Library Csem
Dynamic semantics for the Clight language
Require Import Coqlib.
Require Import Errors.
Require Import Maps.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import AST.
Require Import Mem.
Require Import Events.
Require Import Globalenvs.
Require Import Csyntax.
Require Import Smallstep.
Interpretation of values as truth values.
Non-zero integers, non-zero floats and non-null pointers are
considered as true. The integer zero (which also represents
the null pointer) and the float 0.0 are false.
Inductive is_false: val -> type -> Prop :=
| is_false_int: forall sz sg,
is_false (Vint Int.zero) (Tint sz sg)
| is_false_pointer: forall t,
is_false (Vint Int.zero) (Tpointer t)
| is_false_float: forall sz,
is_false (Vfloat Float.zero) (Tfloat sz).
Inductive is_true: val -> type -> Prop :=
| is_true_int_int: forall n sz sg,
n <> Int.zero ->
is_true (Vint n) (Tint sz sg)
| is_true_pointer_int: forall b ofs sz sg,
is_true (Vptr b ofs) (Tint sz sg)
| is_true_int_pointer: forall n t,
n <> Int.zero ->
is_true (Vint n) (Tpointer t)
| is_true_pointer_pointer: forall b ofs t,
is_true (Vptr b ofs) (Tpointer t)
| is_true_float: forall f sz,
f <> Float.zero ->
is_true (Vfloat f) (Tfloat sz).
Inductive bool_of_val : val -> type -> val -> Prop :=
| bool_of_val_true: forall v ty,
is_true v ty ->
bool_of_val v ty Vtrue
| bool_of_val_false: forall v ty,
is_false v ty ->
bool_of_val v ty Vfalse.
The following
sem_ functions compute the result of an operator
application. Since operators are overloaded, the result depends
both on the static types of the arguments and on their run-time values.
Unlike in C, automatic conversions between integers and floats
are not performed. For instance, e1 + e2 is undefined if e1
is a float and e2 an integer. The Clight producer must have explicitly
promoted e2 to a float.
Function sem_neg (v: val) (ty: type) : option val :=
match ty with
| Tint _ _ =>
match v with
| Vint n => Some (Vint (Int.neg n))
| _ => None
end
| Tfloat _ =>
match v with
| Vfloat f => Some (Vfloat (Float.neg f))
| _ => None
end
| _ => None
end.
Function sem_notint (v: val) : option val :=
match v with
| Vint n => Some (Vint (Int.xor n Int.mone))
| _ => None
end.
Function sem_notbool (v: val) (ty: type) : option val :=
match ty with
| Tint _ _ =>
match v with
| Vint n => Some (Val.of_bool (Int.eq n Int.zero))
| Vptr _ _ => Some Vfalse
| _ => None
end
| Tpointer _ =>
match v with
| Vint n => Some (Val.of_bool (Int.eq n Int.zero))
| Vptr _ _ => Some Vfalse
| _ => None
end
| Tfloat _ =>
match v with
| Vfloat f => Some (Val.of_bool (Float.cmp Ceq f Float.zero))
| _ => None
end
| _ => None
end.
Function sem_add (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
match classify_add t1 t2 with
| add_case_ii =>
integer addition
match v1, v2 with
| Vint n1, Vint n2 => Some (Vint (Int.add n1 n2))
| _, _ => None
end
| add_case_ff =>
float addition
match v1, v2 with
| Vfloat n1, Vfloat n2 => Some (Vfloat (Float.add n1 n2))
| _, _ => None
end
| add_case_pi ty =>
pointer plus integer
match v1,v2 with
| Vptr b1 ofs1, Vint n2 =>
Some (Vptr b1 (Int.add ofs1 (Int.mul (Int.repr (sizeof ty)) n2)))
| _, _ => None
end
| add_case_ip ty =>
integer plus pointer
match v1,v2 with
| Vint n1, Vptr b2 ofs2 =>
Some (Vptr b2 (Int.add ofs2 (Int.mul (Int.repr (sizeof ty)) n1)))
| _, _ => None
end
| add_default => None
end.
Function sem_sub (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
match classify_sub t1 t2 with
| sub_case_ii =>
integer subtraction
match v1,v2 with
| Vint n1, Vint n2 => Some (Vint (Int.sub n1 n2))
| _, _ => None
end
| sub_case_ff =>
float subtraction
match v1,v2 with
| Vfloat f1, Vfloat f2 => Some (Vfloat(Float.sub f1 f2))
| _, _ => None
end
| sub_case_pi ty =>
pointer minus integer
match v1,v2 with
| Vptr b1 ofs1, Vint n2 =>
Some (Vptr b1 (Int.sub ofs1 (Int.mul (Int.repr (sizeof ty)) n2)))
| _, _ => None
end
| sub_case_pp ty =>
pointer minus pointer
match v1,v2 with
| Vptr b1 ofs1, Vptr b2 ofs2 =>
if zeq b1 b2 then
if Int.eq (Int.repr (sizeof ty)) Int.zero then None
else Some (Vint (Int.divu (Int.sub ofs1 ofs2) (Int.repr (sizeof ty))))
else None
| _, _ => None
end
| sub_default => None
end.
Function sem_mul (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
match classify_mul t1 t2 with
| mul_case_ii =>
match v1,v2 with
| Vint n1, Vint n2 => Some (Vint (Int.mul n1 n2))
| _, _ => None
end
| mul_case_ff =>
match v1,v2 with
| Vfloat f1, Vfloat f2 => Some (Vfloat (Float.mul f1 f2))
| _, _ => None
end
| mul_default =>
None
end.
