Module Mach


The Mach intermediate language: abstract syntax. Mach is the last intermediate language before generation of assembly code.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Events.
Require Import Smallstep.
Require Import Op.
Require Import Locations.
Require Import Conventions.
Require Stacklayout.

Abstract syntax


Like Linear, the Mach language is organized as lists of instructions operating over machine registers, with default fall-through behaviour and explicit labels and branch instructions. The main difference with Linear lies in the instructions used to access the activation record. Mach has three such instructions: Mgetstack and Msetstack to read and write within the activation record for the current function, at a given word offset and with a given type; and Mgetparam, to read within the activation record of the caller. These instructions implement a more concrete view of the activation record than the the Lgetstack and Lsetstack instructions of Linear: actual offsets are used instead of abstract stack slots, and the distinction between the caller's frame and the callee's frame is made explicit.

Definition label := positive.

Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.

Definition code := list instruction.

Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.

Definition fundef := AST.fundef function.

Definition program := AST.program fundef unit.

Definition funsig (fd: fundef) :=
  match fd with
  | Internal f => fn_sig f
  | External ef => ef_sig ef
  end.

Definition genv := Genv.t fundef unit.

Operational semantics


The semantics for Mach is close to that of Linear: they differ only on the interpretation of stack slot accesses. In Mach, these accesses are interpreted as memory accesses relative to the stack pointer. More precisely: In addition to this linking of activation records, the semantics also make provisions for storing a back link at offset f.(fn_link_ofs) from the stack pointer, and a return address at offset f.(fn_retaddr_ofs). The latter stack location will be used by the Asm code generated by Asmgen to save the return address into the caller at the beginning of a function, then restore it and jump to it at the end of a function. The Mach concrete semantics does not attach any particular meaning to the pointer stored in this reserved location, but makes sure that it is preserved during execution of a function. The return_address_offset parameter is used to guess the value of the return address that the Asm code generated later will store in the reserved location.

Definition load_stack (m: mem) (sp: val) (ty: typ) (ofs: ptrofs) :=
  Mem.loadv (chunk_of_type ty) m (Val.offset_ptr sp ofs).

Definition store_stack (m: mem) (sp: val) (ty: typ) (ofs: ptrofs) (v: val) :=
  Mem.storev (chunk_of_type ty) m (Val.offset_ptr sp ofs) v.

Module RegEq.
  Definition t := mreg.
  Definition eq := mreg_eq.
End RegEq.

Module Regmap := EMap(RegEq).

Definition regset := Regmap.t val.

Notation "a ## b" := (List.map a b) (at level 1).
Notation "a # b <- c" := (Regmap.set b c a) (at level 1, b at next level).

Fixpoint undef_regs (rl: list mreg) (rs: regset) {struct rl} : regset :=
  match rl with
  | nil => rs
  | r1 :: rl' => Regmap.set r1 Vundef (undef_regs rl' rs)
  end.

Lemma undef_regs_other:
  forall r rl rs, ~In r rl -> undef_regs rl rs r = rs r.
Proof.
  induction rl; simpl; intros. auto. rewrite Regmap.gso. apply IHrl. intuition. intuition.
Qed.

Lemma undef_regs_same:
  forall r rl rs, In r rl -> undef_regs rl rs r = Vundef.
Proof.
  induction rl; simpl; intros. tauto.
  destruct H. subst a. apply Regmap.gss.
  unfold Regmap.set. destruct (RegEq.eq r a); auto.
Qed.

Definition set_pair (p: rpair mreg) (v: val) (rs: regset) : regset :=
  match p with
  | One r => rs#r <- v
  | Twolong rhi rlo => rs#rhi <- (Val.hiword v) #rlo <- (Val.loword v)
  end.

Fixpoint set_res (res: builtin_res mreg) (v: val) (rs: regset) : regset :=
  match res with
  | BR r => Regmap.set r v rs
  | BR_none => rs
  | BR_splitlong hi lo => set_res lo (Val.loword v) (set_res hi (Val.hiword v) rs)
  end.

Definition is_label (lbl: label) (instr: instruction) : bool :=
  match instr with
  | Mlabel lbl' => if peq lbl lbl' then true else false
  | _ => false
  end.

Lemma is_label_correct:
  forall lbl instr,
  if is_label lbl instr then instr = Mlabel lbl else instr <> Mlabel lbl.
Proof.
  intros. destruct instr; simpl; try discriminate.
  case (peq lbl l); intro; congruence.
Qed.

