Dynamic semantics for the Compcert C language

Require Import Coqlib.
Require Import Errors.
Require Import Maps.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import AST.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Ctypes.
Require Import Cop.
Require Import Csyntax.
Require Import Smallstep.

Operational semantics

The semantics uses two environments. The global environment maps names of functions and global variables to memory block references, and function pointers to their definitions. (See module Globalenvs.) It also contains a composite environment, used by type-dependent operations.

Record genv := { genv_genv :> Genv.t fundef type; genv_cenv :> composite_env }.

Definition globalenv (p: program) :=
{| genv_genv := Genv.globalenv p; genv_cenv := p.(prog_comp_env) |}.

The local environment maps local variables to block references and types. The current value of the variable is stored in the associated memory block.

Definition env := PTree.t (block * type).

Definition empty_env: env := (PTree.empty (block * type)).

Section SEMANTICS.

Variable ge: genv.

deref_loc ty m b ofs t v computes the value of a datum of type ty residing in memory m at block b, offset ofs. If the type ty indicates an access by value, the corresponding memory load is performed. If the type ty indicates an access by reference, the pointer Vptr b ofs is returned. v is the value returned, and t the trace of observables (nonempty if this is a volatile access).

Inductive deref_loc (ty: type) (m: mem) (b: block) (ofs: int) : trace -> val -> Prop :=
  | deref_loc_value: forall chunk v,
      access_mode ty = By_value chunk ->
      type_is_volatile ty = false ->
      Mem.loadv chunk m (Vptr b ofs) = Some v ->
      deref_loc ty m b ofs E0 v
  | deref_loc_volatile: forall chunk t v,
      access_mode ty = By_value chunk -> type_is_volatile ty = true ->
      volatile_load ge chunk m b ofs t v ->
      deref_loc ty m b ofs t v
  | deref_loc_reference:
      access_mode ty = By_reference ->
      deref_loc ty m b ofs E0 (Vptr b ofs)
  | deref_loc_copy:
      access_mode ty = By_copy ->
      deref_loc ty m b ofs E0 (Vptr b ofs).

Symmetrically, assign_loc ty m b ofs v t m' returns the memory state after storing the value v in the datum of type ty residing in memory m at block b, offset ofs. This is allowed only if ty indicates an access by value or by copy. m' is the updated memory state and t the trace of observables (nonempty if this is a volatile store).

Inductive assign_loc (ty: type) (m: mem) (b: block) (ofs: int):
                                            val -> trace -> mem -> Prop :=
  | assign_loc_value: forall v chunk m',
      access_mode ty = By_value chunk ->
      type_is_volatile ty = false ->
      Mem.storev chunk m (Vptr b ofs) v = Some m' ->
      assign_loc ty m b ofs v E0 m'
  | assign_loc_volatile: forall v chunk t m',
      access_mode ty = By_value chunk -> type_is_volatile ty = true ->
      volatile_store ge chunk m b ofs v t m' ->
      assign_loc ty m b ofs v t m'
  | assign_loc_copy: forall b' ofs' bytes m',
      access_mode ty = By_copy ->
      (alignof_blockcopy ge ty | Int.unsigned ofs') ->
      (alignof_blockcopy ge ty | Int.unsigned ofs) ->
      b' <> b \/ Int.unsigned ofs' = Int.unsigned ofs
              \/ Int.unsigned ofs' + sizeof ge ty <= Int.unsigned ofs
              \/ Int.unsigned ofs + sizeof ge ty <= Int.unsigned ofs' ->
      Mem.loadbytes m b' (Int.unsigned ofs') (sizeof ge ty) = Some bytes ->
      Mem.storebytes m b (Int.unsigned ofs) bytes = Some m' ->
      assign_loc ty m b ofs (Vptr b' ofs') E0 m'.

Allocation of function-local variables. alloc_variables e1 m1 vars e2 m2 allocates one memory block for each variable declared in vars, and associates the variable name with this block. e1 and m1 are the initial local environment and memory state. e2 and m2 are the final local environment and memory state.