Function sem_div (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
match classify_div t1 t2 with
| div_case_I32unsi =>
match v1,v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero then None else Some (Vint (Int.divu n1 n2))
| _,_ => None
end
| div_case_ii =>
match v1,v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero then None else Some (Vint(Int.divs n1 n2))
| _,_ => None
end
| div_case_ff =>
match v1,v2 with
| Vfloat f1, Vfloat f2 => Some (Vfloat(Float.div f1 f2))
| _, _ => None
end
| div_default =>
None
end.
Function sem_mod (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
match classify_mod t1 t2 with
| mod_case_I32unsi =>
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero then None else Some (Vint (Int.modu n1 n2))
| _, _ => None
end
| mod_case_ii =>
match v1,v2 with
| Vint n1, Vint n2 =>
if Int.eq n2 Int.zero then None else Some (Vint (Int.mods n1 n2))
| _, _ => None
end
| mod_default =>
None
end.
Function sem_and (v1 v2: val) : option val :=
match v1, v2 with
| Vint n1, Vint n2 => Some (Vint(Int.and n1 n2))
| _, _ => None
end .
Function sem_or (v1 v2: val) : option val :=
match v1, v2 with
| Vint n1, Vint n2 => Some (Vint(Int.or n1 n2))
| _, _ => None
end.
Function sem_xor (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 => Some (Vint(Int.xor n1 n2))
| _, _ => None
end.
Function sem_shl (v1 v2: val): option val :=
match v1, v2 with
| Vint n1, Vint n2 =>
if Int.ltu n2 Int.iwordsize then Some (Vint(Int.shl n1 n2)) else None
| _, _ => None
end.
Function sem_shr (v1: val) (t1: type) (v2: val) (t2: type): option val :=
match classify_shr t1 t2 with
| shr_case_I32unsi =>
match v1,v2 with
| Vint n1, Vint n2 =>
if Int.ltu n2 Int.iwordsize then Some (Vint (Int.shru n1 n2)) else None
| _,_ => None
end
| shr_case_ii =>
match v1,v2 with
| Vint n1, Vint n2 =>
if Int.ltu n2 Int.iwordsize then Some (Vint (Int.shr n1 n2)) else None
| _, _ => None
end
| shr_default=>
None
end.
Function sem_cmp_mismatch (c: comparison): option val :=
match c with
| Ceq => Some Vfalse
| Cne => Some Vtrue
| _ => None
end.
Function sem_cmp (c:comparison)
(v1: val) (t1: type) (v2: val) (t2: type)
(m: mem): option val :=
match classify_cmp t1 t2 with
| cmp_case_I32unsi =>
match v1,v2 with
| Vint n1, Vint n2 => Some (Val.of_bool (Int.cmpu c n1 n2))
| _, _ => None
end
| cmp_case_ipip =>
match v1,v2 with
| Vint n1, Vint n2 => Some (Val.of_bool (Int.cmp c n1 n2))
| Vptr b1 ofs1, Vptr b2 ofs2 =>
if valid_pointer m b1 (Int.signed ofs1)
&& valid_pointer m b2 (Int.signed ofs2) then
if zeq b1 b2
then Some (Val.of_bool (Int.cmp c ofs1 ofs2))
else sem_cmp_mismatch c
else None
| Vptr b ofs, Vint n =>
if Int.eq n Int.zero then sem_cmp_mismatch c else None
| Vint n, Vptr b ofs =>
if Int.eq n Int.zero then sem_cmp_mismatch c else None
| _, _ => None
end
| cmp_case_ff =>
match v1,v2 with
| Vfloat f1, Vfloat f2 => Some (Val.of_bool (Float.cmp c f1 f2))
| _, _ => None
end
| cmp_default => None
end.
Definition sem_unary_operation
(op: unary_operation) (v: val) (ty: type): option val :=
match op with
| Onotbool => sem_notbool v ty
| Onotint => sem_notint v
| Oneg => sem_neg v ty
end.
Definition sem_binary_operation
(op: binary_operation)
(v1: val) (t1: type) (v2: val) (t2:type)
(m: mem): option val :=
match op with
| Oadd => sem_add v1 t1 v2 t2
| Osub => sem_sub v1 t1 v2 t2
| Omul => sem_mul v1 t1 v2 t2
| Omod => sem_mod v1 t1 v2 t2
| Odiv => sem_div v1 t1 v2 t2
| Oand => sem_and v1 v2
| Oor => sem_or v1 v2
| Oxor => sem_xor v1 v2
| Oshl => sem_shl v1 v2
| Oshr => sem_shr v1 t1 v2 t2
| Oeq => sem_cmp Ceq v1 t1 v2 t2 m
| One => sem_cmp Cne v1 t1 v2 t2 m
| Olt => sem_cmp Clt v1 t1 v2 t2 m
| Ogt => sem_cmp Cgt v1 t1 v2 t2 m
| Ole => sem_cmp Cle v1 t1 v2 t2 m
| Oge => sem_cmp Cge v1 t1 v2 t2 m
end.
Semantic of casts.
cast v1 t1 t2 v2 holds if value v1,
viewed with static type t1, can be cast to type t2,
resulting in value v2.
Definition cast_int_int (sz: intsize) (sg: signedness) (i: int) : int :=
match sz, sg with
| I8, Signed => Int.sign_ext 8 i
| I8, Unsigned => Int.zero_ext 8 i
| I16, Signed => Int.sign_ext 16 i
| I16, Unsigned => Int.zero_ext 16 i
| I32, _ => i
end.
Definition cast_int_float (si : signedness) (i: int) : float :=
match si with
| Signed => Float.floatofint i
| Unsigned => Float.floatofintu i
end.