Fixpoint find_label (lbl: label) (c: code) {struct c} : option code :=
  match c with
  | nil => None
  | i1 :: il => if is_label lbl i1 then Some il else find_label lbl il
  end.

Lemma find_label_tail:
  forall lbl c c', find_label lbl c = Some c' -> is_tail c' c.
Proof.
  induction c; simpl; intros. discriminate.
  destruct (is_label lbl a). inv H. auto with coqlib. eauto with coqlib.
Qed.

Lemma find_label_incl:
  forall lbl c c', find_label lbl c = Some c' -> incl c' c.
Proof.
  intros; red; intros. eapply is_tail_incl; eauto. eapply find_label_tail; eauto.
Qed.

Section RELSEM.

Variable return_address_offset: function -> code -> ptrofs -> Prop.

Variable ge: genv.

Definition find_function_ptr
        (ge: genv) (ros: mreg + ident) (rs: regset) : option block :=
  match ros with
  | inl r =>
      match rs r with
      | Vptr b ofs => if Ptrofs.eq ofs Ptrofs.zero then Some b else None
      | _ => None
      end
  | inr symb =>
      Genv.find_symbol ge symb
  end.

Extract the values of the arguments to an external call.

Inductive extcall_arg (rs: regset) (m: mem) (sp: val): loc -> val -> Prop :=
  | extcall_arg_reg: forall r,
      extcall_arg rs m sp (R r) (rs r)
  | extcall_arg_stack: forall ofs ty v,
      load_stack m sp ty (Ptrofs.repr (Stacklayout.fe_ofs_arg + 4 * ofs)) = Some v ->
      extcall_arg rs m sp (S Outgoing ofs ty) v.

Inductive extcall_arg_pair (rs: regset) (m: mem) (sp: val): rpair loc -> val -> Prop :=
  | extcall_arg_one: forall l v,
      extcall_arg rs m sp l v ->
      extcall_arg_pair rs m sp (One l) v
  | extcall_arg_twolong: forall hi lo vhi vlo,
      extcall_arg rs m sp hi vhi ->
      extcall_arg rs m sp lo vlo ->
      extcall_arg_pair rs m sp (Twolong hi lo) (Val.longofwords vhi vlo).

Definition extcall_arguments
    (rs: regset) (m: mem) (sp: val) (sg: signature) (args: list val) : Prop :=
  list_forall2 (extcall_arg_pair rs m sp) (loc_arguments sg) args.

Mach execution states.

Mach execution states.

Inductive stackframe: Type :=
  | Stackframe:
      forall (f: block) (* pointer to calling function *)
             (sp: val) (* stack pointer in calling function *)
             (retaddr: val) (* Asm return address in calling function *)
             (c: code), (* program point in calling function *)
      stackframe.

Inductive state: Type :=
  | State:
      forall (stack: list stackframe) (* call stack *)
             (f: block) (* pointer to current function *)
             (sp: val) (* stack pointer *)
             (c: code) (* current program point *)
             (rs: regset) (* register state *)
             (m: mem), (* memory state *)
      state
  | Callstate:
      forall (stack: list stackframe) (* call stack *)
             (f: block) (* pointer to function to call *)
             (rs: regset) (* register state *)
             (m: mem), (* memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe) (* call stack *)
             (rs: regset) (* register state *)
             (m: mem), (* memory state *)
      state.

Definition parent_sp (s: list stackframe) : val :=
  match s with
  | nil => Vnullptr
  | Stackframe f sp ra c :: s' => sp
  end.

Definition parent_ra (s: list stackframe) : val :=
  match s with
  | nil => Vnullptr
  | Stackframe f sp ra c :: s' => ra
  end.

