The RTL intermediate language: abstract syntax and semantics. RTL stands for "Register Transfer Language". This is the first intermediate language after Cminor and CminorSel.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Values.
Require Import Events.
Require Import Memory.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Op.
Require Import Registers.

Abstract syntax

RTL is organized as instructions, functions and programs. Instructions correspond roughly to elementary instructions of the target processor, but take their arguments and leave their results in pseudo-registers (also called temporaries in textbooks). Infinitely many pseudo-registers are available, and each function has its own set of pseudo-registers, unaffected by function calls. Instructions are organized as a control-flow graph: a function is a finite map from ``nodes'' (abstract program points) to instructions, and each instruction lists explicitly the nodes of its successors.

Definition node := positive.

Inductive instruction: Type :=
| Inop: node -> instruction

No operation -- just branch to the successor.

| Iop: operation -> list reg -> reg -> node -> instruction

Iop op args dest succ performs the arithmetic operation op over the values of registers args, stores the result in dest, and branches to succ.

| Iload: memory_chunk -> addressing -> list reg -> reg -> node -> instruction

Iload chunk addr args dest succ loads a chunk quantity from the address determined by the addressing mode addr and the values of the args registers, stores the quantity just read into dest, and branches to succ.

| Istore: memory_chunk -> addressing -> list reg -> reg -> node -> instruction

Istore chunk addr args src succ stores the value of register src in the chunk quantity at the the address determined by the addressing mode addr and the values of the args registers, then branches to succ.

| Icall: signature -> reg + ident -> list reg -> reg -> node -> instruction

Icall sig fn args dest succ invokes the function determined by fn (either a function pointer found in a register or a function name), giving it the values of registers args as arguments. It stores the return value in dest and branches to succ.

| Itailcall: signature -> reg + ident -> list reg -> instruction

Itailcall sig fn args performs a function invocation in tail-call position.

| Ibuiltin: external_function -> list reg -> reg -> node -> instruction

Ibuiltin ef args dest succ calls the built-in function identified by ef, giving it the values of args as arguments. It stores the return value in dest and branches to succ.

| Icond: condition -> list reg -> node -> node -> instruction

Icond cond args ifso ifnot evaluates the boolean condition cond over the values of registers args. If the condition is true, it transitions to ifso. If the condition is false, it transitions to ifnot.

| Ijumptable: reg -> list node -> instruction

Ijumptable arg tbl transitions to the node that is the n-th element of the list tbl, where n is the signed integer value of register arg.

| Ireturn: option reg -> instruction.

Ireturn terminates the execution of the current function (it has no successor). It returns the value of the given register, or Vundef if none is given.

Definition code: Type := PTree.t instruction.

Record function: Type := mkfunction {
  fn_sig: signature;
  fn_params: list reg;
  fn_stacksize: Z;
  fn_code: code;
  fn_entrypoint: node
}.

A function description comprises a control-flow graph (CFG) fn_code (a partial finite mapping from nodes to instructions). As in Cminor, fn_sig is the function signature and fn_stacksize the number of bytes for its stack-allocated activation record. fn_params is the list of registers that are bound to the values of arguments at call time. fn_entrypoint is the node of the first instruction of the function in the CFG. fn_code_wf asserts that all instructions of the function have nodes no greater than fn_nextpc.

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.

Operational semantics

Definition genv := Genv.t fundef unit.
Definition regset := Regmap.t val.

Fixpoint init_regs (vl: list val) (rl: list reg) {struct rl} : regset :=
  match rl, vl with
  | r1 :: rs, v1 :: vs => Regmap.set r1 v1 (init_regs vs rs)
  | _, _ => Regmap.init Vundef
  end.

The dynamic semantics of RTL is given in small-step style, as a set of transitions between states. A state captures the current point in the execution. Three kinds of states appear in the transitions:

State cs f sp pc rs m describes an execution point within a function. f is the current function. sp is the pointer to the stack block for its current activation (as in Cminor). pc is the current program point (CFG node) within the code c. rs gives the current values for the pseudo-registers. m is the current memory state.
Callstate cs f args m is an intermediate state that appears during function calls. f is the function definition that we are calling. args (a list of values) are the arguments for this call. m is the current memory state.
Returnstate cs v m is an intermediate state that appears when a function terminates and returns to its caller. v is the return value and m the current memory state.