Definition cast_float_int (si : signedness) (f: float) : int :=
match si with
| Signed => Float.intoffloat f
| Unsigned => Float.intuoffloat f
end.
Definition cast_float_float (sz: floatsize) (f: float) : float :=
match sz with
| F32 => Float.singleoffloat f
| F64 => f
end.
Inductive neutral_for_cast: type -> Prop :=
| nfc_int: forall sg,
neutral_for_cast (Tint I32 sg)
| nfc_ptr: forall ty,
neutral_for_cast (Tpointer ty)
| nfc_array: forall ty sz,
neutral_for_cast (Tarray ty sz)
| nfc_fun: forall targs tres,
neutral_for_cast (Tfunction targs tres).
Inductive cast : val -> type -> type -> val -> Prop :=
| cast_ii: forall i sz2 sz1 si1 si2,
int to int
cast (Vint i) (Tint sz1 si1) (Tint sz2 si2)
(Vint (cast_int_int sz2 si2 i))
| cast_fi: forall f sz1 sz2 si2,
float to int
cast (Vfloat f) (Tfloat sz1) (Tint sz2 si2)
(Vint (cast_int_int sz2 si2 (cast_float_int si2 f)))
| cast_if: forall i sz1 sz2 si1,
int to float
cast (Vint i) (Tint sz1 si1) (Tfloat sz2)
(Vfloat (cast_float_float sz2 (cast_int_float si1 i)))
| cast_ff: forall f sz1 sz2,
float to float
cast (Vfloat f) (Tfloat sz1) (Tfloat sz2)
(Vfloat (cast_float_float sz2 f))
| cast_nn_p: forall b ofs t1 t2,
no change in data representation
neutral_for_cast t1 -> neutral_for_cast t2 ->
cast (Vptr b ofs) t1 t2 (Vptr b ofs)
| cast_nn_i: forall n t1 t2,
no change in data representation
neutral_for_cast t1 -> neutral_for_cast t2 ->
cast (Vint n) t1 t2 (Vint n).
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.)
Definition genv := Genv.t fundef.
The local environment maps local variables to block references.
The current value of the variable is stored in the associated memory
block.
Definition env := PTree.t block.
Definition empty_env: env := (PTree.empty block).
load_value_of_type ty m b ofs 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.
Definition load_value_of_type (ty: type) (m: mem) (b: block) (ofs: int) : option val :=
match access_mode ty with
| By_value chunk => Mem.loadv chunk m (Vptr b ofs)
| By_reference => Some (Vptr b ofs)
| By_nothing => None
end.
Symmetrically,
store_value_of_type ty m b ofs v 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.
Definition store_value_of_type (ty_dest: type) (m: mem) (loc: block) (ofs: int) (v: val) : option mem :=
match access_mode ty_dest with
| By_value chunk => Mem.storev chunk m (Vptr loc ofs) v
| By_reference => None
| By_nothing => None
end.
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 ty) = (m1, b1) ->
alloc_variables (PTree.set id b1 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: env ->
mem -> list (ident * type) -> list val ->
mem -> Prop :=
| bind_parameters_nil:
forall e m,
bind_parameters e m nil nil m
| bind_parameters_cons:
forall e m id ty params v1 vl b m1 m2,
PTree.get id e = Some b ->
store_value_of_type ty m b Int.zero v1 = Some 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.
Definition blocks_of_env (e: env) : list block :=
List.map (@snd ident block) (PTree.elements e).
Selection of the appropriate case of a
switch, given the value n
of the selector expression.
Fixpoint select_switch (n: int) (sl: labeled_statements)
{struct sl}: labeled_statements :=
match sl with
| LSdefault _ => sl
| LScase c s sl' => if Int.eq c n then sl else select_switch n sl'
end.
Turn a labeled statement into a sequence
Fixpoint seq_of_labeled_statement (sl: labeled_statements) : statement :=
match sl with
| LSdefault s => s
| LScase c s sl' => Ssequence s (seq_of_labeled_statement sl')
end.
Section SEMANTICS.
Variable ge: genv.
Section EXPR.
Variable e: env.
Variable m: mem.
eval_expr ge e m a v defines the evaluation of expression a
in r-value position. v is the value of the expression.
e is the current environment and m is the current memory state.
