Library Linear

The Linear intermediate language: abstract syntax and semantcs

The Linear language is a variant of LTLin where arithmetic instructions operate on machine registers (type mreg) instead of arbitrary locations. Special instructions Lgetstack and Lsetstack are provided to access stack slots.





Require Import Coqlib.

Require Import Maps.

Require Import AST.

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 Import LTL.

Require Import Conventions.

Abstract syntax





Definition label := positive.




Inductive instruction: Type :=

  | Lgetstack: slot -> mreg -> instruction

  | Lsetstack: mreg -> slot -> instruction

  | Lop: operation -> list mreg -> mreg -> instruction

  | Lload: memory_chunk -> addressing -> list mreg -> mreg -> instruction

  | Lstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction

  | Lcall: signature -> mreg + ident -> instruction

  | Ltailcall: signature -> mreg + ident -> instruction

  | Llabel: label -> instruction

  | Lgoto: label -> instruction

  | Lcond: condition -> list mreg -> label -> instruction

  | Ljumptable: mreg -> list label -> instruction

  | Lreturn: instruction.




Definition code: Type := list instruction.




Record function: Type := mkfunction {

  fn_sig: signature;

  fn_stacksize: Z;

  fn_code: code

}.




Definition fundef := AST.fundef function.




Definition program := AST.program fundef unit.




Definition funsig (fd: fundef) :=

  match fd with

  | Internal f => f.(fn_sig)

  | External ef => ef.(ef_sig)

  end.




Definition genv := Genv.t fundef.

Definition locset := Locmap.t.

Operational semantics

Looking up labels in the instruction list.





Definition is_label (lbl: label) (instr: instruction) : bool :=

  match instr with

  | Llabel 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 = Llabel lbl else instr <> Llabel lbl.

find_label lbl c returns a list of instruction, suffix of the code c, that immediately follows the Llabel lbl pseudo-instruction. If the label lbl is multiply-defined, the first occurrence is retained. If the label lbl is not defined, None is returned.





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.




Section RELSEM.




Variable ge: genv.




Definition find_function (ros: mreg + ident) (rs: locset) : option fundef :=

  match ros with

  | inl r => Genv.find_funct ge (rs (R r))

  | inr symb =>

      match Genv.find_symbol ge symb with

      | None => None

      | Some b => Genv.find_funct_ptr ge b

      end

  end.




Definition reglist (rs: locset) (rl: list mreg) : list val :=

  List.map (fun r => rs (R r)) rl.

Calling conventions are reflected at the level of location sets (environments mapping locations to values) by the following two functions.

call_regs caller returns the location set at function entry, as a function of the location set caller of the calling function.

Machine registers have the same values as in the caller.
Incoming stack slots (used for parameter passing) have the same values as the corresponding outgoing stack slots (used for argument passing) in the caller.
Local and outgoing stack slots are initialized to undefined values.





Definition call_regs (caller: locset) : locset :=

  fun (l: loc) =>

    match l with

    | R r => caller (R r)

    | S (Local ofs ty) => Vundef

    | S (Incoming ofs ty) => caller (S (Outgoing ofs ty))

    | S (Outgoing ofs ty) => Vundef

    end.

return_regs caller callee returns the location set after a call instruction, as a function of the location set caller of the caller before the call instruction and of the location set callee of the callee at the return instruction.

Callee-save machine registers have the same values as in the caller before the call.
Caller-save machine registers have the same values as in the callee at the return point.
Stack slots have the same values as in the caller before the call.





Definition return_regs (caller callee: locset) : locset :=

  fun (l: loc) =>

    match l with

    | R r =>

        if In_dec Loc.eq (R r) Conventions.temporaries then

          callee (R r)

        else if In_dec Loc.eq (R r) Conventions.destroyed_at_call then

          callee (R r)

        else

          caller (R r)

    | S s => caller (S s)

    end.

Linear execution states.





Inductive stackframe: Type :=

  | Stackframe:

      forall (f: function)

calling function




             (sp: val)

stack pointer in calling function




             (rs: locset)

location state in calling function




             (c: code),

program point 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: locset)

location state




             (m: mem),

memory state




      state

  | Callstate:

      forall (stack: list stackframe)

call stack




             (f: fundef)

function to call




             (rs: locset)

location state at point of call




             (m: mem),

memory state




      state

  | Returnstate:

      forall (stack: list stackframe)

call stack




             (rs: locset)

location state at point of return




             (m: mem),

memory state




      state.

parent_locset cs returns the mapping of values for locations of the caller function.





Definition parent_locset (stack: list stackframe) : locset :=

  match stack with

  | nil => Locmap.init Vundef

  | Stackframe f sp ls c :: stack' => ls

  end.

The main difference between the Linear transition relation and the LTL transition relation is the handling of function calls. In LTL, arguments and results to calls are transmitted via vargs and vres components of Callstate and Returnstate, respectively. The semantics takes care of transferring these values between the locations of the caller and of the callee.

In Linear and lower-level languages (Mach, PPC), arguments and results are transmitted implicitly: the generated code for the caller arranges for arguments to be left in conventional registers and stack locations, as determined by the calling conventions, where the function being called will find them. Similarly, conventional registers will be used to pass the result value back to the caller. This is reflected in the definition of Callstate and Returnstate above, where a whole location state rs is passed instead of just the values of arguments or returns as in LTL.