In all three kinds of states, the cs parameter represents the call stack. It is a list of frames Stackframe res f sp pc rs. Each frame represents a function call in progress. res is the pseudo-register that will receive the result of the call. f is the calling function. sp is its stack pointer. pc is the program point for the instruction that follows the call. rs is the state of registers in the calling function.

Inductive stackframe : Type :=
  | Stackframe:
      forall (res: reg) (* where to store the result *)
             (f: function) (* calling function *)
             (sp: val) (* stack pointer in calling function *)
             (pc: node) (* program point in calling function *)
             (rs: regset), (* register state in calling function *)
      stackframe.

Inductive state : Type :=
  | State:
      forall (stack: list stackframe) (* call stack *)
             (f: function) (* current function *)
             (sp: val) (* stack pointer *)
             (pc: node) (* current program point in c *)
             (rs: regset) (* register state *)
             (m: mem), (* memory state *)
      state
  | Callstate:
      forall (stack: list stackframe) (* call stack *)
             (f: fundef) (* function to call *)
             (args: list val) (* arguments to the call *)
             (m: mem), (* memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe) (* call stack *)
             (v: val) (* return value for the call *)
             (m: mem), (* memory state *)
      state.

Section RELSEM.

Variable ge: genv.

Definition find_function
      (ros: reg + 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.

The transitions are presented as an inductive predicate step ge st1 t st2, where ge is the global environment, st1 the initial state, st2 the final state, and t the trace of system calls performed during this transition.

Inductive step: state -> trace -> state -> Prop :=
  | exec_Inop:
      forall s f sp pc rs m pc',
      (fn_code f)!pc = Some(Inop pc') ->
      step (State s f sp pc rs m)
        E0 (State s f sp pc' rs m)
  | exec_Iop:
      forall s f sp pc rs m op args res pc' v,
      (fn_code f)!pc = Some(Iop op args res pc') ->
      eval_operation ge sp op rs##args m = Some v ->
      step (State s f sp pc rs m)
        E0 (State s f sp pc' (rs#res <- v) m)
  | exec_Iload:
      forall s f sp pc rs m chunk addr args dst pc' a v,
      (fn_code f)!pc = Some(Iload chunk addr args dst pc') ->
      eval_addressing ge sp addr rs##args = Some a ->
      Mem.loadv chunk m a = Some v ->
      step (State s f sp pc rs m)
        E0 (State s f sp pc' (rs#dst <- v) m)
  | exec_Istore:
      forall s f sp pc rs m chunk addr args src pc' a m',
      (fn_code f)!pc = Some(Istore chunk addr args src pc') ->
      eval_addressing ge sp addr rs##args = Some a ->
      Mem.storev chunk m a rs#src = Some m' ->
      step (State s f sp pc rs m)
        E0 (State s f sp pc' rs m')
  | exec_Icall:
      forall s f sp pc rs m sig ros args res pc' fd,
      (fn_code f)!pc = Some(Icall sig ros args res pc') ->
      find_function ros rs = Some fd ->
      funsig fd = sig ->
      step (State s f sp pc rs m)
        E0 (Callstate (Stackframe res f sp pc' rs :: s) fd rs##args m)
  | exec_Itailcall:
      forall s f stk pc rs m sig ros args fd m',
      (fn_code f)!pc = Some(Itailcall sig ros args) ->
      find_function ros rs = Some fd ->
      funsig fd = sig ->
      Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
      step (State s f (Vptr stk Int.zero) pc rs m)
        E0 (Callstate s fd rs##args m')
  | exec_Ibuiltin:
      forall s f sp pc rs m ef args res pc' t v m',
      (fn_code f)!pc = Some(Ibuiltin ef args res pc') ->
      external_call ef ge rs##args m t v m' ->
      step (State s f sp pc rs m)
         t (State s f sp pc' (rs#res <- v) m')
  | exec_Icond:
      forall s f sp pc rs m cond args ifso ifnot b pc',
      (fn_code f)!pc = Some(Icond cond args ifso ifnot) ->
      eval_condition cond rs##args m = Some b ->
      pc' = (if b then ifso else ifnot) ->
      step (State s f sp pc rs m)
        E0 (State s f sp pc' rs m)
  | exec_Ijumptable:
      forall s f sp pc rs m arg tbl n pc',
      (fn_code f)!pc = Some(Ijumptable arg tbl) ->
      rs#arg = Vint n ->
      list_nth_z tbl (Int.unsigned n) = Some pc' ->
      step (State s f sp pc rs m)
        E0 (State s f sp pc' rs m)
  | exec_Ireturn:
      forall s f stk pc rs m or m',
      (fn_code f)!pc = Some(Ireturn or) ->
      Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
      step (State s f (Vptr stk Int.zero) pc rs m)
        E0 (Returnstate s (regmap_optget or Vundef rs) m')
  | exec_function_internal:
      forall s f args m m' stk,
      Mem.alloc m 0 f.(fn_stacksize) = (m', stk) ->
      step (Callstate s (Internal f) args m)
        E0 (State s
                  f
                  (Vptr stk Int.zero)
                  f.(fn_entrypoint)
                  (init_regs args f.(fn_params))
                  m')
  | exec_function_external:
      forall s ef args res t m m',
      external_call ef ge args m t res m' ->
      step (Callstate s (External ef) args m)
         t (Returnstate s res m')
  | exec_return:
      forall res f sp pc rs s vres m,
      step (Returnstate (Stackframe res f sp pc rs :: s) vres m)
        E0 (State s f sp pc (rs#res <- vres) m).

