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

Abstract syntax

Definition label := positive.

Inductive instruction: Type :=
  | Lgetstack: slot -> Z -> typ -> mreg -> instruction
  | Lsetstack: mreg -> slot -> Z -> typ -> instruction
  | Lop: operation -> list mreg -> mreg -> instruction
  | Lload: annotation -> memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Lstore: annotation -> memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Lcall: signature -> mreg + ident -> instruction
  | Ltailcall: signature -> mreg + ident -> instruction
  | Lbuiltin: external_function -> list (builtin_arg loc) -> builtin_res mreg -> 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 => fn_sig f
  | External ef => ef_sig ef
  end.

Definition genv := Genv.t fundef unit.
Definition locset := Locmap.t.

Operational semantics

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.


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.


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.

Definition parent_locset (stack: list stackframe) : locset :=
  match stack with
  | nil => Locmap.init Vundef
  | Stackframe f sp ls c :: stack' => ls
  end.

Inductive step: state -> trace -> state -> Prop :=
  | exec_Lgetstack:
      forall s f sp sl ofs ty dst b rs m rs',
      rs' = Locmap.set (R dst) (rs (S sl ofs ty)) (undef_regs (destroyed_by_getstack sl) rs) ->
      step (State s f sp (Lgetstack sl ofs ty dst :: b) rs m)
        E0 (State s f sp b rs' m)
  | exec_Lsetstack:
      forall s f sp src sl ofs ty b rs m rs',
      rs' = Locmap.set (S sl ofs ty) (rs (R src)) (undef_regs (destroyed_by_setstack ty) rs) ->
      step (State s f sp (Lsetstack src sl ofs ty :: b) rs m)
        E0 (State s f sp b rs' m)
  | exec_Lop:
      forall s f sp op args res b 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 (State s f sp (Lop op args res :: b) rs m)
        E0 (State s f sp b rs' m)
  | exec_Lload:
      forall s f sp alpha chunk addr args dst b 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 (State s f sp (Lload alpha chunk addr args dst :: b) rs m)
        E0 (State s f sp b rs' m)
  | exec_Lstore:
      forall s f sp alpha chunk addr args src b rs m m' a rs',
      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 (State s f sp (Lstore alpha 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 rs' f' m',
      rs' = return_regs (parent_locset s) rs ->
      find_function ros rs' = Some f' ->
      sig = funsig f' ->
      Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
      step (State s f (Vptr stk Int.zero) (Ltailcall sig ros :: b) rs m)
        E0 (Callstate s f' rs' m')
  | exec_Lbuiltin:
      forall s f sp rs m ef args res b 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 (State s f sp (Lbuiltin ef args res :: b) rs m)
         t (State s f sp b rs' m')
  | 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 rs' b',
      eval_condition cond (reglist rs args) m = Some true ->
      rs' = undef_regs (destroyed_by_cond cond) rs ->
      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 rs',
      eval_condition cond (reglist rs args) m = Some false ->
      rs' = undef_regs (destroyed_by_cond cond) rs ->
      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',
      rs (R arg) = Vint n ->
      list_nth_z tbl (Int.unsigned n) = Some lbl ->
      find_label lbl f.(fn_code) = Some b' ->
      rs' = undef_regs (destroyed_by_jumptable) rs ->
      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 m',
      Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
      step (State s f (Vptr stk Int.zero) (Lreturn :: b) rs m)
        E0 (Returnstate s (return_regs (parent_locset s) rs) m')
  | exec_function_internal:
      forall s f rs m rs' m' stk,
      Mem.alloc m 0 f.(fn_stacksize) = (m', stk) ->
      rs' = undef_regs destroyed_at_function_entry (call_regs rs) ->
      step (Callstate s (Internal f) rs m)
        E0 (State s f (Vptr stk Int.zero) f.(fn_code) rs' m')
  | exec_function_external:
      forall s ef args res rs1 rs2 m t m',
      args = map rs1 (loc_arguments (ef_sig ef)) ->
      external_call' ef ge args m t res m' ->
      rs2 = Locmap.setlist (map R (loc_result (ef_sig ef))) 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).

Definition step_safe (s: state) (t: trace) (s': state) :=
  step s t s' /\
  match s with
  | State s _ sp (Lload alpha chunk addr args dst :: b) rs m =>
    forall a, eval_addressing ge sp addr (reglist rs args) = Some a ->
         annot_sem (Genv.find_symbol ge) (sp::(List.map (fun s => match s with Stackframe f sp rs c => sp end) s)) (snd alpha) a
  | State s _ sp (Lstore alpha chunk addr args src :: b) rs m =>
    forall a, eval_addressing ge sp addr (reglist rs args) = Some a ->
         annot_sem (Genv.find_symbol ge) (sp::(List.map (fun s => match s with Stackframe f sp rs c => sp end) s)) (snd alpha) a
  | _ => True
  end.

End RELSEM.

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 r retcode,
      loc_result signature_main = r :: nil ->
      rs (R r) = Vint retcode ->
      final_state (Returnstate nil rs m) retcode.

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

Definition semantics_safe (p: program) :=
  Semantics step_safe (initial_state p) final_state (Genv.globalenv p).