(** 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: Set :=
  | 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
  | Lalloc: instruction
  | Llabel: label -> instruction
  | Lgoto: label -> instruction
  | Lcond: condition -> list mreg -> label -> instruction
  | Lreturn: instruction.

Definition code: Set := list instruction.

Record function: Set := 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.
Proof.
  intros.  destruct instr; simpl; try discriminate.
  case (peq lbl l); intro; congruence.
Qed.

(** [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.

(** The components of a Linear execution state are:

- [State cs f sp c rs m]: [f] is the function currently executing.
  [sp] is the stack pointer.  [c] is the sequence of instructions
  that remain to be executed.
  [rs] maps locations to their current values. [m] is the current
  memory state.

- [Callstate cs f rs m]:
  [f] is the function definition that we are calling.
  [rs] is the values of locations just before the call.
  [m] is the current memory state.

- [Returnstate cs rs m]:
  [rs] is the values of locations just before the return.
  [m] is the current memory state.

[cs] is a list of stack frames [Stackframe res f rs pc].
[f] is the calling function, [sp] its stack pointer.
[rs] the values of locations just before the call.
[c] is the sequence of instructions following the call in the code of [f].
*)

Inductive stackframe: Set :=
  | Stackframe:
      forall (f: function) (sp: val) (rs: locset) (c: code),
      stackframe.

Inductive state: Set :=
  | State:
      forall (stack: list stackframe) (f: function) (sp: val)
             (c: code) (rs: locset) (m: mem),
      state
  | Callstate:
      forall (stack: list stackframe) (f: fundef) (rs: locset) (m: mem),
      state
  | Returnstate:
      forall (stack: list stackframe) (rs: locset) (m: mem),
      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.

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) m = 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_Lalloc:
      forall s f sp b rs m sz m' blk,
      rs (R Conventions.loc_alloc_argument) = Vint sz ->
      Mem.alloc m 0 (Int.signed sz) = (m', blk) ->
      step (State s f sp (Lalloc :: b) rs m)
        E0 (State s f sp b
                    (Locmap.set (R Conventions.loc_alloc_result)
                                (Vptr blk Int.zero) 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 b',
      eval_condition cond (reglist rs args) m = 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) m = Some false ->
      step (State s f sp (Lcond cond args lbl :: 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 R3) = 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.