Lemma exec_Iop':
  forall s f sp pc rs m op args res pc' rs' v,
  (fn_code f)!pc = Some(Iop op args res pc') ->
  eval_operation ge sp op rs##args m = Some v ->
  rs' = (rs#res <- v) ->
  step (State s f sp pc rs m)
    E0 (State s f sp pc' rs' m).

Proof.

intros. subst rs'. eapply exec_Iop; eauto.
Qed.

Lemma exec_Iload':
  forall s f sp pc rs m chunk addr args dst pc' rs' a v,
  (fn_code f)!pc = Some(Iload chunk addr args dst pc') ->
  eval_addressing ge sp addr rs##args = Some a ->
  Mem.loadv chunk m a = Some v ->
  rs' = (rs#dst <- v) ->
  step (State s f sp pc rs m)
    E0 (State s f sp pc' rs' m).

Proof.

intros. subst rs'. eapply exec_Iload; eauto.
Qed.

End RELSEM.

Execution of whole programs are described as sequences of transitions from an initial state to a final state. An initial state is a Callstate corresponding to the invocation of the ``main'' function of the program without arguments and with an empty call stack.

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

A final state is a Returnstate with an empty call stack.

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

The small-step semantics for a program.

Definition semantics (p: program) :=
Semantics step (initial_state p) final_state (Genv.globalenv p).

This semantics is receptive to changes in events.

Lemma semantics_receptive:
forall (p: program), receptive (semantics p).

Proof.

  intros. constructor; simpl; intros.
  assert (t1 = E0 -> exists s2, step (Genv.globalenv p) s t2 s2).
    intros. subst. inv H0. exists s1; auto.
  inversion H; subst; auto.
  exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]].
  exists (State s0 f sp pc' (rs#res <- vres2) m2). eapply exec_Ibuiltin; eauto.
  exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]].
  exists (Returnstate s0 vres2 m2). econstructor; eauto.
  red; intros; inv H; simpl; try omega.
  eapply external_call_trace_length; eauto.
  eapply external_call_trace_length; eauto.
Qed.

Operations on RTL abstract syntax

Computation of the possible successors of an instruction. This is used in particular for dataflow analyses.

Definition successors_instr (i: instruction) : list node :=
  match i with
  | Inop s => s :: nil
  | Iop op args res s => s :: nil
  | Iload chunk addr args dst s => s :: nil
  | Istore chunk addr args src s => s :: nil
  | Icall sig ros args res s => s :: nil
  | Itailcall sig ros args => nil
  | Ibuiltin ef args res s => s :: nil
  | Icond cond args ifso ifnot => ifso :: ifnot :: nil
  | Ijumptable arg tbl => tbl
  | Ireturn optarg => nil
  end.

Definition successors (f: function) : PTree.t (list node) :=
  PTree.map1 successors_instr f.(fn_code).

Transformation of a RTL function instruction by instruction. This applies a given transformation function to all instructions of a function and constructs a transformed function from that.

Section TRANSF.

Variable transf: node -> instruction -> instruction.

Definition transf_function (f: function) : function :=
  mkfunction
    f.(fn_sig)
    f.(fn_params)
    f.(fn_stacksize)
    (PTree.map transf f.(fn_code))
    f.(fn_entrypoint).

End TRANSF.

Module RTL

Abstract syntax

Operational semantics

Operations on RTL abstract syntax