Module RTL


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 Maps.
Require Import AST Integers Values Events Memory Globalenvs Smallstep.
Require Import Op 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 (builtin_arg reg) -> builtin_res 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.

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: 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 Ptrofs.zero) pc rs m)
        E0 (Callstate s fd rs##args m')
  | exec_Ibuiltin:
      forall s f sp pc rs m ef args res pc' vargs t vres m',
      (fn_code f)!pc = Some(Ibuiltin ef args res pc') ->
      eval_builtin_args ge (fun r => rs#r) sp m args vargs ->
      external_call ef ge vargs m t vres m' ->
      step (State s f sp pc rs m)
         t (State s f sp pc' (regmap_setres res vres rs) 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 Ptrofs.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 Ptrofs.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 = signature_main ->
      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.
(* receptiveness *)
  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' (regmap_setres res vres2 rs) m2). eapply exec_Ibuiltin; eauto.
  exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]].
  exists (Returnstate s0 vres2 m2). econstructor; eauto.
(* trace length *)
  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


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.

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

The registers used by an instruction

Definition instr_uses (i: instruction) : list reg :=
  match i with
  | Inop s => nil
  | Iop op args res s => args
  | Iload chunk addr args dst s => args
  | Istore chunk addr args src s => src :: args
  | Icall sig (inl r) args res s => r :: args
  | Icall sig (inr id) args res s => args
  | Itailcall sig (inl r) args => r :: args
  | Itailcall sig (inr id) args => args
  | Ibuiltin ef args res s => params_of_builtin_args args
  | Icond cond args ifso ifnot => args
  | Ijumptable arg tbl => arg :: nil
  | Ireturn None => nil
  | Ireturn (Some arg) => arg :: nil
  end.

The register defined by an instruction, if any

Definition instr_defs (i: instruction) : option reg :=
  match i with
  | Inop s => None
  | Iop op args res s => Some res
  | Iload chunk addr args dst s => Some dst
  | Istore chunk addr args src s => None
  | Icall sig ros args res s => Some res
  | Itailcall sig ros args => None
  | Ibuiltin ef args res s =>
      match res with BR r => Some r | _ => None end
  | Icond cond args ifso ifnot => None
  | Ijumptable arg tbl => None
  | Ireturn optarg => None
  end.

Maximum PC (node number) in the CFG of a function. All nodes of the CFG of f are between 1 and max_pc_function f (inclusive).

Definition max_pc_function (f: function) :=
  PTree.fold (fun m pc i => Pos.max m pc) f.(fn_code) 1%positive.

Lemma max_pc_function_sound:
  forall f pc i, f.(fn_code)!pc = Some i -> Ple pc (max_pc_function f).
Proof.
  intros until i. unfold max_pc_function.
  apply PTree_Properties.fold_rec with (P := fun c m => c!pc = Some i -> Ple pc m).
  (* extensionality *)
  intros. apply H0. rewrite H; auto.
  (* base case *)
  rewrite PTree.gempty. congruence.
  (* inductive case *)
  intros. rewrite PTree.gsspec in H2. destruct (peq pc k).
  inv H2. xomega.
  apply Ple_trans with a. auto. xomega.
Qed.

Maximum pseudo-register mentioned in a function. All results or arguments of an instruction of f, as well as all parameters of f, are between 1 and max_reg_function (inclusive).

Definition max_reg_instr (m: positive) (pc: node) (i: instruction) :=
  match i with
  | Inop s => m
  | Iop op args res s => fold_left Pos.max args (Pos.max res m)
  | Iload chunk addr args dst s => fold_left Pos.max args (Pos.max dst m)
  | Istore chunk addr args src s => fold_left Pos.max args (Pos.max src m)
  | Icall sig (inl r) args res s => fold_left Pos.max args (Pos.max r (Pos.max res m))
  | Icall sig (inr id) args res s => fold_left Pos.max args (Pos.max res m)
  | Itailcall sig (inl r) args => fold_left Pos.max args (Pos.max r m)
  | Itailcall sig (inr id) args => fold_left Pos.max args m
  | Ibuiltin ef args res s =>
      fold_left Pos.max (params_of_builtin_args args)
        (fold_left Pos.max (params_of_builtin_res res) m)
  | Icond cond args ifso ifnot => fold_left Pos.max args m
  | Ijumptable arg tbl => Pos.max arg m
  | Ireturn None => m
  | Ireturn (Some arg) => Pos.max arg m
  end.