Inductive eval_expr: expr -> val -> Prop :=
| eval_Econst_int: forall i ty,
eval_expr (Expr (Econst_int i) ty) (Vint i)
| eval_Econst_float: forall f ty,
eval_expr (Expr (Econst_float f) ty) (Vfloat f)
| eval_Elvalue: forall a ty loc ofs v,
eval_lvalue (Expr a ty) loc ofs ->
load_value_of_type ty m loc ofs = Some v ->
eval_expr (Expr a ty) v
| eval_Eaddrof: forall a ty loc ofs,
eval_lvalue a loc ofs ->
eval_expr (Expr (Eaddrof a) ty) (Vptr loc ofs)
| eval_Esizeof: forall ty' ty,
eval_expr (Expr (Esizeof ty') ty) (Vint (Int.repr (sizeof ty')))
| eval_Eunop: forall op a ty v1 v,
eval_expr a v1 ->
sem_unary_operation op v1 (typeof a) = Some v ->
eval_expr (Expr (Eunop op a) ty) v
| eval_Ebinop: forall op a1 a2 ty v1 v2 v,
eval_expr a1 v1 ->
eval_expr a2 v2 ->
sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some v ->
eval_expr (Expr (Ebinop op a1 a2) ty) v
| eval_Econdition_true: forall a1 a2 a3 ty v1 v2,
eval_expr a1 v1 ->
is_true v1 (typeof a1) ->
eval_expr a2 v2 ->
eval_expr (Expr (Econdition a1 a2 a3) ty) v2
| eval_Econdition_false: forall a1 a2 a3 ty v1 v3,
eval_expr a1 v1 ->
is_false v1 (typeof a1) ->
eval_expr a3 v3 ->
eval_expr (Expr (Econdition a1 a2 a3) ty) v3
| eval_Eorbool_1: forall a1 a2 ty v1,
eval_expr a1 v1 ->
is_true v1 (typeof a1) ->
eval_expr (Expr (Eorbool a1 a2) ty) Vtrue
| eval_Eorbool_2: forall a1 a2 ty v1 v2 v,
eval_expr a1 v1 ->
is_false v1 (typeof a1) ->
eval_expr a2 v2 ->
bool_of_val v2 (typeof a2) v ->
eval_expr (Expr (Eorbool a1 a2) ty) v
| eval_Eandbool_1: forall a1 a2 ty v1,
eval_expr a1 v1 ->
is_false v1 (typeof a1) ->
eval_expr (Expr (Eandbool a1 a2) ty) Vfalse
| eval_Eandbool_2: forall a1 a2 ty v1 v2 v,
eval_expr a1 v1 ->
is_true v1 (typeof a1) ->
eval_expr a2 v2 ->
bool_of_val v2 (typeof a2) v ->
eval_expr (Expr (Eandbool a1 a2) ty) v
| eval_Ecast: forall a ty ty' v1 v,
eval_expr a v1 ->
cast v1 (typeof a) ty v ->
eval_expr (Expr (Ecast ty a) ty') v
eval_lvalue ge e m a b ofs defines the evaluation of expression a
in l-value position. The result is the memory location b, ofs
that contains the value of the expression a.
with eval_lvalue: expr -> block -> int -> Prop :=
| eval_Evar_local: forall id l ty,
e!id = Some l ->
eval_lvalue (Expr (Evar id) ty) l Int.zero
| eval_Evar_global: forall id l ty,
e!id = None ->
Genv.find_symbol ge id = Some l ->
eval_lvalue (Expr (Evar id) ty) l Int.zero
| eval_Ederef: forall a ty l ofs,
eval_expr a (Vptr l ofs) ->
eval_lvalue (Expr (Ederef a) ty) l ofs
| eval_Efield_struct: forall a i ty l ofs id fList delta,
eval_lvalue a l ofs ->
typeof a = Tstruct id fList ->
field_offset i fList = OK delta ->
eval_lvalue (Expr (Efield a i) ty) l (Int.add ofs (Int.repr delta))
| eval_Efield_union: forall a i ty l ofs id fList,
eval_lvalue a l ofs ->
typeof a = Tunion id fList ->
eval_lvalue (Expr (Efield a i) ty) l ofs.
Scheme eval_expr_ind2 := Minimality for eval_expr Sort Prop
with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop.
eval_exprlist ge e m al vl evaluates a list of r-value
expressions al to their values vl.
Inductive eval_exprlist: list expr -> list val -> Prop :=
| eval_Enil:
eval_exprlist nil nil
| eval_Econs: forall a bl v vl,
eval_expr a v ->
eval_exprlist bl vl ->
eval_exprlist (a :: bl) (v :: vl).
End EXPR.
Continuations
Inductive cont: Type :=
| Kstop: cont
| Kseq: statement -> cont -> cont
Kseq s2 k = after s1 in s1;s2
| Kwhile: expr -> statement -> cont -> cont
Kwhile e s k = after s in while (e) s
| Kdowhile: expr -> statement -> cont -> cont
Kdowhile e s k = after s in do s while (e)
| Kfor2: expr -> statement -> statement -> cont -> cont
Kfor2 e2 e3 s k = after s in for(e1;e2;e3) s
| Kfor3: expr -> statement -> statement -> cont -> cont
Kfor3 e2 e3 s k = after e3 in for(e1;e2;e3) s
| Kswitch: cont -> cont
catches break statements arising out of switch
| Kcall: option (block * int * type) ->
where to store result
function ->
calling function
env ->
local env of calling function
cont -> cont.
Pop continuation until a call or stop
Fixpoint call_cont (k: cont) : cont :=
match k with
| Kseq s k => call_cont k
| Kwhile e s k => call_cont k
| Kdowhile e s k => call_cont k
| Kfor2 e2 e3 s k => call_cont k
| Kfor3 e2 e3 s k => call_cont k
| Kswitch k => call_cont k
| _ => k
end.
Definition is_call_cont (k: cont) : Prop :=
match k with
| Kstop => True
| Kcall _ _ _ _ => True
| _ => False
end.
States
Inductive state: Type :=
| State
(f: function)
(s: statement)
(k: cont)
(e: env)
(m: mem) : state
| Callstate
(fd: fundef)
(args: list val)
(k: cont)
(m: mem) : state
| Returnstate
(res: val)
(k: cont)
(m: mem) : state.
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 (Kwhile a s1 k)
| Sdowhile a s1 =>
find_label lbl s1 (Kdowhile 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 (Kfor2 a2 a3 s1 k) with
| Some sk => Some sk
| None => find_label lbl a3 (Kfor3 a2 a3 s1 k)
end
end
| Sswitch e sl =>
find_label_ls lbl sl (Kswitch 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
| LSdefault s => find_label lbl s k
| LScase _ 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.