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

Initialization of local variables that are parameters to a function. bind_parameters e m1 params args m2 stores the values args in the memory blocks corresponding to the variables params. m1 is the initial memory state and m2 the final memory state.

Inductive bind_parameters (e: env):
                           mem -> list (ident * type) -> list val ->
                           mem -> Prop :=
  | bind_parameters_nil:
      forall m,
      bind_parameters e m nil nil m
  | bind_parameters_cons:
      forall m id ty params v1 vl b m1 m2,
      PTree.get id e = Some(b, ty) ->
      assign_loc ty m b Int.zero v1 E0 m1 ->
      bind_parameters e m1 params vl m2 ->
      bind_parameters e m ((id, ty) :: params) (v1 :: vl) m2.

Return the list of blocks in the codomain of e, with low and high bounds.

Definition block_of_binding (id_b_ty: ident * (block * type)) :=
match id_b_ty with (id, (b, ty)) => (b, 0, sizeof ge ty) end.

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

Selection of the appropriate case of a switch, given the value n of the selector expression.

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

Fixpoint select_switch_case (n: Z) (sl: labeled_statements): option labeled_statements :=
  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'
  end.

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

Turn a labeled statement into a sequence

Fixpoint seq_of_labeled_statement (sl: labeled_statements) : statement :=
  match sl with
  | LSnil => Sskip
  | LScons _ s sl' => Ssequence s (seq_of_labeled_statement sl')
  end.

Extract the values from a list of function arguments

Inductive cast_arguments: exprlist -> typelist -> list val -> Prop :=
  | cast_args_nil:
      cast_arguments Enil Tnil nil
  | cast_args_cons: forall v ty el targ1 targs v1 vl,
      sem_cast v ty targ1 = Some v1 -> cast_arguments el targs vl ->
      cast_arguments (Econs (Eval v ty) el) (Tcons targ1 targs) (v1 :: vl).

Reduction semantics for expressions

Section EXPR.

Variable e: env.

The semantics of expressions follows the popular Wright-Felleisen style. It is a small-step semantics that reduces one redex at a time. We first define head reductions (at the top of an expression, then use reduction contexts to define reduction within an expression.

Head reduction for l-values.

Inductive lred: expr -> mem -> expr -> mem -> Prop :=
  | red_var_local: forall x ty m b,
      e!x = Some(b, ty) ->
      lred (Evar x ty) m
           (Eloc b Int.zero ty) m
  | red_var_global: forall x ty m b,
      e!x = None ->
      Genv.find_symbol ge x = Some b ->
      lred (Evar x ty) m
           (Eloc b Int.zero ty) m
  | red_deref: forall b ofs ty1 ty m,
      lred (Ederef (Eval (Vptr b ofs) ty1) ty) m
           (Eloc b ofs ty) m
  | red_field_struct: forall b ofs id co a f ty m delta,
      ge.(genv_cenv)!id = Some co ->
      field_offset ge f (co_members co) = OK delta ->
      lred (Efield (Eval (Vptr b ofs) (Tstruct id a)) f ty) m
           (Eloc b (Int.add ofs (Int.repr delta)) ty) m
  | red_field_union: forall b ofs id co a f ty m,
      ge.(genv_cenv)!id = Some co ->
      lred (Efield (Eval (Vptr b ofs) (Tunion id a)) f ty) m
           (Eloc b ofs ty) m.