Definition max_reg_function (f: function) :=
  Pos.max
    (PTree.fold max_reg_instr f.(fn_code) 1%positive)
    (fold_left Pos.max f.(fn_params) 1%positive).

Remark max_reg_instr_ge:
  forall m pc i, Ple m (max_reg_instr m pc i).
Proof.
  intros.
  assert (X: forall l n, Ple m n -> Ple m (fold_left Pos.max l n)).
  { induction l; simpl; intros.
    auto.
    apply IHl. xomega. }
  destruct i; simpl; try (destruct s0); repeat (apply X); try xomega.
  destruct o; xomega.
Qed.

Remark max_reg_instr_def:
  forall m pc i r, instr_defs i = Some r -> Ple r (max_reg_instr m pc i).
Proof.
  intros.
  assert (X: forall l n, Ple r n -> Ple r (fold_left Pos.max l n)).
  { induction l; simpl; intros. xomega. apply IHl. xomega. }
  destruct i; simpl in *; inv H.
- apply X. xomega.
- apply X. xomega.
- destruct s0; apply X; xomega.
- destruct b; inv H1. apply X. simpl. xomega.
Qed.

Remark max_reg_instr_uses:
  forall m pc i r, In r (instr_uses i) -> Ple r (max_reg_instr m pc i).
Proof.
  intros.
  assert (X: forall l n, In r l \/ Ple r n -> Ple r (fold_left Pos.max l n)).
  { induction l; simpl; intros.
    tauto.
    apply IHl. destruct H0 as [[A|A]|A]. right; subst; xomega. auto. right; xomega. }
  destruct i; simpl in *; try (destruct s0); try (apply X; auto).
- contradiction.
- destruct H. right; subst; xomega. auto.
- destruct H. right; subst; xomega. auto.
- destruct H. right; subst; xomega. auto.
- intuition. subst; xomega.
- destruct o; simpl in H; intuition. subst; xomega.
Qed.

Lemma max_reg_function_def:
  forall f pc i r,
  f.(fn_code)!pc = Some i -> instr_defs i = Some r -> Ple r (max_reg_function f).
Proof.
  intros.
  assert (Ple r (PTree.fold max_reg_instr f.(fn_code) 1%positive)).
  { revert H.
     apply PTree_Properties.fold_rec with
       (P := fun c m => c!pc = Some i -> Ple r m).
   - intros. rewrite H in H1; auto.
   - rewrite PTree.gempty; congruence.
   - intros. rewrite PTree.gsspec in H3. destruct (peq pc k).
     + inv H3. eapply max_reg_instr_def; eauto.
     + apply Ple_trans with a. auto. apply max_reg_instr_ge.
  }
  unfold max_reg_function. xomega.
Qed.

Lemma max_reg_function_use:
  forall f pc i r,
  f.(fn_code)!pc = Some i -> In r (instr_uses i) -> Ple r (max_reg_function f).
Proof.
  intros.
  assert (Ple r (PTree.fold max_reg_instr f.(fn_code) 1%positive)).
  { revert H.
     apply PTree_Properties.fold_rec with
       (P := fun c m => c!pc = Some i -> Ple r m).
   - intros. rewrite H in H1; auto.
   - rewrite PTree.gempty; congruence.
   - intros. rewrite PTree.gsspec in H3. destruct (peq pc k).
     + inv H3. eapply max_reg_instr_uses; eauto.
     + apply Ple_trans with a. auto. apply max_reg_instr_ge.
  }
  unfold max_reg_function. xomega.
Qed.

Lemma max_reg_function_params:
  forall f r, In r f.(fn_params) -> Ple r (max_reg_function f).
Proof.
  intros.
  assert (X: forall l n, In r l \/ Ple r n -> Ple r (fold_left Pos.max l n)).
  { induction l; simpl; intros.
    tauto.
    apply IHl. destruct H0 as [[A|A]|A]. right; subst; xomega. auto. right; xomega. }
  assert (Y: Ple r (fold_left Pos.max f.(fn_params) 1%positive)).
  { apply X; auto. }
  unfold max_reg_function. xomega.
Qed.