Transition relation
Inductive step: state -> trace -> state -> Prop :=
| step_assign: forall f a1 a2 k e m loc ofs v2 m',
eval_lvalue e m a1 loc ofs ->
eval_expr e m a2 v2 ->
store_value_of_type (typeof a1) m loc ofs v2 = Some m' ->
step (State f (Sassign a1 a2) k e m)
E0 (State f Sskip k e m')
| step_call_none: forall f a al k e m vf vargs fd,
eval_expr e m a vf ->
eval_exprlist e m al vargs ->
Genv.find_funct ge vf = Some fd ->
type_of_fundef fd = typeof a ->
step (State f (Scall None a al) k e m)
E0 (Callstate fd vargs (Kcall None f e k) m)
| step_call_some: forall f lhs a al k e m loc ofs vf vargs fd,
eval_lvalue e m lhs loc ofs ->
eval_expr e m a vf ->
eval_exprlist e m al vargs ->
Genv.find_funct ge vf = Some fd ->
type_of_fundef fd = typeof a ->
step (State f (Scall (Some lhs) a al) k e m)
E0 (Callstate fd vargs (Kcall (Some(loc, ofs, typeof lhs)) f e k) m)
| step_seq: forall f s1 s2 k e m,
step (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,
step (State f Sskip (Kseq s k) e m)
E0 (State f s k e m)
| step_continue_seq: forall f s k e m,
step (State f Scontinue (Kseq s k) e m)
E0 (State f Scontinue k e m)
| step_break_seq: forall f s k e m,
step (State f Sbreak (Kseq s k) e m)
E0 (State f Sbreak k e m)
| step_ifthenelse_true: forall f a s1 s2 k e m v1,
eval_expr e m a v1 ->
is_true v1 (typeof a) ->
step (State f (Sifthenelse a s1 s2) k e m)
E0 (State f s1 k e m)
| step_ifthenelse_false: forall f a s1 s2 k e m v1,
eval_expr e m a v1 ->
is_false v1 (typeof a) ->
step (State f (Sifthenelse a s1 s2) k e m)
E0 (State f s2 k e m)
| step_while_false: forall f a s k e m v,
eval_expr e m a v ->
is_false v (typeof a) ->
step (State f (Swhile a s) k e m)
E0 (State f Sskip k e m)
| step_while_true: forall f a s k e m v,
eval_expr e m a v ->
is_true v (typeof a) ->
step (State f (Swhile a s) k e m)
E0 (State f s (Kwhile a s k) e m)
| step_skip_or_continue_while: forall f x a s k e m,
x = Sskip \/ x = Scontinue ->
step (State f x (Kwhile a s k) e m)
E0 (State f (Swhile a s) k e m)
| step_break_while: forall f a s k e m,
step (State f Sbreak (Kwhile a s k) e m)
E0 (State f Sskip k e m)
| step_dowhile: forall f a s k e m,
step (State f (Sdowhile a s) k e m)
E0 (State f s (Kdowhile a s k) e m)
| step_skip_or_continue_dowhile_false: forall f x a s k e m v,
x = Sskip \/ x = Scontinue ->
eval_expr e m a v ->
is_false v (typeof a) ->
step (State f x (Kdowhile a s k) e m)
E0 (State f Sskip k e m)
| step_skip_or_continue_dowhile_true: forall f x a s k e m v,
x = Sskip \/ x = Scontinue ->
eval_expr e m a v ->
is_true v (typeof a) ->
step (State f x (Kdowhile a s k) e m)
E0 (State f (Sdowhile a s) k e m)
| step_break_dowhile: forall f a s k e m,
step (State f Sbreak (Kdowhile 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 ->
step (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_false: forall f a2 a3 s k e m v,
eval_expr e m a2 v ->
is_false v (typeof a2) ->
step (State f (Sfor Sskip a2 a3 s) k e m)
E0 (State f Sskip k e m)
| step_for_true: forall f a2 a3 s k e m v,
eval_expr e m a2 v ->
is_true v (typeof a2) ->
step (State f (Sfor Sskip a2 a3 s) k e m)
E0 (State f s (Kfor2 a2 a3 s k) e m)
| step_skip_or_continue_for2: forall f x a2 a3 s k e m,
x = Sskip \/ x = Scontinue ->
step (State f x (Kfor2 a2 a3 s k) e m)
E0 (State f a3 (Kfor3 a2 a3 s k) e m)
| step_break_for2: forall f a2 a3 s k e m,
step (State f Sbreak (Kfor2 a2 a3 s k) e m)
E0 (State f Sskip k e m)
| step_skip_for3: forall f a2 a3 s k e m,
step (State f Sskip (Kfor3 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,
f.(fn_return) = Tvoid ->
step (State f (Sreturn None) k e m)
E0 (Returnstate Vundef (call_cont k) (Mem.free_list m (blocks_of_env e)))
| step_return_1: forall f a k e m v,
f.(fn_return) <> Tvoid ->
eval_expr e m a v ->
step (State f (Sreturn (Some a)) k e m)
E0 (Returnstate v (call_cont k) (Mem.free_list m (blocks_of_env e)))
| step_skip_call: forall f k e m,
is_call_cont k ->
f.(fn_return) = Tvoid ->
step (State f Sskip k e m)
E0 (Returnstate Vundef k (Mem.free_list m (blocks_of_env e)))
| step_switch: forall f a sl k e m n,
eval_expr e m a (Vint n) ->
step (State f (Sswitch a sl) k e m)
E0 (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m)
| step_skip_break_switch: forall f x k e m,
x = Sskip \/ x = Sbreak ->
step (State f x (Kswitch k) e m)
E0 (State f Sskip k e m)
| step_continue_switch: forall f k e m,
step (State f Scontinue (Kswitch k) e m)
E0 (State f Scontinue k e m)
| step_label: forall f lbl s k e m,
step (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') ->
step (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,
alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
bind_parameters e m1 f.(fn_params) vargs m2 ->
step (Callstate (Internal f) vargs k m)
E0 (State f f.(fn_body) k e m2)
| step_external_function: forall id targs tres vargs k m vres t,
event_match (external_function id targs tres) vargs t vres ->
step (Callstate (External id targs tres) vargs k m)
t (Returnstate vres k m)
| step_returnstate_0: forall v f e k m,
step (Returnstate v (Kcall None f e k) m)
E0 (State f Sskip k e m)
| step_returnstate_1: forall v f e k m m' loc ofs ty,
store_value_of_type ty m loc ofs v = Some m' ->
step (Returnstate v (Kcall (Some(loc, ofs, ty)) f e k) m)
E0 (State f Sskip k e m').
