Module LTL


The LTL intermediate language: abstract syntax and semantics. LTL (``Location Transfer Language'') is the target language for register allocation and the source language for linearization.

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

Abstract syntax


LTL is close to RTL, but uses machine registers and stack slots instead of pseudo-registers. Also, the nodes of the control-flow graph are basic blocks instead of single instructions.

Definition node := positive.

Inductive instruction: Type :=
  | Lop (op: operation) (args: list mreg) (res: mreg)
  | Lload (trap : trapping_mode) (chunk: memory_chunk) (addr: addressing) (args: list mreg) (dst: mreg)
  | Lgetstack (sl: slot) (ofs: Z) (ty: typ) (dst: mreg)
  | Lsetstack (src: mreg) (sl: slot) (ofs: Z) (ty: typ)
  | Lstore (chunk: memory_chunk) (addr: addressing) (args: list mreg) (src: mreg)
  | Lcall (sg: signature) (ros: mreg + ident)
  | Ltailcall (sg: signature) (ros: mreg + ident)
  | Lbuiltin (ef: external_function) (args: list (builtin_arg loc)) (res: builtin_res mreg)
  | Lbranch (s: node)
  | Lcond (cond: condition) (args: list mreg) (s1 s2: node) (info: option bool)
  | Ljumptable (arg: mreg) (tbl: list node)
  | Lreturn.

Definition bblock := list instruction.

Definition code: Type := PTree.t bblock.

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

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 locset := Locmap.t.

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.

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.

Definition return_regs (caller callee: locset) : locset :=
  fun (l: loc) =>
    match l with
    | R r => if is_callee_save r then caller (R r) else callee (R r)
    | S Outgoing ofs ty => Vundef
    | S sl ofs ty => caller (S sl ofs ty)
    end.

undef_caller_save_regs ls models the effect of calling an external function: caller-save registers and outgoing locations can change unpredictably, hence we set them to Vundef.

Definition undef_caller_save_regs (ls: locset) : locset :=
  fun (l: loc) =>
    match l with
    | R r => if is_callee_save r then ls (R r) else Vundef
    | S Outgoing ofs ty => Vundef
    | S sl ofs ty => ls (S sl ofs ty)
    end.

LTL execution states.

Inductive stackframe : Type :=
  | Stackframe:
      forall (f: function) (* calling function *)
             (sp: val) (* stack pointer in calling function *)
             (ls: locset) (* location state in calling function *)
             (bb: bblock), (* continuation in calling function *)
      stackframe.

Inductive state : Type :=
  | State:
      forall (stack: list stackframe) (* call stack *)
             (f: function) (* function currently executing *)
             (sp: val) (* stack pointer *)
             (pc: node) (* current program point *)
             (ls: locset) (* location state *)
             (m: mem), (* memory state *)
      state
  | Block:
      forall (stack: list stackframe) (* call stack *)
             (f: function) (* function currently executing *)
             (sp: val) (* stack pointer *)
             (bb: bblock) (* current basic block *)
             (ls: locset) (* location state *)
             (m: mem), (* memory state *)
      state
  | Callstate:
      forall (stack: list stackframe) (* call stack *)
             (f: fundef) (* function to call *)
             (ls: locset) (* location state of caller *)
             (m: mem), (* memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe) (* call stack *)
             (ls: locset) (* location state of callee *)
             (m: mem), (* memory state *)
      state.


Section RELSEM.

Variable ge: genv.

Definition reglist (rs: locset) (rl: list mreg) : list val :=
  List.map (fun r => rs (R r)) rl.

Fixpoint undef_regs (rl: list mreg) (rs: locset) : locset :=
  match rl with
  | nil => rs
  | r1 :: rl => Locmap.set (R r1) Vundef (undef_regs rl rs)
  end.

Definition destroyed_by_getstack (s: slot): list mreg :=
  match s with
  | Incoming => temp_for_parent_frame :: nil
  | _ => nil
  end.

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.

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 bb :: stack' => ls
  end.

