Module Csharpminor

Abstract syntax and semantics for the Csharpminor language.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Switch.
Require Cminor.
Require Import Smallstep.

Abstract syntax

Csharpminor is a low-level imperative language structured in expressions, statements, functions and programs. Expressions include reading temporary variables, taking the address of a variable, constants, arithmetic operations, and dereferencing addresses.

Inductive constant : Type :=
  | Ointconst: int -> constant (* integer constant *)
  | Ofloatconst: float -> constant (* double-precision floating-point constant *)
  | Osingleconst: float32 -> constant (* single-precision floating-point constant *)
  | Olongconst: int64 -> constant. (* long integer constant *)

Definition unary_operation : Type := Cminor.unary_operation.
Definition binary_operation : Type := Cminor.binary_operation.

Inductive expr : Type :=
  | Evar : ident -> expr (* reading a temporary variable *)
  | Eaddrof : ident -> expr (* taking the address of a variable *)
  | Econst : constant -> expr (* constants *)
  | Eunop : unary_operation -> expr -> expr (* unary operation *)
  | Ebinop : binary_operation -> expr -> expr -> expr (* binary operation *)
  | Eload : memory_chunk -> expr -> expr. (* memory read *)

Statements include expression evaluation, temporary variable assignment, memory stores, function calls, an if/then/else conditional, infinite loops, blocks and early block exits, and early function returns. Sexit n terminates prematurely the execution of the n+1 enclosing Sblock statements.

Definition label := ident.

Inductive stmt : Type :=
  | Sskip: stmt
  | Sset : ident -> expr -> stmt
  | Sstore : memory_chunk -> expr -> expr -> stmt
  | Scall : option ident -> signature -> expr -> list expr -> stmt
  | Sbuiltin : option ident -> external_function -> list expr -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: expr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: bool -> expr -> lbl_stmt -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt

with lbl_stmt : Type :=
  | LSnil: lbl_stmt
  | LScons: option Z -> stmt -> lbl_stmt -> lbl_stmt.

Functions are composed of a return type, a list of parameter names, a list of local variables with their sizes, a list of temporary variables, and a statement representing the function body.

Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list (ident * Z);
  fn_temps: list ident;
  fn_body: stmt

Definition fundef := AST.fundef function.

Definition program : Type := AST.program fundef unit.

Definition funsig (fd: fundef) :=
  match fd with
  | Internal f => fn_sig f
  | External ef => ef_sig ef

Operational semantics

Three evaluation environments are involved:

Definition genv := Genv.t fundef unit.
Definition env := PTree.t (block * Z).
Definition temp_env := PTree.t val.

Definition empty_env : env := PTree.empty (block * Z).
Definition empty_temp_env : temp_env := PTree.empty val.

Initialization of temporary variables

Fixpoint create_undef_temps (temps: list ident) : temp_env :=
  match temps with
  | nil => PTree.empty val
  | id :: temps' => PTree.set id Vundef (create_undef_temps temps')

Initialization of temporaries that are parameters.

Fixpoint bind_parameters (formals: list ident) (args: list val)
                         (le: temp_env) : option temp_env :=
 match formals, args with
 | nil, nil => Some le
 | id :: xl, v :: vl => bind_parameters xl vl (PTree.set id v le)
 | _, _ => None


Inductive cont: Type :=
  | Kstop: cont (* stop program execution *)
  | Kseq: stmt -> cont -> cont (* execute stmt, then cont *)
  | Kblock: cont -> cont (* exit a block, then do cont *)
  | Kcall: option ident -> function -> env -> temp_env -> cont -> cont.


Inductive state: Type :=
  | State: (* Execution within a function *)
      forall (f: function) (* currently executing function *)
             (s: stmt) (* statement under consideration *)
             (k: cont) (* its continuation -- what to do next *)
             (e: env) (* current local environment *)
             (le: temp_env) (* current temporary environment *)
             (m: mem), (* current memory state *)
  | Callstate: (* Invocation of a function *)
      forall (f: fundef) (* function to invoke *)
             (args: list val) (* arguments provided by caller *)
             (k: cont) (* what to do next *)
             (m: mem), (* memory state *)
  | Returnstate: (* Return from a function *)
      forall (v: val) (* Return value *)
             (k: cont) (* what to do next *)
             (m: mem), (* memory state *)

Pop continuation until a call or stop

Fixpoint call_cont (k: cont) : cont :=
  match k with
  | Kseq s k => call_cont k
  | Kblock k => call_cont k
  | _ => k

Definition is_call_cont (k: cont) : Prop :=
  match k with
  | Kstop => True
  | Kcall _ _ _ _ _ => True
  | _ => False

Resolve switch statements.