Head reductions for r-values

Inductive rred: expr -> mem -> trace -> expr -> mem -> Prop :=
  | red_rvalof: forall b ofs ty m t v,
      deref_loc ty m b ofs t v ->
      rred (Evalof (Eloc b ofs ty) ty) m
         t (Eval v ty) m
  | red_addrof: forall b ofs ty1 ty m,
      rred (Eaddrof (Eloc b ofs ty1) ty) m
        E0 (Eval (Vptr b ofs) ty) m
  | red_unop: forall op v1 ty1 ty m v,
      sem_unary_operation op v1 ty1 m = Some v ->
      rred (Eunop op (Eval v1 ty1) ty) m
        E0 (Eval v ty) m
  | red_binop: forall op v1 ty1 v2 ty2 ty m v,
      sem_binary_operation ge op v1 ty1 v2 ty2 m = Some v ->
      rred (Ebinop op (Eval v1 ty1) (Eval v2 ty2) ty) m
        E0 (Eval v ty) m
  | red_cast: forall ty v1 ty1 m v,
      sem_cast v1 ty1 ty = Some v ->
      rred (Ecast (Eval v1 ty1) ty) m
        E0 (Eval v ty) m
  | red_seqand_true: forall v1 ty1 r2 ty m,
      bool_val v1 ty1 m = Some true ->
      rred (Eseqand (Eval v1 ty1) r2 ty) m
        E0 (Eparen r2 type_bool ty) m
  | red_seqand_false: forall v1 ty1 r2 ty m,
      bool_val v1 ty1 m = Some false ->
      rred (Eseqand (Eval v1 ty1) r2 ty) m
        E0 (Eval (Vint Int.zero) ty) m
  | red_seqor_true: forall v1 ty1 r2 ty m,
      bool_val v1 ty1 m = Some true ->
      rred (Eseqor (Eval v1 ty1) r2 ty) m
        E0 (Eval (Vint Int.one) ty) m
  | red_seqor_false: forall v1 ty1 r2 ty m,
      bool_val v1 ty1 m = Some false ->
      rred (Eseqor (Eval v1 ty1) r2 ty) m
        E0 (Eparen r2 type_bool ty) m
  | red_condition: forall v1 ty1 r1 r2 ty b m,
      bool_val v1 ty1 m = Some b ->
      rred (Econdition (Eval v1 ty1) r1 r2 ty) m
        E0 (Eparen (if b then r1 else r2) ty ty) m
  | red_sizeof: forall ty1 ty m,
      rred (Esizeof ty1 ty) m
        E0 (Eval (Vint (Int.repr (sizeof ge ty1))) ty) m
  | red_alignof: forall ty1 ty m,
      rred (Ealignof ty1 ty) m
        E0 (Eval (Vint (Int.repr (alignof ge ty1))) ty) m
  | red_assign: forall b ofs ty1 v2 ty2 m v t m',
      sem_cast v2 ty2 ty1 = Some v ->
      assign_loc ty1 m b ofs v t m' ->
      rred (Eassign (Eloc b ofs ty1) (Eval v2 ty2) ty1) m
         t (Eval v ty1) m'
  | red_assignop: forall op b ofs ty1 v2 ty2 tyres m t v1,
      deref_loc ty1 m b ofs t v1 ->
      rred (Eassignop op (Eloc b ofs ty1) (Eval v2 ty2) tyres ty1) m
         t (Eassign (Eloc b ofs ty1)
                    (Ebinop op (Eval v1 ty1) (Eval v2 ty2) tyres) ty1) m
  | red_postincr: forall id b ofs ty m t v1 op,
      deref_loc ty m b ofs t v1 ->
      op = match id with Incr => Oadd | Decr => Osub end ->
      rred (Epostincr id (Eloc b ofs ty) ty) m
         t (Ecomma (Eassign (Eloc b ofs ty)
                            (Ebinop op (Eval v1 ty)
                                       (Eval (Vint Int.one) type_int32s)
                                       (incrdecr_type ty))
                           ty)
                   (Eval v1 ty) ty) m
  | red_comma: forall v ty1 r2 ty m,
      typeof r2 = ty ->
      rred (Ecomma (Eval v ty1) r2 ty) m
        E0 r2 m
  | red_paren: forall v1 ty1 ty2 ty m v,
      sem_cast v1 ty1 ty2 = Some v ->
      rred (Eparen (Eval v1 ty1) ty2 ty) m
        E0 (Eval v ty) m
  | red_builtin: forall ef tyargs el ty m vargs t vres m',
      cast_arguments el tyargs vargs ->
      external_call ef ge vargs m t vres m' ->
      rred (Ebuiltin ef tyargs el ty) m
         t (Eval vres ty) m'.

