Module AST


This file defines a number of data types and operations used in the abstract syntax trees of many of the intermediate languages.

Require Import Coqlib.
Require String.
Require Import Errors.
Require Import Integers.
Require Import Floats.

Set Implicit Arguments.

Syntactic elements


Identifiers (names of local variables, of global symbols and functions, etc) are represented by the type positive of positive integers.

Definition ident := positive.

Definition ident_eq := peq.

Parameter ident_of_string : String.string -> ident.

The intermediate languages are weakly typed, using the following types:

Inductive typ : Type :=
  | Tint (* 32-bit integers or pointers *)
  | Tfloat (* 64-bit double-precision floats *)
  | Tlong (* 64-bit integers *)
  | Tsingle (* 32-bit single-precision floats *)
  | Tany32 (* any 32-bit value *)
  | Tany64. (* any 64-bit value, i.e. any value *)

Lemma typ_eq: forall (t1 t2: typ), {t1=t2} + {t1<>t2}.
Proof.
decide equality. Defined.
Global Opaque typ_eq.

Definition opt_typ_eq: forall (t1 t2: option typ), {t1=t2} + {t1<>t2}
                     := option_eq typ_eq.

Definition list_typ_eq: forall (l1 l2: list typ), {l1=l2} + {l1<>l2}
                     := list_eq_dec typ_eq.

Definition typesize (ty: typ) : Z :=
  match ty with
  | Tint => 4
  | Tfloat => 8
  | Tlong => 8
  | Tsingle => 4
  | Tany32 => 4
  | Tany64 => 8
  end.

Lemma typesize_pos: forall ty, typesize ty > 0.
Proof.
destruct ty; simpl; omega. Qed.

All values of size 32 bits are also of type Tany32. All values are of type Tany64. This corresponds to the following subtyping relation over types.

Definition subtype (ty1 ty2: typ) : bool :=
  match ty1, ty2 with
  | Tint, Tint => true
  | Tlong, Tlong => true
  | Tfloat, Tfloat => true
  | Tsingle, Tsingle => true
  | (Tint | Tsingle | Tany32), Tany32 => true
  | _, Tany64 => true
  | _, _ => false
  end.

Fixpoint subtype_list (tyl1 tyl2: list typ) : bool :=
  match tyl1, tyl2 with
  | nil, nil => true
  | ty1::tys1, ty2::tys2 => subtype ty1 ty2 && subtype_list tys1 tys2
  | _, _ => false
  end.

Additionally, function definitions and function calls are annotated by function signatures indicating: These signatures are used in particular to determine appropriate calling conventions for the function.

Record calling_convention : Type := mkcallconv {
  cc_vararg: bool;
  cc_structret: bool
}.

Definition cc_default :=
  {| cc_vararg := false; cc_structret := false |}.

Record signature : Type := mksignature {
  sig_args: list typ;
  sig_res: option typ;
  sig_cc: calling_convention
}.

Definition proj_sig_res (s: signature) : typ :=
  match s.(sig_res) with
  | None => Tint
  | Some t => t
  end.

Definition signature_eq: forall (s1 s2: signature), {s1=s2} + {s1<>s2}.
Proof.
  generalize opt_typ_eq, list_typ_eq; intros; decide equality.
  generalize bool_dec; intros. decide equality.
Defined.
Global Opaque signature_eq.

Definition signature_main :=
  {| sig_args := nil; sig_res := Some Tint; sig_cc := cc_default |}.

Memory accesses (load and store instructions) are annotated by a ``memory chunk'' indicating the type, size and signedness of the chunk of memory being accessed.

Inductive memory_chunk : Type :=
  | Mint8signed (* 8-bit signed integer *)
  | Mint8unsigned (* 8-bit unsigned integer *)
  | Mint16signed (* 16-bit signed integer *)
  | Mint16unsigned (* 16-bit unsigned integer *)
  | Mint32 (* 32-bit integer, or pointer *)
  | Mint64 (* 64-bit integer *)
  | Mfloat32 (* 32-bit single-precision float *)
  | Mfloat64 (* 64-bit double-precision float *)
  | Many32 (* any value that fits in 32 bits *)
  | Many64. (* any value *)

Definition chunk_eq: forall (c1 c2: memory_chunk), {c1=c2} + {c1<>c2}.
Proof.
decide equality. Defined.
Global Opaque chunk_eq.