These location states passed across calls are treated in a way that reflects the callee-save/caller-save convention:

The exec_Lcall transition from State to Callstate saves the current location state ls in the call stack, and passes it to the callee.
The exec_function_internal transition from Callstate to State changes the view of stack slots (Outgoing slots slide to Incoming slots as per call_regs).
The exec_Lreturn transition from State to Returnstate restores the values of callee-save locations from the location state of the caller, using return_regs.

This protocol makes it much easier to later prove the correctness of the Stacking pass, which inserts actual code that performs the saving and restoring of callee-save registers described above.





Inductive step: state -> trace -> state -> Prop :=

  | exec_Lgetstack:

      forall s f sp sl r b rs m,

      step (State s f sp (Lgetstack sl r :: b) rs m)

        E0 (State s f sp b (Locmap.set (R r) (rs (S sl)) rs) m)

  | exec_Lsetstack:

      forall s f sp r sl b rs m,

      step (State s f sp (Lsetstack r sl :: b) rs m)

        E0 (State s f sp b (Locmap.set (S sl) (rs (R r)) rs) m)

  | exec_Lop:

      forall s f sp op args res b rs m v,

      eval_operation ge sp op (reglist rs args) = Some v ->

      step (State s f sp (Lop op args res :: b) rs m)

        E0 (State s f sp b (Locmap.set (R res) v rs) m)

  | exec_Lload:

      forall s f sp chunk addr args dst b rs m a v,

      eval_addressing ge sp addr (reglist rs args) = Some a ->

      loadv chunk m a = Some v ->

      step (State s f sp (Lload chunk addr args dst :: b) rs m)

        E0 (State s f sp b (Locmap.set (R dst) v rs) m)

  | exec_Lstore:

      forall s f sp chunk addr args src b rs m m' a,

      eval_addressing ge sp addr (reglist rs args) = Some a ->

      storev chunk m a (rs (R src)) = Some m' ->

      step (State s f sp (Lstore chunk addr args src :: b) rs m)

        E0 (State s f sp b rs m')

  | exec_Lcall:

      forall s f sp sig ros b rs m f',

      find_function ros rs = Some f' ->

      sig = funsig f' ->

      step (State s f sp (Lcall sig ros :: b) rs m)

        E0 (Callstate (Stackframe f sp rs b:: s) f' rs m)

  | exec_Ltailcall:

      forall s f stk sig ros b rs m f',

      find_function ros rs = Some f' ->

      sig = funsig f' ->

      step (State s f (Vptr stk Int.zero) (Ltailcall sig ros :: b) rs m)

        E0 (Callstate s f' (return_regs (parent_locset s) rs) (Mem.free m stk))

  | exec_Llabel:

      forall s f sp lbl b rs m,

      step (State s f sp (Llabel lbl :: b) rs m)

        E0 (State s f sp b rs m)

  | exec_Lgoto:

      forall s f sp lbl b rs m b',

      find_label lbl f.(fn_code) = Some b' ->

      step (State s f sp (Lgoto lbl :: b) rs m)

        E0 (State s f sp b' rs m)

  | exec_Lcond_true:

      forall s f sp cond args lbl b rs m b',

      eval_condition cond (reglist rs args) = Some true ->

      find_label lbl f.(fn_code) = Some b' ->

      step (State s f sp (Lcond cond args lbl :: b) rs m)

        E0 (State s f sp b' rs m)

  | exec_Lcond_false:

      forall s f sp cond args lbl b rs m,

      eval_condition cond (reglist rs args) = Some false ->

      step (State s f sp (Lcond cond args lbl :: b) rs m)

        E0 (State s f sp b rs m)

  | exec_Ljumptable:

      forall s f sp arg tbl b rs m n lbl b',

      rs (R arg) = Vint n ->

      list_nth_z tbl (Int.signed n) = Some lbl ->

      find_label lbl f.(fn_code) = Some b' ->

      step (State s f sp (Ljumptable arg tbl :: b) rs m)

        E0 (State s f sp b' rs m)

  | exec_Lreturn:

      forall s f stk b rs m,

      step (State s f (Vptr stk Int.zero) (Lreturn :: b) rs m)

        E0 (Returnstate s (return_regs (parent_locset s) rs) (Mem.free m stk))

  | exec_function_internal:

      forall s f rs m m' stk,

      alloc m 0 f.(fn_stacksize) = (m', stk) ->

      step (Callstate s (Internal f) rs m)

        E0 (State s f (Vptr stk Int.zero) f.(fn_code) (call_regs rs) m')

  | exec_function_external:

      forall s ef args res rs1 rs2 m t,

      event_match ef args t res ->

      args = List.map rs1 (Conventions.loc_arguments ef.(ef_sig)) ->

      rs2 = Locmap.set (R (Conventions.loc_result ef.(ef_sig))) res rs1 ->

      step (Callstate s (External ef) rs1 m)

         t (Returnstate s rs2 m)

  | exec_return:

      forall s f sp rs0 c rs m,

      step (Returnstate (Stackframe f sp rs0 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 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 ->

      funsig f = mksignature nil (Some Tint) ->

      initial_state p (Callstate nil f (Locmap.init Vundef) m0).




Inductive final_state: state -> int -> Prop :=

  | final_state_intro: forall rs m r,

      rs (R (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.