Head reduction for function calls. (More exactly, identification of function calls that can reduce.)

Inductive callred: expr -> fundef -> list val -> type -> Prop :=
  | red_call: forall vf tyf tyargs tyres cconv el ty fd vargs,
      Genv.find_funct ge vf = Some fd ->
      cast_arguments el tyargs vargs ->
      type_of_fundef fd = Tfunction tyargs tyres cconv ->
      classify_fun tyf = fun_case_f tyargs tyres cconv ->
      callred (Ecall (Eval vf tyf) el ty)
              fd vargs ty.

Reduction contexts. In accordance with C's nondeterministic semantics, we allow reduction both to the left and to the right of a binary operator. To enforce C's notion of sequence point, reductions within a conditional a ? b : c can only take place in a, not in b nor c; reductions within a sequential "or" / "and" a && b or a || b can only take place in a, not in b; and reductions within a sequence a, b can only take place in a, not in b. Reduction contexts are represented by functions C from expressions to expressions, suitably constrained by the context from to C predicate below. Contexts are "kinded" with respect to l-values and r-values: from is the kind of the hole in the context and to is the kind of the term resulting from filling the hole.

Inductive kind : Type := LV | RV.

Inductive context: kind -> kind -> (expr -> expr) -> Prop :=
  | ctx_top: forall k,
      context k k (fun x => x)
  | ctx_deref: forall k C ty,
      context k RV C -> context k LV (fun x => Ederef (C x) ty)
  | ctx_field: forall k C f ty,
      context k RV C -> context k LV (fun x => Efield (C x) f ty)
  | ctx_rvalof: forall k C ty,
      context k LV C -> context k RV (fun x => Evalof (C x) ty)
  | ctx_addrof: forall k C ty,
      context k LV C -> context k RV (fun x => Eaddrof (C x) ty)
  | ctx_unop: forall k C op ty,
      context k RV C -> context k RV (fun x => Eunop op (C x) ty)
  | ctx_binop_left: forall k C op e2 ty,
      context k RV C -> context k RV (fun x => Ebinop op (C x) e2 ty)
  | ctx_binop_right: forall k C op e1 ty,
      context k RV C -> context k RV (fun x => Ebinop op e1 (C x) ty)
  | ctx_cast: forall k C ty,
      context k RV C -> context k RV (fun x => Ecast (C x) ty)
  | ctx_seqand: forall k C r2 ty,
      context k RV C -> context k RV (fun x => Eseqand (C x) r2 ty)
  | ctx_seqor: forall k C r2 ty,
      context k RV C -> context k RV (fun x => Eseqor (C x) r2 ty)
  | ctx_condition: forall k C r2 r3 ty,
      context k RV C -> context k RV (fun x => Econdition (C x) r2 r3 ty)
  | ctx_assign_left: forall k C e2 ty,
      context k LV C -> context k RV (fun x => Eassign (C x) e2 ty)
  | ctx_assign_right: forall k C e1 ty,
      context k RV C -> context k RV (fun x => Eassign e1 (C x) ty)
  | ctx_assignop_left: forall k C op e2 tyres ty,
      context k LV C -> context k RV (fun x => Eassignop op (C x) e2 tyres ty)
  | ctx_assignop_right: forall k C op e1 tyres ty,
      context k RV C -> context k RV (fun x => Eassignop op e1 (C x) tyres ty)
  | ctx_postincr: forall k C id ty,
      context k LV C -> context k RV (fun x => Epostincr id (C x) ty)
  | ctx_call_left: forall k C el ty,
      context k RV C -> context k RV (fun x => Ecall (C x) el ty)
  | ctx_call_right: forall k C e1 ty,
      contextlist k C -> context k RV (fun x => Ecall e1 (C x) ty)
  | ctx_builtin: forall k C ef tyargs ty,
      contextlist k C -> context k RV (fun x => Ebuiltin ef tyargs (C x) ty)
  | ctx_comma: forall k C e2 ty,
      context k RV C -> context k RV (fun x => Ecomma (C x) e2 ty)
  | ctx_paren: forall k C tycast ty,
      context k RV C -> context k RV (fun x => Eparen (C x) tycast ty)

with contextlist: kind -> (expr -> exprlist) -> Prop :=
  | ctx_list_head: forall k C el,
      context k RV C -> contextlist k (fun x => Econs (C x) el)
  | ctx_list_tail: forall k C e1,
      contextlist k C -> contextlist k (fun x => Econs e1 (C x)).