The execution of a statement produces an ``outcome'', indicating
how the execution terminated: either normally or prematurely
through the execution of a
break, continue or return statement.
Inductive outcome: Type :=
| Out_break: outcome
terminated by break
| Out_continue: outcome
terminated by continue
| Out_normal: outcome
terminated normally
| Out_return: option val -> outcome.
terminated by return
Inductive out_normal_or_continue : outcome -> Prop :=
| Out_normal_or_continue_N: out_normal_or_continue Out_normal
| Out_normal_or_continue_C: out_normal_or_continue Out_continue.
Inductive out_break_or_return : outcome -> outcome -> Prop :=
| Out_break_or_return_B: out_break_or_return Out_break Out_normal
| Out_break_or_return_R: forall ov,
out_break_or_return (Out_return ov) (Out_return ov).
Definition outcome_switch (out: outcome) : outcome :=
match out with
| Out_break => Out_normal
| o => o
end.
Definition outcome_result_value (out: outcome) (t: type) (v: val) : Prop :=
match out, t with
| Out_normal, Tvoid => v = Vundef
| Out_return None, Tvoid => v = Vundef
| Out_return (Some v'), ty => ty <> Tvoid /\ v'=v
| _, _ => False
end.
exec_stmt ge e m1 s t m2 out describes the execution of
the statement s. out is the outcome for this execution.
m1 is the initial memory state, m2 the final memory state.
t is the trace of input/output events performed during this
evaluation.
Inductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop :=
| exec_Sskip: forall e m,
exec_stmt e m Sskip
E0 m Out_normal
| exec_Sassign: forall e m a1 a2 loc ofs v2 m',
eval_lvalue e m a1 loc ofs ->
eval_expr e m a2 v2 ->
store_value_of_type (typeof a1) m loc ofs v2 = Some m' ->
exec_stmt e m (Sassign a1 a2)
E0 m' Out_normal
| exec_Scall_none: forall e m a al vf vargs f t m' vres,
eval_expr e m a vf ->
eval_exprlist e m al vargs ->
Genv.find_funct ge vf = Some f ->
type_of_fundef f = typeof a ->
eval_funcall m f vargs t m' vres ->
exec_stmt e m (Scall None a al)
t m' Out_normal
| exec_Scall_some: forall e m lhs a al loc ofs vf vargs f t m' vres m'',
eval_lvalue e m lhs loc ofs ->
eval_expr e m a vf ->
eval_exprlist e m al vargs ->
Genv.find_funct ge vf = Some f ->
type_of_fundef f = typeof a ->
eval_funcall m f vargs t m' vres ->
store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' ->
exec_stmt e m (Scall (Some lhs) a al)
t m'' Out_normal
| exec_Sseq_1: forall e m s1 s2 t1 m1 t2 m2 out,
exec_stmt e m s1 t1 m1 Out_normal ->
exec_stmt e m1 s2 t2 m2 out ->
exec_stmt e m (Ssequence s1 s2)
(t1 ** t2) m2 out
| exec_Sseq_2: forall e m s1 s2 t1 m1 out,
exec_stmt e m s1 t1 m1 out ->
out <> Out_normal ->
exec_stmt e m (Ssequence s1 s2)
t1 m1 out
| exec_Sifthenelse_true: forall e m a s1 s2 v1 t m' out,
eval_expr e m a v1 ->
is_true v1 (typeof a) ->
exec_stmt e m s1 t m' out ->
exec_stmt e m (Sifthenelse a s1 s2)
t m' out
| exec_Sifthenelse_false: forall e m a s1 s2 v1 t m' out,
eval_expr e m a v1 ->
is_false v1 (typeof a) ->
exec_stmt e m s2 t m' out ->
exec_stmt e m (Sifthenelse a s1 s2)
t m' out
| exec_Sreturn_none: forall e m,
exec_stmt e m (Sreturn None)
E0 m (Out_return None)
| exec_Sreturn_some: forall e m a v,
eval_expr e m a v ->
exec_stmt e m (Sreturn (Some a))
E0 m (Out_return (Some v))
| exec_Sbreak: forall e m,
exec_stmt e m Sbreak
E0 m Out_break
| exec_Scontinue: forall e m,
exec_stmt e m Scontinue
E0 m Out_continue
| exec_Swhile_false: forall e m a s v,
eval_expr e m a v ->
is_false v (typeof a) ->
exec_stmt e m (Swhile a s)
E0 m Out_normal
| exec_Swhile_stop: forall e m a v s t m' out' out,
eval_expr e m a v ->
is_true v (typeof a) ->
exec_stmt e m s t m' out' ->
out_break_or_return out' out ->
exec_stmt e m (Swhile a s)
t m' out
| exec_Swhile_loop: forall e m a s v t1 m1 out1 t2 m2 out,
eval_expr e m a v ->
is_true v (typeof a) ->
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
exec_stmt e m1 (Swhile a s) t2 m2 out ->
exec_stmt e m (Swhile a s)
(t1 ** t2) m2 out
| exec_Sdowhile_false: forall e m s a t m1 out1 v,
exec_stmt e m s t m1 out1 ->
out_normal_or_continue out1 ->
eval_expr e m1 a v ->
is_false v (typeof a) ->
exec_stmt e m (Sdowhile a s)
t m1 Out_normal
| exec_Sdowhile_stop: forall e m s a t m1 out1 out,
exec_stmt e m s t m1 out1 ->
out_break_or_return out1 out ->
exec_stmt e m (Sdowhile a s)
t m1 out
| exec_Sdowhile_loop: forall e m s a m1 m2 t1 t2 out out1 v,
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
eval_expr e m1 a v ->
is_true v (typeof a) ->
exec_stmt e m1 (Sdowhile a s) t2 m2 out ->
exec_stmt e m (Sdowhile a s)
(t1 ** t2) m2 out
