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 Cminor.
Require Import Smallstep.

Abstract syntax

Csharpminor is a low-level imperative language structured in expressions, statements, functions and programs. Expressions include reading global or local variables, reading store locations, arithmetic operations, function calls, and conditional expressions (similar to e1 ? e2 : e3 in C). Unlike in Cminor (the next intermediate language of the back-end), Csharpminor local variables reside in memory, and their addresses can be taken using Eaddrof expressions.

Inductive constant : Type :=
  | Ointconst: int -> constant (* integer constant *)
  | Ofloatconst: float -> constant. (* floating-point constant *)

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

Inductive expr : Type :=
  | Evar : ident -> expr (* reading a scalar variable *)
  | Etempvar : 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 *)
  | Econdition : expr -> expr -> expr -> expr. (* conditional expression *)

Statements include expression evaluation, 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
  | Sassign : ident -> expr -> 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: expr -> lbl_stmt -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt

with lbl_stmt : Type :=
  | LSdefault: stmt -> lbl_stmt
  | LScase: int -> stmt -> lbl_stmt -> lbl_stmt.

The variables can be either scalar variables (whose type, size and signedness are given by a memory_chunk or array variables (of the indicated sizes and alignment). The only operation permitted on an array variable is taking its address.

Inductive var_kind : Type :=
  | Vscalar(chunk: memory_chunk)
  | Varray(sz al: Z).

Definition sizeof (lv: var_kind) : Z :=
  match lv with
  | Vscalar chunk => size_chunk chunk
  | Varray sz al => Zmax 0 sz

Definition type_of_kind (lv: var_kind) : typ :=
  match lv with
  | Vscalar chunk => type_of_chunk chunk
  | Varray _ _ => Tint

Functions are composed of a return type, a list of parameter names with associated var_kind descriptions, a list of local variables with associated var_kind descriptions, and a statement representing the function body.

Definition variable_name (v: ident * var_kind) := fst v.
Definition variable_kind (v: ident * var_kind) := snd v.

Record function : Type := mkfunction {
  fn_return: option typ;
  fn_params: list (ident * var_kind);
  fn_vars: list (ident * var_kind);
  fn_temps: list ident;
  fn_body: stmt

Definition fundef := AST.fundef function.

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

Definition fn_sig (f: function) :=
  mksignature ( type_of_kind ( variable_kind f.(fn_params)))

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

Definition fn_variables (f: function) := f.(fn_params) ++ f.(fn_vars).

Definition fn_params_names (f: function) := variable_name f.(fn_params).
Definition fn_vars_names (f: function) := variable_name f.(fn_vars).

Operational semantics

Three evaluation environments are involved:

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

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


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 (n: int) (sl: lbl_stmt) {struct sl} : lbl_stmt :=
  match sl with
  | LSdefault _ => sl
  | LScase c s sl' => if Int.eq c n then sl else select_switch n sl'

Fixpoint seq_of_lbl_stmt (sl: lbl_stmt) : stmt :=
  match sl with
  | LSdefault s => s
  | LScase 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 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
  | LSdefault s => find_label lbl s k
  | LScase _ 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)

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

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

Definition block_of_binding (id_b_lv: ident * (block * var_kind)) :=
  match id_b_lv with (id, (b, lv)) => (b, 0, sizeof lv) end.

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

Section RELSEM.

Variable ge: genv.

Initialization of local variables that are parameters. The value of the corresponding argument is stored into the memory block bound to the parameter.

Definition val_normalized (v: val) (chunk: memory_chunk) : Prop :=
  Val.load_result chunk v = v.

Inductive bind_parameters: env ->
                           mem -> list (ident * var_kind) -> list val ->
                           mem -> Prop :=
  | bind_parameters_nil:
      forall e m,
      bind_parameters e m nil nil m
  | bind_parameters_scalar:
      forall e m id chunk params v1 vl b m1 m2,
      PTree.get id e = Some (b, Vscalar chunk) ->
      val_normalized v1 chunk -> chunk m b 0 v1 = Some m1 ->
      bind_parameters e m1 params vl m2 ->
      bind_parameters e m ((id, Vscalar chunk) :: params) (v1 :: vl) m2
  | bind_parameters_array:
      forall e m id sz al params v1 vl b m1 m2,
      PTree.get id e = Some (b, Varray sz al) ->
      extcall_memcpy_sem sz (Zmin al 4)
                   ge (Vptr b :: v1 :: nil) m E0 Vundef m1 ->
      bind_parameters e m1 params vl m2 ->
      bind_parameters e m ((id, Varray sz al) :: params) (v1 :: vl) m2.

Inductive eval_var_addr: env -> ident -> block -> Prop :=
  | eval_var_addr_local:
      forall e id b vi,
      PTree.get id e = Some (b, vi) ->
      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.

Inductive eval_var_ref: env -> ident -> block -> memory_chunk -> Prop :=
  | eval_var_ref_local:
      forall e id b chunk,
      PTree.get id e = Some (b, Vscalar chunk) ->
      eval_var_ref e id b chunk
  | eval_var_ref_global:
      forall e id b gv chunk,
      PTree.get id e = None ->
      Genv.find_symbol ge id = Some b ->
      Genv.find_var_info ge b = Some gv ->
      gvar_info gv = Vscalar chunk ->
      eval_var_ref e id b chunk.

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 b chunk v,
      eval_var_ref e id b chunk ->
      Mem.load chunk m b 0 = Some v ->
      eval_expr (Evar id) v
  | eval_Etempvar: forall id v,
      le!id = Some v ->
      eval_expr (Etempvar 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
  | eval_Econdition: forall a b c v1 vb1 v2,
      eval_expr a v1 ->
      Val.bool_of_val v1 vb1 ->
      eval_expr (if vb1 then b else c) v2 ->
      eval_expr (Econdition a b c) v2.

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).


Execution of an assignment to a variable.

Inductive exec_assign: env -> mem -> ident -> val -> mem -> Prop :=
  exec_assign_intro: forall e m id v b chunk m',
    eval_var_ref e id b chunk ->
    val_normalized v chunk -> chunk m b 0 v = Some m' ->
    exec_assign e m id v m'.

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 ->
      f.(fn_return) = None ->
      Mem.free_list m (blocks_of_env e) = Some m' ->
      step (State f Sskip k e le m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k e le m m' v,
      eval_expr e le m a v ->
      exec_assign e m id v m' ->
      step (State f (Sassign id a) k e le m)
        E0 (State f Sskip k e le 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 a cases k e le m n,
      eval_expr e le m a (Vint n) ->
      step (State f (Sswitch 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 m2 e,
      list_norepet (fn_params_names f ++ fn_vars_names f) ->
      alloc_variables empty_env m (fn_variables f) 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 empty_temp_env m2)

  | 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 = mksignature nil (Some Tint) ->
      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).