Library Machabstr

The Mach intermediate language: abstract semantics.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Mem.
Require Import Integers.
Require Import Values.
Require Import Mem.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Op.
Require Import Locations.
Require Conventions.
Require Import Mach.
Require Stacklayout.
This file defines the "abstract" semantics for the Mach intermediate language, as opposed to the more concrete semantics given in module Machconcr.

The only difference with the concrete semantics is the interpretation of the stack access instructions Mgetstack, Msetstack and Mgetparam. In the concrete semantics, these are interpreted as memory accesses relative to the stack pointer. In the abstract semantics, these instructions are interpreted as accesses in a frame environment, not resident in memory. The frame environment is an additional component of the state.

Not having the frame data in memory facilitates the proof of the Stacking pass, which shows that the generated code executes correctly with the abstract semantics.

Structure of abstract stack frames
An abstract stack frame is a mapping from (type, offset) pairs to values. Like location sets (see module Locations), overlap can occur.

Definition frame : Type := typ -> Z -> val.

Definition typ_eq: forall (ty1 ty2: typ), {ty1=ty2} + {ty1<>ty2}.


Definition update (ty: typ) (ofs: Z) (v: val) (f: frame) : frame :=
  fun ty' ofs' =>
    if zeq ofs ofs' then
      if typ_eq ty ty' then v else Vundef
    else
      if zle (ofs' + AST.typesize ty') ofs then f ty' ofs'
      else if zle (ofs + AST.typesize ty) ofs' then f ty' ofs'
      else Vundef.

Lemma update_same:
  forall ty ofs v fr,
  update ty ofs v fr ty ofs = v.


Lemma update_other:
  forall ty ofs v fr ty' ofs',
  ofs + AST.typesize ty <= ofs' \/ ofs' + AST.typesize ty' <= ofs ->
  update ty ofs v fr ty' ofs' = fr ty' ofs'.


Definition empty_frame : frame := fun ty ofs => Vundef.

Section FRAME_ACCESSES.

Variable f: function.
A slot (ty, ofs) within a frame is valid in function f if it lies within the bounds of f's frame, it does not overlap with the slots reserved for the return address and the back link, and it is aligned on a 4-byte boundary.

Inductive slot_valid: typ -> Z -> Prop :=
  slot_valid_intro:
    forall ty ofs,
    0 <= ofs ->
    ofs + AST.typesize ty <= f.(fn_framesize) ->
    (ofs + AST.typesize ty <= Int.signed f.(fn_link_ofs)
       \/ Int.signed f.(fn_link_ofs) + 4 <= ofs) ->
    (ofs + AST.typesize ty <= Int.signed f.(fn_retaddr_ofs)
       \/ Int.signed f.(fn_retaddr_ofs) + 4 <= ofs) ->
    (4 | ofs) ->
    slot_valid ty ofs.
get_slot f fr ty ofs v holds if the frame fr contains value v with type ty at word offset ofs.

Inductive get_slot: frame -> typ -> Z -> val -> Prop :=
  | get_slot_intro:
      forall fr ty ofs v,
      slot_valid ty ofs ->
      v = fr ty (ofs - f.(fn_framesize)) ->
      get_slot fr ty ofs v.
set_slot f fr ty ofs v fr' holds if frame fr' is obtained from frame fr by storing value v with type ty at word offset ofs.

Inductive set_slot: frame -> typ -> Z -> val -> frame -> Prop :=
  | set_slot_intro:
      forall fr ty ofs v fr',
      slot_valid ty ofs ->
      fr' = update ty (ofs - f.(fn_framesize)) v fr ->
      set_slot fr ty ofs v fr'.
Extract the values of the arguments of an external call.