| exec_Sfor_start: forall e m s a1 a2 a3 out m1 m2 t1 t2,
a1 <> Sskip ->
exec_stmt e m a1 t1 m1 Out_normal ->
exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out ->
exec_stmt e m (Sfor a1 a2 a3 s)
(t1 ** t2) m2 out
| exec_Sfor_false: forall e m s a2 a3 v,
eval_expr e m a2 v ->
is_false v (typeof a2) ->
exec_stmt e m (Sfor Sskip a2 a3 s)
E0 m Out_normal
| exec_Sfor_stop: forall e m s a2 a3 v m1 t out1 out,
eval_expr e m a2 v ->
is_true v (typeof a2) ->
exec_stmt e m s t m1 out1 ->
out_break_or_return out1 out ->
exec_stmt e m (Sfor Sskip a2 a3 s)
t m1 out
| exec_Sfor_loop: forall e m s a2 a3 v m1 m2 m3 t1 t2 t3 out1 out,
eval_expr e m a2 v ->
is_true v (typeof a2) ->
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
exec_stmt e m1 a3 t2 m2 Out_normal ->
exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out ->
exec_stmt e m (Sfor Sskip a2 a3 s)
(t1 ** t2 ** t3) m3 out
| exec_Sswitch: forall e m a t n sl m1 out,
eval_expr e m a (Vint n) ->
exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out ->
exec_stmt e m (Sswitch a sl)
t m1 (outcome_switch out)
eval_funcall m1 fd args t m2 res describes the invocation of
function fd with arguments args. res is the value returned
by the call.
with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
| eval_funcall_internal: forall m f vargs t e m1 m2 m3 out vres,
alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
bind_parameters e m1 f.(fn_params) vargs m2 ->
exec_stmt e m2 f.(fn_body) t m3 out ->
outcome_result_value out f.(fn_return) vres ->
eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres
| eval_funcall_external: forall m id targs tres vargs t vres,
event_match (external_function id targs tres) vargs t vres ->
eval_funcall m (External id targs tres) vargs t m vres.
Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop
with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop.
Coinductive semantics for divergence.
execinf_stmt ge e m s t holds if the execution of statement s
diverges, i.e. loops infinitely. t is the possibly infinite
trace of observable events performed during the execution.
CoInductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop :=
| execinf_Scall_none: forall e m a al vf vargs f t,
eval_expr e m a vf ->
eval_exprlist e m al vargs ->
Genv.find_funct ge vf = Some f ->
type_of_fundef f = typeof a ->
evalinf_funcall m f vargs t ->
execinf_stmt e m (Scall None a al) t
| execinf_Scall_some: forall e m lhs a al loc ofs vf vargs f t,
eval_lvalue e m lhs loc ofs ->
eval_expr e m a vf ->
eval_exprlist e m al vargs ->
Genv.find_funct ge vf = Some f ->
type_of_fundef f = typeof a ->
evalinf_funcall m f vargs t ->
execinf_stmt e m (Scall (Some lhs) a al) t
| execinf_Sseq_1: forall e m s1 s2 t,
execinf_stmt e m s1 t ->
execinf_stmt e m (Ssequence s1 s2) t
| execinf_Sseq_2: forall e m s1 s2 t1 m1 t2,
exec_stmt e m s1 t1 m1 Out_normal ->
execinf_stmt e m1 s2 t2 ->
execinf_stmt e m (Ssequence s1 s2) (t1 *** t2)
| execinf_Sifthenelse_true: forall e m a s1 s2 v1 t,
eval_expr e m a v1 ->
is_true v1 (typeof a) ->
execinf_stmt e m s1 t ->
execinf_stmt e m (Sifthenelse a s1 s2) t
| execinf_Sifthenelse_false: forall e m a s1 s2 v1 t,
eval_expr e m a v1 ->
is_false v1 (typeof a) ->
execinf_stmt e m s2 t ->
execinf_stmt e m (Sifthenelse a s1 s2) t
| execinf_Swhile_body: forall e m a v s t,
eval_expr e m a v ->
is_true v (typeof a) ->
execinf_stmt e m s t ->
execinf_stmt e m (Swhile a s) t
| execinf_Swhile_loop: forall e m a s v t1 m1 out1 t2,
eval_expr e m a v ->
is_true v (typeof a) ->
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
execinf_stmt e m1 (Swhile a s) t2 ->
execinf_stmt e m (Swhile a s) (t1 *** t2)
| execinf_Sdowhile_body: forall e m s a t,
execinf_stmt e m s t ->
execinf_stmt e m (Sdowhile a s) t
| execinf_Sdowhile_loop: forall e m s a m1 t1 t2 out1 v,
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
eval_expr e m1 a v ->
is_true v (typeof a) ->
execinf_stmt e m1 (Sdowhile a s) t2 ->
execinf_stmt e m (Sdowhile a s) (t1 *** t2)
| execinf_Sfor_start_1: forall e m s a1 a2 a3 t,
execinf_stmt e m a1 t ->
execinf_stmt e m (Sfor a1 a2 a3 s) t
| execinf_Sfor_start_2: forall e m s a1 a2 a3 m1 t1 t2,
a1 <> Sskip ->
exec_stmt e m a1 t1 m1 Out_normal ->
execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 ->
execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2)
| execinf_Sfor_body: forall e m s a2 a3 v t,
eval_expr e m a2 v ->
is_true v (typeof a2) ->
execinf_stmt e m s t ->
execinf_stmt e m (Sfor Sskip a2 a3 s) t
| execinf_Sfor_next: forall e m s a2 a3 v m1 t1 t2 out1,
eval_expr e m a2 v ->