Inductive step: state -> trace -> state -> Prop :=
  | exec_Mlabel:
      forall s f sp lbl c rs m,
      step (State s f sp (Mlabel lbl :: c) rs m)
        E0 (State s f sp c rs m)
  | exec_Mgetstack:
      forall s f sp ofs ty dst c rs m v,
      load_stack m sp ty ofs = Some v ->
      step (State s f sp (Mgetstack ofs ty dst :: c) rs m)
        E0 (State s f sp c (rs#dst <- v) m)
  | exec_Msetstack:
      forall s f sp src ofs ty c rs m m' rs',
      store_stack m sp ty ofs (rs src) = Some m' ->
      rs' = undef_regs (destroyed_by_setstack ty) rs ->
      step (State s f sp (Msetstack src ofs ty :: c) rs m)
        E0 (State s f sp c rs' m')
  | exec_Mgetparam:
      forall s fb f sp ofs ty dst c rs m v rs',
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      load_stack m sp Tptr f.(fn_link_ofs) = Some (parent_sp s) ->
      load_stack m (parent_sp s) ty ofs = Some v ->
      rs' = (rs # temp_for_parent_frame <- Vundef # dst <- v) ->
      step (State s fb sp (Mgetparam ofs ty dst :: c) rs m)
        E0 (State s fb sp c rs' m)
  | exec_Mop:
      forall s f sp op args res c rs m v rs',
      eval_operation ge sp op rs##args m = Some v ->
      rs' = ((undef_regs (destroyed_by_op op) rs)#res <- v) ->
      step (State s f sp (Mop op args res :: c) rs m)
        E0 (State s f sp c rs' m)
  | exec_Mload:
      forall s f sp chunk addr args dst c rs m a v rs',
      eval_addressing ge sp addr rs##args = Some a ->
      Mem.loadv chunk m a = Some v ->
      rs' = ((undef_regs (destroyed_by_load chunk addr) rs)#dst <- v) ->
      step (State s f sp (Mload chunk addr args dst :: c) rs m)
        E0 (State s f sp c rs' m)
  | exec_Mstore:
      forall s f sp chunk addr args src c rs m m' a rs',
      eval_addressing ge sp addr rs##args = Some a ->
      Mem.storev chunk m a (rs src) = Some m' ->
      rs' = undef_regs (destroyed_by_store chunk addr) rs ->
      step (State s f sp (Mstore chunk addr args src :: c) rs m)
        E0 (State s f sp c rs' m')
  | exec_Mcall:
      forall s fb sp sig ros c rs m f f' ra,
      find_function_ptr ge ros rs = Some f' ->
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      return_address_offset f c ra ->
      step (State s fb sp (Mcall sig ros :: c) rs m)
        E0 (Callstate (Stackframe fb sp (Vptr fb ra) c :: s)
                       f' rs m)
  | exec_Mtailcall:
      forall s fb stk soff sig ros c rs m f f' m',
      find_function_ptr ge ros rs = Some f' ->
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      load_stack m (Vptr stk soff) Tptr f.(fn_link_ofs) = Some (parent_sp s) ->
      load_stack m (Vptr stk soff) Tptr f.(fn_retaddr_ofs) = Some (parent_ra s) ->
      Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
      step (State s fb (Vptr stk soff) (Mtailcall sig ros :: c) rs m)
        E0 (Callstate s f' rs m')
  | exec_Mbuiltin:
      forall s f sp rs m ef args res b vargs t vres rs' m',
      eval_builtin_args ge rs sp m args vargs ->
      external_call ef ge vargs m t vres m' ->
      rs' = set_res res vres (undef_regs (destroyed_by_builtin ef) rs) ->
      step (State s f sp (Mbuiltin ef args res :: b) rs m)
         t (State s f sp b rs' m')
  | exec_Mgoto:
      forall s fb f sp lbl c rs m c',
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      find_label lbl f.(fn_code) = Some c' ->
      step (State s fb sp (Mgoto lbl :: c) rs m)
        E0 (State s fb sp c' rs m)
  | exec_Mcond_true:
      forall s fb f sp cond args lbl c rs m c' rs',
      eval_condition cond rs##args m = Some true ->
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      find_label lbl f.(fn_code) = Some c' ->
      rs' = undef_regs (destroyed_by_cond cond) rs ->
      step (State s fb sp (Mcond cond args lbl :: c) rs m)
        E0 (State s fb sp c' rs' m)
  | exec_Mcond_false:
      forall s f sp cond args lbl c rs m rs',
      eval_condition cond rs##args m = Some false ->
      rs' = undef_regs (destroyed_by_cond cond) rs ->
      step (State s f sp (Mcond cond args lbl :: c) rs m)
        E0 (State s f sp c rs' m)
  | exec_Mjumptable:
      forall s fb f sp arg tbl c rs m n lbl c' rs',
      rs arg = Vint n ->
      list_nth_z tbl (Int.unsigned n) = Some lbl ->
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      find_label lbl f.(fn_code) = Some c' ->
      rs' = undef_regs destroyed_by_jumptable rs ->
      step (State s fb sp (Mjumptable arg tbl :: c) rs m)
        E0 (State s fb sp c' rs' m)
  | exec_Mreturn:
      forall s fb stk soff c rs m f m',
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      load_stack m (Vptr stk soff) Tptr f.(fn_link_ofs) = Some (parent_sp s) ->
      load_stack m (Vptr stk soff) Tptr f.(fn_retaddr_ofs) = Some (parent_ra s) ->
      Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
      step (State s fb (Vptr stk soff) (Mreturn :: c) rs m)
        E0 (Returnstate s rs m')
  | exec_function_internal:
      forall s fb rs m f m1 m2 m3 stk rs',
      Genv.find_funct_ptr ge fb = Some (Internal f) ->
      Mem.alloc m 0 f.(fn_stacksize) = (m1, stk) ->
      let sp := Vptr stk Ptrofs.zero in
      store_stack m1 sp Tptr f.(fn_link_ofs) (parent_sp s) = Some m2 ->
      store_stack m2 sp Tptr f.(fn_retaddr_ofs) (parent_ra s) = Some m3 ->
      rs' = undef_regs destroyed_at_function_entry rs ->
      step (Callstate s fb rs m)
        E0 (State s fb sp f.(fn_code) rs' m3)
  | exec_function_external:
      forall s fb rs m t rs' ef args res m',
      Genv.find_funct_ptr ge fb = Some (External ef) ->
      extcall_arguments rs m (parent_sp s) (ef_sig ef) args ->
      external_call ef ge args m t res m' ->
      rs' = set_pair (loc_result (ef_sig ef)) res rs ->
      step (Callstate s fb rs m)
         t (Returnstate s rs' m')
  | exec_return:
      forall s f sp ra c rs m,
      step (Returnstate (Stackframe f sp ra c :: s) rs m)
        E0 (State s f sp c rs m).