Inductive extcall_arg: regset -> frame -> loc -> val -> Prop :=
  | extcall_arg_reg: forall rs fr r,
      extcall_arg rs fr (R r) (rs r)
  | extcall_arg_stack: forall rs fr ofs ty v,
      get_slot fr ty (Int.signed (Int.repr (Stacklayout.fe_ofs_arg + 4 * ofs))) v ->
      extcall_arg rs fr (S (Outgoing ofs ty)) v.

Inductive extcall_args: regset -> frame -> list loc -> list val -> Prop :=
  | extcall_args_nil: forall rs fr,
      extcall_args rs fr nil nil
  | extcall_args_cons: forall rs fr l1 ll v1 vl,
      extcall_arg rs fr l1 v1 -> extcall_args rs fr ll vl ->
      extcall_args rs fr (l1 :: ll) (v1 :: vl).

Definition extcall_arguments
   (rs: regset) (fr: frame) (sg: signature) (args: list val) : Prop :=
  extcall_args rs fr (Conventions.loc_arguments sg) args.

End FRAME_ACCESSES.
Mach execution states.

Inductive stackframe: Type :=
  | Stackframe:
      forall (f: function) calling function
             (sp: val) stack pointer in calling function
             (c: code) program point in calling function
             (fr: frame), frame state in calling function
      stackframe.

Inductive state: Type :=
  | State:
      forall (stack: list stackframe) call stack
             (f: function) function currently executing
             (sp: val) stack pointer
             (c: code) current program point
             (rs: regset) register state
             (fr: frame) frame state
             (m: mem), memory state
      state
  | Callstate:
      forall (stack: list stackframe) call stack
             (f: fundef) 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.
parent_frame s returns the frame of the calling function. It is used to access function parameters that are passed on the stack (the Mgetparent instruction). parent_function s returns the calling function itself. Suitable defaults are used if there are no calling function.

Definition parent_frame (s: list stackframe) : frame :=
  match s with
  | nil => empty_frame
  | Stackframe f sp c fr :: s => fr
  end.

Definition empty_function :=
  mkfunction (mksignature nil None) nil 0 0 Int.zero Int.zero.

Definition parent_function (s: list stackframe) : function :=
  match s with
  | nil => empty_function
  | Stackframe f sp c fr :: s => f
  end.

Section RELSEM.

Variable ge: genv.

Definition find_function (ros: mreg + ident) (rs: regset) : option fundef :=
  match ros with
  | inl r => Genv.find_funct ge (rs r)
  | inr symb =>
      match Genv.find_symbol ge symb with
      | None => None
      | Some b => Genv.find_funct_ptr ge b
      end
  end.