is_true v (typeof a2) ->
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
execinf_stmt e m1 a3 t2 ->
execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2)
| execinf_Sfor_loop: forall e m s a2 a3 v m1 m2 t1 t2 t3 out1,
eval_expr e m a2 v ->
is_true v (typeof a2) ->
exec_stmt e m s t1 m1 out1 ->
out_normal_or_continue out1 ->
exec_stmt e m1 a3 t2 m2 Out_normal ->
execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 ->
execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3)
| execinf_Sswitch: forall e m a t n sl,
eval_expr e m a (Vint n) ->
execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t ->
execinf_stmt e m (Sswitch a sl) t
evalinf_funcall ge m fd args t holds if the invocation of function
fd on arguments args diverges, with observable trace t.
with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop :=
| evalinf_funcall_internal: forall m f vargs t e m1 m2,
alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
bind_parameters e m1 f.(fn_params) vargs m2 ->
execinf_stmt e m2 f.(fn_body) t ->
evalinf_funcall m (Internal f) vargs t.
End 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,
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 ->
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.
Execution of a whole program:
exec_program p beh
holds if the application of p's main function to no arguments
in the initial memory state for p has beh as observable
behavior.
Definition exec_program (p: program) (beh: program_behavior) : Prop :=
program_behaves step (initial_state p) final_state (Genv.globalenv p) beh.
Big-step execution of a whole program.
Inductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
| bigstep_program_terminates_intro: forall b f m1 t r,
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 ->
eval_funcall ge m0 f nil t m1 (Vint r) ->
bigstep_program_terminates p t r.
Inductive bigstep_program_diverges (p: program): traceinf -> Prop :=
| bigstep_program_diverges_intro: forall b f t,
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 ->
evalinf_funcall ge m0 f nil t ->
bigstep_program_diverges p t.
Section BIGSTEP_TO_TRANSITIONS.
Variable prog: program.
Let ge : genv := Genv.globalenv prog.
Definition exec_stmt_eval_funcall_ind
(PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop)
(PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) :=
fun a b c d e f g h i j k l m n o p q r s t u v w x y =>
conj (exec_stmt_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y)
(eval_funcall_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y).
Inductive outcome_state_match
(e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop :=
| osm_normal:
outcome_state_match e m f k Out_normal (State f Sskip k e m)
| osm_break:
outcome_state_match e m f k Out_break (State f Sbreak k e m)
| osm_continue:
outcome_state_match e m f k Out_continue (State f Scontinue k e m)
| osm_return_none: forall k',
call_cont k' = call_cont k ->
outcome_state_match e m f k
(Out_return None) (State f (Sreturn None) k' e m)
| osm_return_some: forall a v k',
call_cont k' = call_cont k ->
eval_expr ge e m a v ->
outcome_state_match e m f k
(Out_return (Some v)) (State f (Sreturn (Some a)) k' e m).
Lemma is_call_cont_call_cont:
forall k, is_call_cont k -> call_cont k = k.
Lemma exec_stmt_eval_funcall_steps:
(forall e m s t m' out,
exec_stmt ge e m s t m' out ->
forall f k, exists S,
star step ge (State f s k e m) t S
/\ outcome_state_match e m' f k out S)
/\
(forall m fd args t m' res,
eval_funcall ge m fd args t m' res ->
forall k,
is_call_cont k ->
star step ge (Callstate fd args k m) t (Returnstate res k m')).
Lemma exec_stmt_steps:
forall e m s t m' out,
exec_stmt ge e m s t m' out ->
forall f k, exists S,
star step ge (State f s k e m) t S
/\ outcome_state_match e m' f k out S.
Proof (proj1 exec_stmt_eval_funcall_steps).
Lemma eval_funcall_steps:
forall m fd args t m' res,
eval_funcall ge m fd args t m' res ->
forall k,
is_call_cont k ->
star step ge (Callstate fd args k m) t (Returnstate res k m').
Proof (proj2 exec_stmt_eval_funcall_steps).
Definition order (x y: unit) := False.
Lemma evalinf_funcall_forever:
forall m fd args T k,
evalinf_funcall ge m fd args T ->
forever_N step order ge tt (Callstate fd args k m) T.
Theorem bigstep_program_terminates_exec:
forall t r, bigstep_program_terminates prog t r -> exec_program prog (Terminates t r).
Theorem bigstep_program_diverges_exec:
forall T, bigstep_program_diverges prog T ->
exec_program prog (Reacts T) \/
exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T.
End BIGSTEP_TO_TRANSITIONS.