - rewrite H0 in Heqb. rewrite H1 in Heqb. discriminate. *) +(* - rewrite Heqb in Heqb0. discriminate. *) +(* - admit. *) +(* - admit. (* ror *) *) +(* - admit. (* addressing *) *) +(* - admit. (* eval_condition *) *) +(* - admit. (* select *) *) +(* 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[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. *) +(* inv CONST; assumption. *) +(* inv CONST; assumption. *) +(* (* processing of state *) *) +(* econstructor. *) +(* crush. *) +(* econstructor. *) +(* econstructor. *) +(* econstructor. *) + +(* all: invert MARR; big_tac. *) + +(* inv CONST; constructor; simplify; rewrite AssocMap.gso; auto; lia. *) + +(* Unshelve. auto. *) +(* 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. inv MARR. *) +(* exploit eval_correct; eauto. intros. inversion H1. inversion H2. *) +(* econstructor. split. *) +(* apply Smallstep.plus_one. *) +(* eapply HTL.step_module; eauto. *) +(* inv CONST. assumption. *) +(* inv CONST. assumption. *) +(* econstructor; simpl; trivial. *) +(* constructor; trivial. *) +(* econstructor; simpl; eauto. *) +(* simpl. econstructor. econstructor. *) +(* apply H5. simplify. *) + +(* all: big_tac. *) + +(* assert (HPle: Ple res0 (RTL.max_reg_function f)) *) +(* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) + +(* unfold Ple in HPle. lia. *) +(* apply regs_lessdef_add_match. assumption. *) +(* apply regs_lessdef_add_greater. unfold Plt; lia. assumption. *) +(* assert (HPle: Ple res0 (RTL.max_reg_function f)) *) +(* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) +(* unfold Ple in HPle; lia. *) +(* eapply op_stack_based; eauto. *) +(* inv CONST. constructor; simplify. rewrite AssocMap.gso. rewrite AssocMap.gso. *) +(* assumption. lia. *) +(* assert (HPle: Ple res0 (RTL.max_reg_function f)) *) +(* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) +(* unfold Ple in HPle. lia. *) +(* rewrite AssocMap.gso. rewrite AssocMap.gso. *) +(* assumption. lia. *) +(* assert (HPle: Ple res0 (RTL.max_reg_function f)) *) +(* by (eapply RTL.max_reg_function_def; eauto; simpl; auto). *) +(* unfold Ple in HPle. lia. *) +(* Unshelve. trivial. *) +(* Qed. *) +(* 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. *) +(* inv CONST; assumption. *) +(* inv CONST; assumption. *) +(* (* 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.*) *) +(* Admitted. *) +(* 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. *) +(* inv CONST; assumption. *) +(* inv CONST; assumption. *) +(* constructor. *) +(* econstructor; simpl; trivial. *) +(* econstructor; simpl; trivial. *) +(* constructor. *) +(* econstructor; simpl; trivial. *) +(* constructor. *) + +(* constructor. constructor. *) + +(* unfold state_st_wf in WF; big_tac; eauto. *) +(* destruct wf as [HCTRL HDATA]. apply HCTRL. *) +(* apply AssocMapExt.elements_iff. eexists. *) +(* match goal with H: control ! pc = Some _ |- _ => apply H end. *) + +(* apply HTL.step_finish. *) +(* unfold Verilog.merge_regs. *) +(* unfold_merge; simpl. *) +(* rewrite AssocMap.gso. *) +(* apply AssocMap.gss. lia. *) +(* apply AssocMap.gss. *) +(* rewrite Events.E0_left. reflexivity. *) + +(* constructor; auto. *) +(* constructor. *) + +(* (* FIXME: Duplication *) *) +(* - econstructor. split. *) +(* eapply Smallstep.plus_two. *) +(* eapply HTL.step_module; eauto. *) +(* inv CONST; assumption. *) +(* inv CONST; assumption. *) +(* constructor. *) +(* econstructor; simpl; trivial. *) +(* econstructor; simpl; trivial. *) +(* constructor. constructor. constructor. *) +(* constructor. constructor. constructor. *) + +(* unfold state_st_wf in WF; big_tac; eauto. *) + +(* destruct wf as [HCTRL HDATA]. apply HCTRL. *) +(* apply AssocMapExt.elements_iff. eexists. *) +(* match goal with H: control ! pc = Some _ |- _ => apply H end. *) + +(* apply HTL.step_finish. *) +(* unfold Verilog.merge_regs. *) +(* unfold_merge. *) +(* 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 regs_lessdef_add_greater. unfold Plt; lia. *) +(* apply regs_lessdef_add_greater. unfold Plt; lia. *) +(* apply init_reg_assoc_empty. *) + +(* constructor. *) + +(* destruct (Mem.load AST.Mint32 m' stk *) +(* (Integers.Ptrofs.unsigned (Integers.Ptrofs.add *) +(* Integers.Ptrofs.zero *) +(* (Integers.Ptrofs.repr (4 * ptr))))) eqn:LOAD. *) +(* pose proof Mem.load_alloc_same as LOAD_ALLOC. *) +(* pose proof H as ALLOC. *) +(* eapply LOAD_ALLOC in ALLOC. *) +(* 2: { exact LOAD. } *) +(* ptrofs. rewrite LOAD. *) +(* rewrite ALLOC. *) +(* repeat constructor. *) + +(* ptrofs. rewrite LOAD. *) +(* repeat constructor. *) + +(* unfold reg_stack_based_pointers. intros. *) +(* unfold RTL.init_regs; crush. *) +(* destruct (RTL.fn_params f); *) +(* rewrite Registers.Regmap.gi; constructor. *) + +(* unfold arr_stack_based_pointers. intros. *) +(* crush. *) +(* destruct (Mem.load AST.Mint32 m' stk *) +(* (Integers.Ptrofs.unsigned (Integers.Ptrofs.add *) +(* Integers.Ptrofs.zero *) +(* (Integers.Ptrofs.repr (4 * ptr))))) eqn:LOAD. *) +(* pose proof Mem.load_alloc_same as LOAD_ALLOC. *) +(* pose proof H as ALLOC. *) +(* eapply LOAD_ALLOC in ALLOC. *) +(* 2: { exact LOAD. } *) +(* rewrite ALLOC. *) +(* repeat constructor. *) +(* constructor. *) + +(* Transparent Mem.alloc. (* TODO: Since there are opaque there's probably a lemma. *) *) +(* Transparent Mem.load. *) +(* Transparent Mem.store. *) +(* unfold stack_bounds. *) +(* split. *) + +(* unfold Mem.alloc in H. *) +(* 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. *) +(* constructor. simplify. rewrite AssocMap.gss. *) +(* simplify. rewrite AssocMap.gso. apply AssocMap.gss. simplify. 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. + (* eapply Smallstep.forward_simulation_plus; eauto with htlproof. *) + (* apply senv_preserved. *) + (* Qed. *) + Admitted. End CORRECTNESS. |