In a nondeterministic semantics, expressions can go wrong according to one reduction order while being defined according to another. Consider for instance (x = 1) + (10 / x) where x is initially 0. This expression goes wrong if evaluated right-to-left, but is defined if evaluated left-to-right. Since our compiler is going to pick one particular evaluation order, we must make sure that all orders are safe, i.e. never evaluate a subexpression that goes wrong. Being safe is a stronger requirement than just not getting stuck during reductions. Consider f() + (10 / x), where f() does not terminate. This expression is never stuck because the evaluation of f() can make infinitely many transitions. Yet it contains a subexpression 10 / x that can go wrong if x = 0, and the compiler may choose to evaluate 10 / x first, before calling f(). Therefore, we must make sure that not only an expression cannot get stuck, but none of its subexpressions can either. We say that a subexpression is not immediately stuck if it is a value (of the appropriate kind) or it can reduce (at head or within).

Inductive imm_safe: kind -> expr -> mem -> Prop :=
  | imm_safe_val: forall v ty m,
      imm_safe RV (Eval v ty) m
  | imm_safe_loc: forall b ofs ty m,
      imm_safe LV (Eloc b ofs ty) m
  | imm_safe_lred: forall to C e m e' m',
      lred e m e' m' ->
      context LV to C ->
      imm_safe to (C e) m
  | imm_safe_rred: forall to C e m t e' m',
      rred e m t e' m' ->
      context RV to C ->
      imm_safe to (C e) m
  | imm_safe_callred: forall to C e m fd args ty,
      callred e fd args ty ->
      context RV to C ->
      imm_safe to (C e) m.

End EXPR.

Transition semantics.

Continuations describe the computations that remain to be performed after the statement or expression under consideration has evaluated completely.

Inductive cont: Type :=
  | Kstop: cont
  | Kdo: cont -> cont (* Kdo k = after x in x; *)
  | Kseq: statement -> cont -> cont (* Kseq s2 k = after s1 in s1;s2 *)
  | Kifthenelse: statement -> statement -> cont -> cont (* Kifthenelse s1 s2 k = after x in if (x) { s1 } else { s2 } *)
  | Kwhile1: expr -> statement -> cont -> cont (* Kwhile1 x s k = after x in while(x) s *)
  | Kwhile2: expr -> statement -> cont -> cont (* Kwhile x s k = after s in while (x) s *)
  | Kdowhile1: expr -> statement -> cont -> cont (* Kdowhile1 x s k = after s in do s while (x) *)
  | Kdowhile2: expr -> statement -> cont -> cont (* Kdowhile2 x s k = after x in do s while (x) *)
  | Kfor2: expr -> statement -> statement -> cont -> cont (* Kfor2 e2 e3 s k = after e2 in for(e1;e2;e3) s *)
  | Kfor3: expr -> statement -> statement -> cont -> cont (* Kfor3 e2 e3 s k = after s in for(e1;e2;e3) s *)
  | Kfor4: expr -> statement -> statement -> cont -> cont (* Kfor4 e2 e3 s k = after e3 in for(e1;e2;e3) s *)
  | Kswitch1: labeled_statements -> cont -> cont (* Kswitch1 ls k = after e in switch(e) { ls } *)
  | Kswitch2: cont -> cont (* catches break statements arising out of switch *)
  | Kreturn: cont -> cont (* Kreturn k = after e in return e; *)
  | Kcall: function -> (* calling function *)
           env -> (* local env of calling function *)
           (expr -> expr) -> (* context of the call *)
           type -> (* type of call expression *)
           cont -> cont.