Fixpoint select_switch_default (sl: lbl_stmt): lbl_stmt :=
  match sl with
  | LSnil => sl
  | LScons None s sl' => sl
  | LScons (Some i) s sl' => select_switch_default sl'

Fixpoint select_switch_case (n: Z) (sl: lbl_stmt): option lbl_stmt :=
  match sl with
  | LSnil => None
  | LScons None s sl' => select_switch_case n sl'
  | LScons (Some c) s sl' => if zeq c n then Some sl else select_switch_case n sl'

Definition select_switch (n: Z) (sl: lbl_stmt): lbl_stmt :=
  match select_switch_case n sl with
  | Some sl' => sl'
  | None => select_switch_default sl

Fixpoint seq_of_lbl_stmt (sl: lbl_stmt) : stmt :=
  match sl with
  | LSnil => Sskip
  | LScons c s sl' => Sseq s (seq_of_lbl_stmt sl')

Find the statement and manufacture the continuation corresponding to a label

Fixpoint find_label (lbl: label) (s: stmt) (k: cont)
                    {struct s}: option (stmt * cont) :=
  match s with
  | Sseq s1 s2 =>
      match find_label lbl s1 (Kseq s2 k) with
      | Some sk => Some sk
      | None => find_label lbl s2 k
  | Sifthenelse a s1 s2 =>
      match find_label lbl s1 k with
      | Some sk => Some sk
      | None => find_label lbl s2 k
  | Sloop s1 =>
      find_label lbl s1 (Kseq (Sloop s1) k)
  | Sblock s1 =>
      find_label lbl s1 (Kblock k)
  | Sswitch long a sl =>
      find_label_ls lbl sl k
  | Slabel lbl' s' =>
      if ident_eq lbl lbl' then Some(s', k) else find_label lbl s' k
  | _ => None

with find_label_ls (lbl: label) (sl: lbl_stmt) (k: cont)
                   {struct sl}: option (stmt * cont) :=
  match sl with
  | LSnil => None
  | LScons _ s sl' =>
      match find_label lbl s (Kseq (seq_of_lbl_stmt sl') k) with
      | Some sk => Some sk
      | None => find_label_ls lbl sl' k

Evaluation of operator applications.

Definition eval_constant (cst: constant) : option val :=
  match cst with
  | Ointconst n => Some (Vint n)
  | Ofloatconst n => Some (Vfloat n)
  | Osingleconst n => Some (Vsingle n)
  | Olongconst n => Some (Vlong n)

Definition eval_unop := Cminor.eval_unop.

Definition eval_binop := Cminor.eval_binop.

Allocation of local variables at function entry. Each variable is bound to the reference to a fresh block of the appropriate size.

Inductive alloc_variables: env -> mem ->
                           list (ident * Z) ->
                           env -> mem -> Prop :=
  | alloc_variables_nil:
      forall e m,
      alloc_variables e m nil e m
  | alloc_variables_cons:
      forall e m id sz vars m1 b1 m2 e2,
      Mem.alloc m 0 sz = (m1, b1) ->
      alloc_variables (PTree.set id (b1, sz) e) m1 vars e2 m2 ->
      alloc_variables e m ((id, sz) :: vars) e2 m2.

List of blocks mentioned in an environment, with low and high bounds

Definition block_of_binding (id_b_sz: ident * (block * Z)) :=
  match id_b_sz with (id, (b, sz)) => (b, 0, sz) end.

Definition blocks_of_env (e: env) : list (block * Z * Z) := block_of_binding (PTree.elements e).

Section RELSEM.

Variable ge: genv.

Inductive eval_var_addr: env -> ident -> block -> Prop :=
  | eval_var_addr_local:
      forall e id b sz,
      PTree.get id e = Some (b, sz) ->
      eval_var_addr e id b
  | eval_var_addr_global:
      forall e id b,
      PTree.get id e = None ->
      Genv.find_symbol ge id = Some b ->
      eval_var_addr e id b.

Evaluation of an expression: eval_expr prg e m a v states that expression a, in initial memory state m and local environment e, evaluates to value v.

Section EVAL_EXPR.

Variable e: env.
Variable le: temp_env.
Variable m: mem.

Inductive eval_expr: expr -> val -> Prop :=
  | eval_Evar: forall id v,
      le!id = Some v ->
      eval_expr (Evar id) v
  | eval_Eaddrof: forall id b,
      eval_var_addr e id b ->
      eval_expr (Eaddrof id) (Vptr b
  | eval_Econst: forall cst v,
      eval_constant cst = Some v ->
      eval_expr (Econst cst) v
  | eval_Eunop: forall op a1 v1 v,
      eval_expr a1 v1 ->
      eval_unop op v1 = Some v ->
      eval_expr (Eunop op a1) v
  | eval_Ebinop: forall op a1 a2 v1 v2 v,
      eval_expr a1 v1 ->
      eval_expr a2 v2 ->
      eval_binop op v1 v2 m = Some v ->
      eval_expr (Ebinop op a1 a2) v
  | eval_Eload: forall chunk a v1 v,
      eval_expr a v1 ->
      Mem.loadv chunk m v1 = Some v ->
      eval_expr (Eload chunk a) v.