Inductive step: state -> trace -> state -> Prop :=
  | exec_start_block: forall s f sp pc rs m bb,
      (fn_code f)!pc = Some bb ->
      step (State s f sp pc rs m)
        E0 (Block s f sp bb rs m)
  | exec_Lop: forall s f sp op args res bb rs m v rs',
      eval_operation ge sp op (reglist rs args) m = Some v ->
      rs' = Locmap.set (R res) v (undef_regs (destroyed_by_op op) rs) ->
      step (Block s f sp (Lop op args res :: bb) rs m)
        E0 (Block s f sp bb rs' m)
  | exec_Lload: forall s f sp trap chunk addr args dst bb rs m a v rs',
      eval_addressing ge sp addr (reglist rs args) = Some a ->
      Mem.loadv chunk m a = Some v ->
      rs' = Locmap.set (R dst) v (undef_regs (destroyed_by_load chunk addr) rs) ->
      step (Block s f sp (Lload trap chunk addr args dst :: bb) rs m)
        E0 (Block s f sp bb rs' m)
  | exec_Lload_notrap1: forall s f sp chunk addr args dst bb rs m rs',
      eval_addressing ge sp addr (reglist rs args) = None ->
      rs' = Locmap.set (R dst) (default_notrap_load_value chunk)
                       (undef_regs (destroyed_by_load chunk addr) rs) ->
      step (Block s f sp (Lload NOTRAP chunk addr args dst :: bb) rs m)
        E0 (Block s f sp bb rs' m)
  | exec_Lload_notrap2: forall s f sp chunk addr args dst bb rs m a rs',
      eval_addressing ge sp addr (reglist rs args) = Some a ->
      Mem.loadv chunk m a = None ->
      rs' = Locmap.set (R dst) (default_notrap_load_value chunk)
                       (undef_regs (destroyed_by_load chunk addr) rs) ->
      step (Block s f sp (Lload NOTRAP chunk addr args dst :: bb) rs m)
        E0 (Block s f sp bb rs' m)
  | exec_Lgetstack: forall s f sp sl ofs ty dst bb rs m rs',
      rs' = Locmap.set (R dst) (rs (S sl ofs ty)) (undef_regs (destroyed_by_getstack sl) rs) ->
      step (Block s f sp (Lgetstack sl ofs ty dst :: bb) rs m)
        E0 (Block s f sp bb rs' m)
  | exec_Lsetstack: forall s f sp src sl ofs ty bb rs m rs',
      rs' = Locmap.set (S sl ofs ty) (rs (R src)) (undef_regs (destroyed_by_setstack ty) rs) ->
      step (Block s f sp (Lsetstack src sl ofs ty :: bb) rs m)
        E0 (Block s f sp bb rs' m)
  | exec_Lstore: forall s f sp chunk addr args src bb rs m a rs' m',
      eval_addressing ge sp addr (reglist rs args) = Some a ->
      Mem.storev chunk m a (rs (R src)) = Some m' ->
      rs' = undef_regs (destroyed_by_store chunk addr) rs ->
      step (Block s f sp (Lstore chunk addr args src :: bb) rs m)
        E0 (Block s f sp bb rs' m')
  | exec_Lcall: forall s f sp sig ros bb rs m fd,
      find_function ros rs = Some fd ->
      funsig fd = sig ->
      step (Block s f sp (Lcall sig ros :: bb) rs m)
        E0 (Callstate (Stackframe f sp rs bb :: s) fd rs m)
  | exec_Ltailcall: forall s f sp sig ros bb rs m fd rs' m',
      rs' = return_regs (parent_locset s) rs ->
      find_function ros rs' = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stacksize) = Some m' ->
      step (Block s f (Vptr sp Ptrofs.zero) (Ltailcall sig ros :: bb) rs m)
        E0 (Callstate s fd rs' m')
  | exec_Lbuiltin: forall s f sp ef args res bb rs m vargs t vres rs' m',
      eval_builtin_args ge rs sp m args vargs ->
      external_call ef ge vargs m t vres m' ->
      rs' = Locmap.setres res vres (undef_regs (destroyed_by_builtin ef) rs) ->
      step (Block s f sp (Lbuiltin ef args res :: bb) rs m)
         t (Block s f sp bb rs' m')
  | exec_Lbranch: forall s f sp pc bb rs m,
      step (Block s f sp (Lbranch pc :: bb) rs m)
        E0 (State s f sp pc rs m)
  | exec_Lcond: forall s f sp cond args pc1 pc2 bb rs b pc rs' m i,
      eval_condition cond (reglist rs args) m = Some b ->
      pc = (if b then pc1 else pc2) ->
      rs' = undef_regs (destroyed_by_cond cond) rs ->
      step (Block s f sp (Lcond cond args pc1 pc2 i :: bb) rs m)
        E0 (State s f sp pc rs' m)
  | exec_Ljumptable: forall s f sp arg tbl bb rs m n pc rs',
      rs (R arg) = Vint n ->
      list_nth_z tbl (Int.unsigned n) = Some pc ->
      rs' = undef_regs (destroyed_by_jumptable) rs ->
      step (Block s f sp (Ljumptable arg tbl :: bb) rs m)
        E0 (State s f sp pc rs' m)
  | exec_Lreturn: forall s f sp bb rs m m',
      Mem.free m sp 0 f.(fn_stacksize) = Some m' ->
      step (Block s f (Vptr sp Ptrofs.zero) (Lreturn :: bb) rs m)
        E0 (Returnstate s (return_regs (parent_locset s) rs) m')
  | exec_function_internal: forall s f rs m m' sp rs',
      Mem.alloc m 0 f.(fn_stacksize) = (m', sp) ->
      rs' = undef_regs destroyed_at_function_entry (call_regs rs) ->
      step (Callstate s (Internal f) rs m)
        E0 (State s f (Vptr sp Ptrofs.zero) f.(fn_entrypoint) rs' m')
  | exec_function_external: forall s ef t args res rs m rs' m',
      args = map (fun p => Locmap.getpair p rs) (loc_arguments (ef_sig ef)) ->
      external_call ef ge args m t res m' ->
      rs' = Locmap.setpair (loc_result (ef_sig ef)) res (undef_caller_save_regs rs) ->
      step (Callstate s (External ef) rs m)
         t (Returnstate s rs' m')
  | exec_return: forall f sp rs1 bb s rs m,
      step (Returnstate (Stackframe f sp rs1 bb :: s) rs m)
        E0 (Block s f sp bb rs m).

End RELSEM.

Execution of a whole program boils down to invoking its main function. The result of the program is the return value of the main function, to be found in the machine register dictated by the calling conventions.

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 (Locmap.init Vundef) m0).

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

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

Operations over LTL


Computation of the possible successors of a block. This is used in particular for dataflow analyses.

Fixpoint successors_block (b: bblock) : list node :=
  match b with
  | nil => nil (* should never happen *)
  | Ltailcall _ _ :: _ => nil
  | Lbranch s :: _ => s :: nil
  | Lcond _ _ s1 s2 _ :: _ => s1 :: s2 :: nil
  | Ljumptable _ tbl :: _ => tbl
  | Lreturn :: _ => nil
  | instr :: b' => successors_block b'
  end.