Pop continuation until a call or stop

Execution states of the program are grouped in 4 classes corresponding to the part of the program we are currently executing. It can be a statement (State), an expression (ExprState), a transition from a calling function to a called function (Callstate), or the symmetrical transition from a function back to its caller (Returnstate).

Inductive state: Type :=
  | State (* execution of a statement *)
      (f: function)
      (s: statement)
      (k: cont)
      (e: env)
      (m: mem) : state
  | ExprState (* reduction of an expression *)
      (f: function)
      (r: expr)
      (k: cont)
      (e: env)
      (m: mem) : state
  | Callstate (* calling a function *)
      (fd: fundef)
      (args: list val)
      (k: cont)
      (m: mem) : state
  | Returnstate (* returning from a function *)
      (res: val)
      (k: cont)
      (m: mem) : state
  | Stuckstate. (* undefined behavior occurred *)

Find the statement and manufacture the continuation corresponding to a label.

Fixpoint find_label (lbl: label) (s: statement) (k: cont)
                    {struct s}: option (statement * cont) :=
  match s with
  | Ssequence s1 s2 =>
      match find_label lbl s1 (Kseq s2 k) with
      | Some sk => Some sk
      | None => find_label lbl s2 k
      end
  | Sifthenelse a s1 s2 =>
      match find_label lbl s1 k with
      | Some sk => Some sk
      | None => find_label lbl s2 k
      end
  | Swhile a s1 =>
      find_label lbl s1 (Kwhile2 a s1 k)
  | Sdowhile a s1 =>
      find_label lbl s1 (Kdowhile1 a s1 k)
  | Sfor a1 a2 a3 s1 =>
      match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with
      | Some sk => Some sk
      | None =>
          match find_label lbl s1 (Kfor3 a2 a3 s1 k) with
          | Some sk => Some sk
          | None => find_label lbl a3 (Kfor4 a2 a3 s1 k)
          end
      end
  | Sswitch e sl =>
      find_label_ls lbl sl (Kswitch2 k)
  | Slabel lbl' s' =>
      if ident_eq lbl lbl' then Some(s', k) else find_label lbl s' k
  | _ => None
  end

with find_label_ls (lbl: label) (sl: labeled_statements) (k: cont)
                    {struct sl}: option (statement * cont) :=
  match sl with
  | LSnil => None
  | LScons _ s sl' =>
      match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with
      | Some sk => Some sk
      | None => find_label_ls lbl sl' k
      end
  end.

We separate the transition rules in two groups:

one group that deals with reductions over expressions;
the other group that deals with everything else: statements, function calls, etc.

This makes it easy to express different reduction strategies for expressions: the second group of rules can be reused as is.

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

  | step_lred: forall C f a k e m a' m',
      lred e a m a' m' ->
      context LV RV C ->
      estep (ExprState f (C a) k e m)
         E0 (ExprState f (C a') k e m')

  | step_rred: forall C f a k e m t a' m',
      rred a m t a' m' ->
      context RV RV C ->
      estep (ExprState f (C a) k e m)
          t (ExprState f (C a') k e m')

  | step_call: forall C f a k e m fd vargs ty,
      callred a fd vargs ty ->
      context RV RV C ->
      estep (ExprState f (C a) k e m)
         E0 (Callstate fd vargs (Kcall f e C ty k) m)

  | step_stuck: forall C f a k e m K,
      context K RV C -> ~(imm_safe e K a m) ->
      estep (ExprState f (C a) k e m)
         E0 Stuckstate.

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

  | step_do_1: forall f x k e m,
      sstep (State f (Sdo x) k e m)
         E0 (ExprState f x (Kdo k) e m)
  | step_do_2: forall f v ty k e m,
      sstep (ExprState f (Eval v ty) (Kdo k) e m)
         E0 (State f Sskip k e m)