End RELSEM.

Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall fb m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some fb ->
      initial_state p (Callstate nil fb (Regmap.init Vundef) m0).

Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall rs m r retcode,
      loc_result signature_main = One r ->
      rs r = Vint retcode ->
      final_state (Returnstate nil rs m) retcode.

Definition semantics (rao: function -> code -> ptrofs -> Prop) (p: program) :=
  Semantics (step rao) (initial_state p) final_state (Genv.globalenv p).

Leaf functions


A leaf function is a function that contains no Mcall instruction.

Definition is_leaf_function (f: function) : bool :=
  List.forallb
    (fun i => match i with Mcall _ _ => false | _ => true end)
    f.(fn_code).

Semantic characterization of leaf functions: functions in the call stack are never leaf functions.

Section WF_STATES.

Variable rao: function -> code -> ptrofs -> Prop.

Variable ge: genv.

Inductive wf_frame: stackframe -> Prop :=
  | wf_stackframe_intro: forall fb sp ra c f
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (LEAF: is_leaf_function f = false)
        (TAIL: is_tail c f.(fn_code)),
      wf_frame (Stackframe fb sp ra c).

Inductive wf_state: state -> Prop :=
  | wf_normal_state: forall s fb sp c rs m f
        (STACK: Forall wf_frame s)
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (TAIL: is_tail c f.(fn_code)),
      wf_state (State s fb sp c rs m)
  | wf_call_state: forall s fb rs m
        (STACK: Forall wf_frame s),
      wf_state (Callstate s fb rs m)
  | wf_return_state: forall s rs m
        (STACK: Forall wf_frame s),
      wf_state (Returnstate s rs m).

Lemma wf_step:
  forall S1 t S2, step rao ge S1 t S2 -> wf_state S1 -> wf_state S2.
Proof.
  induction 1; intros WF; inv WF; try (econstructor; now eauto with coqlib).
- (* call *)
  assert (f0 = f) by congruence. subst f0.
  constructor.
  constructor; auto. econstructor; eauto with coqlib.
  destruct (is_leaf_function f) eqn:E; auto.
  unfold is_leaf_function in E; rewrite forallb_forall in E.
  symmetry. apply (E (Mcall sig ros)). eapply is_tail_in; eauto.
- (* goto *)
  assert (f0 = f) by congruence. subst f0. econstructor; eauto using find_label_tail.
- (* cond *)
  assert (f0 = f) by congruence. subst f0. econstructor; eauto using find_label_tail.
- (* jumptable *)
  assert (f0 = f) by congruence. subst f0. econstructor; eauto using find_label_tail.
- (* return *)
  inv STACK. inv H1. econstructor; eauto.
Qed.

End WF_STATES.

Lemma wf_initial:
  forall p S, initial_state p S -> wf_state (Genv.globalenv p) S.
Proof.
  intros. inv H. fold ge. constructor. constructor.
Qed.