Evaluation of a list of expressions: eval_exprlist prg e m al vl states that the list al of expressions evaluate to the list vl of values. The other parameters are as in eval_expr.

Inductive eval_exprlist: list expr -> list val -> Prop :=
  | eval_Enil:
      eval_exprlist nil nil
  | eval_Econs: forall a1 al v1 vl,
      eval_expr a1 v1 -> eval_exprlist al vl ->
      eval_exprlist (a1 :: al) (v1 :: vl).


One step of execution

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k e le m,
      step (State f Sskip (Kseq s k) e le m)
        E0 (State f s k e le m)
  | step_skip_block: forall f k e le m,
      step (State f Sskip (Kblock k) e le m)
        E0 (State f Sskip k e le m)
  | step_skip_call: forall f k e le m m',
      is_call_cont k ->
      Mem.free_list m (blocks_of_env e) = Some m' ->
      step (State f Sskip k e le m)
        E0 (Returnstate Vundef k m')

  | step_set: forall f id a k e le m v,
      eval_expr e le m a v ->
      step (State f (Sset id a) k e le m)
        E0 (State f Sskip k e (PTree.set id v le) m)

  | step_store: forall f chunk addr a k e le m vaddr v m',
      eval_expr e le m addr vaddr ->
      eval_expr e le m a v ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr a) k e le m)
        E0 (State f Sskip k e le m')

  | step_call: forall f optid sig a bl k e le m vf vargs fd,
      eval_expr e le m a vf ->
      eval_exprlist e le m bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k e le m)
        E0 (Callstate fd vargs (Kcall optid f e le k) m)

  | step_builtin: forall f optid ef bl k e le m vargs t vres m',
      eval_exprlist e le m bl vargs ->
      external_call ef ge vargs m t vres m' ->
      step (State f (Sbuiltin optid ef bl) k e le m)
         t (State f Sskip k e (Cminor.set_optvar optid vres le) m')

  | step_seq: forall f s1 s2 k e le m,
      step (State f (Sseq s1 s2) k e le m)
        E0 (State f s1 (Kseq s2 k) e le m)

  | step_ifthenelse: forall f a s1 s2 k e le m v b,
      eval_expr e le m a v ->
      Val.bool_of_val v b ->
      step (State f (Sifthenelse a s1 s2) k e le m)
        E0 (State f (if b then s1 else s2) k e le m)

  | step_loop: forall f s k e le m,
      step (State f (Sloop s) k e le m)
        E0 (State f s (Kseq (Sloop s) k) e le m)

  | step_block: forall f s k e le m,
      step (State f (Sblock s) k e le m)
        E0 (State f s (Kblock k) e le m)

  | step_exit_seq: forall f n s k e le m,
      step (State f (Sexit n) (Kseq s k) e le m)
        E0 (State f (Sexit n) k e le m)
  | step_exit_block_0: forall f k e le m,
      step (State f (Sexit O) (Kblock k) e le m)
        E0 (State f Sskip k e le m)
  | step_exit_block_S: forall f n k e le m,
      step (State f (Sexit (S n)) (Kblock k) e le m)
        E0 (State f (Sexit n) k e le m)

  | step_switch: forall f islong a cases k e le m v n,
      eval_expr e le m a v ->
      switch_argument islong v n ->
      step (State f (Sswitch islong a cases) k e le m)
        E0 (State f (seq_of_lbl_stmt (select_switch n cases)) k e le m)

  | step_return_0: forall f k e le m m',
      Mem.free_list m (blocks_of_env e) = Some m' ->
      step (State f (Sreturn None) k e le m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k e le m v m',
      eval_expr e le m a v ->
      Mem.free_list m (blocks_of_env e) = Some m' ->
      step (State f (Sreturn (Some a)) k e le m)
        E0 (Returnstate v (call_cont k) m')
  | step_label: forall f lbl s k e le m,
      step (State f (Slabel lbl s) k e le m)
        E0 (State f s k e le m)

  | step_goto: forall f lbl k e le m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k e le m)
        E0 (State f s' k' e le m)

  | step_internal_function: forall f vargs k m m1 e le,
      list_norepet (map fst f.(fn_vars)) ->
      list_norepet f.(fn_params) ->
      list_disjoint f.(fn_params) f.(fn_temps) ->
      alloc_variables empty_env m (fn_vars f) e m1 ->
      bind_parameters f.(fn_params) vargs (create_undef_temps f.(fn_temps)) = Some le ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k e le m1)

  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f e le k m,
      step (Returnstate v (Kcall optid f e le k) m)
        E0 (State f Sskip k e (Cminor.set_optvar optid v le) m).


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 continuation.

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 f nil Kstop m0).

A final state is a Returnstate with an empty continuation.

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

Wrapping up these definitions in a small-step semantics.

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