  | step_seq: forall f s1 s2 k e m,
      sstep (State f (Ssequence s1 s2) k e m)
         E0 (State f s1 (Kseq s2 k) e m)
  | step_skip_seq: forall f s k e m,
      sstep (State f Sskip (Kseq s k) e m)
         E0 (State f s k e m)
  | step_continue_seq: forall f s k e m,
      sstep (State f Scontinue (Kseq s k) e m)
         E0 (State f Scontinue k e m)
  | step_break_seq: forall f s k e m,
      sstep (State f Sbreak (Kseq s k) e m)
         E0 (State f Sbreak k e m)

  | step_ifthenelse_1: forall f a s1 s2 k e m,
      sstep (State f (Sifthenelse a s1 s2) k e m)
         E0 (ExprState f a (Kifthenelse s1 s2 k) e m)
  | step_ifthenelse_2: forall f v ty s1 s2 k e m b,
      bool_val v ty m = Some b ->
      sstep (ExprState f (Eval v ty) (Kifthenelse s1 s2 k) e m)
         E0 (State f (if b then s1 else s2) k e m)

  | step_while: forall f x s k e m,
      sstep (State f (Swhile x s) k e m)
        E0 (ExprState f x (Kwhile1 x s k) e m)
  | step_while_false: forall f v ty x s k e m,
      bool_val v ty m = Some false ->
      sstep (ExprState f (Eval v ty) (Kwhile1 x s k) e m)
        E0 (State f Sskip k e m)
  | step_while_true: forall f v ty x s k e m ,
      bool_val v ty m = Some true ->
      sstep (ExprState f (Eval v ty) (Kwhile1 x s k) e m)
        E0 (State f s (Kwhile2 x s k) e m)
  | step_skip_or_continue_while: forall f s0 x s k e m,
      s0 = Sskip \/ s0 = Scontinue ->
      sstep (State f s0 (Kwhile2 x s k) e m)
        E0 (State f (Swhile x s) k e m)
  | step_break_while: forall f x s k e m,
      sstep (State f Sbreak (Kwhile2 x s k) e m)
        E0 (State f Sskip k e m)

  | step_dowhile: forall f a s k e m,
      sstep (State f (Sdowhile a s) k e m)
        E0 (State f s (Kdowhile1 a s k) e m)
  | step_skip_or_continue_dowhile: forall f s0 x s k e m,
      s0 = Sskip \/ s0 = Scontinue ->
      sstep (State f s0 (Kdowhile1 x s k) e m)
         E0 (ExprState f x (Kdowhile2 x s k) e m)
  | step_dowhile_false: forall f v ty x s k e m,
      bool_val v ty m = Some false ->
      sstep (ExprState f (Eval v ty) (Kdowhile2 x s k) e m)
         E0 (State f Sskip k e m)
  | step_dowhile_true: forall f v ty x s k e m,
      bool_val v ty m = Some true ->
      sstep (ExprState f (Eval v ty) (Kdowhile2 x s k) e m)
         E0 (State f (Sdowhile x s) k e m)
  | step_break_dowhile: forall f a s k e m,
      sstep (State f Sbreak (Kdowhile1 a s k) e m)
         E0 (State f Sskip k e m)

  | step_for_start: forall f a1 a2 a3 s k e m,
      a1 <> Sskip ->
      sstep (State f (Sfor a1 a2 a3 s) k e m)
         E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m)
  | step_for: forall f a2 a3 s k e m,
      sstep (State f (Sfor Sskip a2 a3 s) k e m)
         E0 (ExprState f a2 (Kfor2 a2 a3 s k) e m)
  | step_for_false: forall f v ty a2 a3 s k e m,
      bool_val v ty m = Some false ->
      sstep (ExprState f (Eval v ty) (Kfor2 a2 a3 s k) e m)
         E0 (State f Sskip k e m)
  | step_for_true: forall f v ty a2 a3 s k e m,
      bool_val v ty m = Some true ->
      sstep (ExprState f (Eval v ty) (Kfor2 a2 a3 s k) e m)
         E0 (State f s (Kfor3 a2 a3 s k) e m)
  | step_skip_or_continue_for3: forall f x a2 a3 s k e m,
      x = Sskip \/ x = Scontinue ->
      sstep (State f x (Kfor3 a2 a3 s k) e m)
         E0 (State f a3 (Kfor4 a2 a3 s k) e m)
  | step_break_for3: forall f a2 a3 s k e m,
      sstep (State f Sbreak (Kfor3 a2 a3 s k) e m)
         E0 (State f Sskip k e m)
  | step_skip_for4: forall f a2 a3 s k e m,
      sstep (State f Sskip (Kfor4 a2 a3 s k) e m)
         E0 (State f (Sfor Sskip a2 a3 s) k e m)