Inductive step: state -> trace -> state -> Prop :=
  | exec_Mlabel:
      forall s f sp lbl c rs fr m,
      step (State s f sp (Mlabel lbl :: c) rs fr m)
        E0 (State s f sp c rs fr m)
  | exec_Mgetstack:
      forall s f sp ofs ty dst c rs fr m v,
      get_slot f fr ty (Int.signed ofs) v ->
      step (State s f sp (Mgetstack ofs ty dst :: c) rs fr m)
        E0 (State s f sp c (rs#dst <- v) fr m)
  | exec_Msetstack:
     forall s f sp src ofs ty c rs fr m fr',
      set_slot f fr ty (Int.signed ofs) (rs src) fr' ->
      step (State s f sp (Msetstack src ofs ty :: c) rs fr m)
        E0 (State s f sp c rs fr' m)
  | exec_Mgetparam:
      forall s f sp ofs ty dst c rs fr m v,
      get_slot (parent_function s) (parent_frame s) ty (Int.signed ofs) v ->
      step (State s f sp (Mgetparam ofs ty dst :: c) rs fr m)
        E0 (State s f sp c (rs#dst <- v) fr m)
  | exec_Mop:
      forall s f sp op args res c rs fr m v,
      eval_operation ge sp op rs##args = Some v ->
      step (State s f sp (Mop op args res :: c) rs fr m)
        E0 (State s f sp c (rs#res <- v) fr m)
  | exec_Mload:
      forall s f sp chunk addr args dst c rs fr m a v,
      eval_addressing ge sp addr rs##args = Some a ->
      Mem.loadv chunk m a = Some v ->
      step (State s f sp (Mload chunk addr args dst :: c) rs fr m)
        E0 (State s f sp c (rs#dst <- v) fr m)
  | exec_Mstore:
      forall s f sp chunk addr args src c rs fr m m' a,
      eval_addressing ge sp addr rs##args = Some a ->
      Mem.storev chunk m a (rs src) = Some m' ->
      step (State s f sp (Mstore chunk addr args src :: c) rs fr m)
        E0 (State s f sp c rs fr m')
  | exec_Mcall:
      forall s f sp sig ros c rs fr m f',
      find_function ros rs = Some f' ->
      step (State s f sp (Mcall sig ros :: c) rs fr m)
        E0 (Callstate (Stackframe f sp c fr :: s) f' rs m)
  | exec_Mtailcall:
      forall s f stk soff sig ros c rs fr m f',
      find_function ros rs = Some f' ->
      step (State s f (Vptr stk soff) (Mtailcall sig ros :: c) rs fr m)
        E0 (Callstate s f' rs (Mem.free m stk))
  | exec_Mgoto:
      forall s f sp lbl c rs fr m c',
      find_label lbl f.(fn_code) = Some c' ->
      step (State s f sp (Mgoto lbl :: c) rs fr m)
        E0 (State s f sp c' rs fr m)
  | exec_Mcond_true:
      forall s f sp cond args lbl c rs fr m c',
      eval_condition cond rs##args = Some true ->
      find_label lbl f.(fn_code) = Some c' ->
      step (State s f sp (Mcond cond args lbl :: c) rs fr m)
        E0 (State s f sp c' rs fr m)
  | exec_Mcond_false:
      forall s f sp cond args lbl c rs fr m,
      eval_condition cond rs##args = Some false ->
      step (State s f sp (Mcond cond args lbl :: c) rs fr m)
        E0 (State s f sp c rs fr m)
  | exec_Mjumptable:
      forall s f sp arg tbl c rs fr m n lbl c',
      rs arg = Vint n ->
      list_nth_z tbl (Int.signed n) = Some lbl ->
      find_label lbl f.(fn_code) = Some c' ->
      step (State s f sp (Mjumptable arg tbl :: c) rs fr m)
        E0 (State s f sp c' rs fr m)
  | exec_Mreturn:
      forall s f stk soff c rs fr m,
      step (State s f (Vptr stk soff) (Mreturn :: c) rs fr m)
        E0 (Returnstate s rs (Mem.free m stk))
  | exec_function_internal:
      forall s f rs m m' stk,
      Mem.alloc m 0 f.(fn_stacksize) = (m', stk) ->
      step (Callstate s (Internal f) rs m)
        E0 (State s f (Vptr stk (Int.repr (-f.(fn_framesize))))
                  f.(fn_code) rs empty_frame m')
  | exec_function_external:
      forall s ef args res rs1 rs2 m t,
      event_match ef args t res ->
      extcall_arguments (parent_function s) rs1 (parent_frame s) ef.(ef_sig) args ->
      rs2 = (rs1#(Conventions.loc_result ef.(ef_sig)) <- res) ->
      step (Callstate s (External ef) rs1 m)
         t (Returnstate s rs2 m)
  | exec_return:
      forall f sp c fr s rs m,
      step (Returnstate (Stackframe f sp c fr :: s) rs m)
        E0 (State s f sp c rs fr m).

End RELSEM.

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

Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall rs m r,
      rs (Conventions.loc_result (mksignature nil (Some Tint))) = Vint r ->
      final_state (Returnstate nil rs m) r.

Definition exec_program (p: program) (beh: program_behavior) : Prop :=
  program_behaves step (initial_state p) final_state (Genv.globalenv p) beh.