  | step_return_0: forall f k e m m',
      Mem.free_list m (blocks_of_env e) = Some m' ->
      sstep (State f (Sreturn None) k e m)
         E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f x k e m,
      sstep (State f (Sreturn (Some x)) k e m)
         E0 (ExprState f x (Kreturn k) e m)
  | step_return_2: forall f v1 ty k e m v2 m',
      sem_cast v1 ty f.(fn_return) = Some v2 ->
      Mem.free_list m (blocks_of_env e) = Some m' ->
      sstep (ExprState f (Eval v1 ty) (Kreturn k) e m)
         E0 (Returnstate v2 (call_cont k) m')
  | step_skip_call: forall f k e m m',
      is_call_cont k ->
      Mem.free_list m (blocks_of_env e) = Some m' ->
      sstep (State f Sskip k e m)
         E0 (Returnstate Vundef k m')

  | step_switch: forall f x sl k e m,
      sstep (State f (Sswitch x sl) k e m)
         E0 (ExprState f x (Kswitch1 sl k) e m)
  | step_expr_switch: forall f ty sl k e m v n,
      sem_switch_arg v ty = Some n ->
      sstep (ExprState f (Eval v ty) (Kswitch1 sl k) e m)
         E0 (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch2 k) e m)
  | step_skip_break_switch: forall f x k e m,
      x = Sskip \/ x = Sbreak ->
      sstep (State f x (Kswitch2 k) e m)
         E0 (State f Sskip k e m)
  | step_continue_switch: forall f k e m,
      sstep (State f Scontinue (Kswitch2 k) e m)
         E0 (State f Scontinue k e m)

  | step_label: forall f lbl s k e m,
      sstep (State f (Slabel lbl s) k e m)
         E0 (State f s k e m)

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

  | step_internal_function: forall f vargs k m e m1 m2,
      list_norepet (var_names (fn_params f) ++ var_names (fn_vars f)) ->
      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
      bind_parameters e m1 f.(fn_params) vargs m2 ->
      sstep (Callstate (Internal f) vargs k m)
         E0 (State f f.(fn_body) k e m2)

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

  | step_returnstate: forall v f e C ty k m,
      sstep (Returnstate v (Kcall f e C ty k) m)
         E0 (ExprState f (C (Eval v ty)) k e m).

Definition step (S: state) (t: trace) (S': state) : Prop :=
  estep S t S' \/ sstep S t S'.

End SEMANTICS.

Whole-program semantics

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 := 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 ->
      type_of_fundef f = Tfunction Tnil type_int32s cc_default ->
      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_gen step (initial_state p) final_state (globalenv p) (globalenv p).

This semantics has the single-event property.

Lemma semantics_single_events:
forall p, single_events (semantics p).

Proof.

  unfold semantics; intros; red; simpl; intros.
  set (ge := globalenv p) in *.
  assert (DEREF: forall chunk m b ofs t v, deref_loc ge chunk m b ofs t v -> (length t <= 1)%nat).
    intros. inv H0; simpl; try omega. inv H3; simpl; try omega.
  assert (ASSIGN: forall chunk m b ofs t v m', assign_loc ge chunk m b ofs v t m' -> (length t <= 1)%nat).
    intros. inv H0; simpl; try omega. inv H3; simpl; try omega.
  destruct H.
  inv H; simpl; try omega. inv H0; eauto; simpl; try omega.
  eapply external_call_trace_length; eauto.
  inv H; simpl; try omega. eapply external_call_trace_length; eauto.
Qed.

Module Csem

Operational semantics

Reduction semantics for expressions

Transition semantics.

Whole-program semantics