The type (integer/pointer or float) of a chunk.

Definition type_of_chunk (c: memory_chunk) : typ :=
  match c with
  | Mint8signed => Tint
  | Mint8unsigned => Tint
  | Mint16signed => Tint
  | Mint16unsigned => Tint
  | Mint32 => Tint
  | Mint64 => Tlong
  | Mfloat32 => Tsingle
  | Mfloat64 => Tfloat
  | Many32 => Tany32
  | Many64 => Tany64
  end.

The chunk that is appropriate to store and reload a value of the given type, without losing information.

Definition chunk_of_type (ty: typ) :=
  match ty with
  | Tint => Mint32
  | Tfloat => Mfloat64
  | Tlong => Mint64
  | Tsingle => Mfloat32
  | Tany32 => Many32
  | Tany64 => Many64
  end.

Initialization data for global variables.

Inductive init_data: Type :=
  | Init_int8: int -> init_data
  | Init_int16: int -> init_data
  | Init_int32: int -> init_data
  | Init_int64: int64 -> init_data
  | Init_float32: float32 -> init_data
  | Init_float64: float -> init_data
  | Init_space: Z -> nat -> init_data
  | Init_addrof: ident -> int -> init_data. (* address of symbol + offset *)

Information attached to global variables.

Record globvar (V: Type) : Type := mkglobvar {
  gvar_info: V; (* language-dependent info, e.g. a type *)
  gvar_init: list init_data; (* initialization data *)
  gvar_readonly: bool; (* read-only variable? (const) *)
  gvar_volatile: bool (* volatile variable? *)
}.

Whole programs consist of: A global definition is either a global function or a global variable. The type of function descriptions and that of additional information for variables vary among the various intermediate languages and are taken as parameters to the program type. The other parts of whole programs are common to all languages.

Inductive globdef (F V: Type) : Type :=
  | Gfun (f: F)
  | Gvar (v: globvar V).

Implicit Arguments Gfun [F V].
Implicit Arguments Gvar [F V].

Record program (F V: Type) : Type := mkprogram {
  prog_defs: list (ident * globdef F V);
  prog_main: ident
}.

Definition prog_defs_names (F V: Type) (p: program F V) : list ident :=
  List.map fst p.(prog_defs).

Generic transformations over programs


We now define a general iterator over programs that applies a given code transformation function to all function descriptions and leaves the other parts of the program unchanged.

Section TRANSF_PROGRAM.

Variable A B V: Type.
Variable transf: A -> B.

Definition transform_program_globdef (idg: ident * globdef A V) : ident * globdef B V :=
  match idg with
  | (id, Gfun f) => (id, Gfun (transf f))
  | (id, Gvar v) => (id, Gvar v)
  end.

Definition transform_program (p: program A V) : program B V :=
  mkprogram
    (List.map transform_program_globdef p.(prog_defs))
    p.(prog_main).

Lemma transform_program_function:
  forall p i tf,
  In (i, Gfun tf) (transform_program p).(prog_defs) ->
  exists f, In (i, Gfun f) p.(prog_defs) /\ transf f = tf.
Proof.
  simpl. unfold transform_program. intros.
  exploit list_in_map_inv; eauto.
  intros [[i' gd] [EQ IN]]. simpl in EQ. destruct gd; inv EQ.
  exists f; auto.
Qed.

End TRANSF_PROGRAM.

The following is a more general presentation of transform_program where global variable information can be transformed, in addition to function definitions. Moreover, the transformation functions can fail and return an error message.

Open Local Scope error_monad_scope.
Open Local Scope string_scope.

Section TRANSF_PROGRAM_GEN.

Variables A B V W: Type.
Variable transf_fun: A -> res B.
Variable transf_var: V -> res W.

Definition transf_globvar (g: globvar V) : res (globvar W) :=
  do info' <- transf_var g.(gvar_info);
  OK (mkglobvar info' g.(gvar_init) g.(gvar_readonly) g.(gvar_volatile)).

Fixpoint transf_globdefs (l: list (ident * globdef A V)) : res (list (ident * globdef B W)) :=
  match l with
  | nil => OK nil
  | (id, Gfun f) :: l' =>
      match transf_fun f with
      | Error msg => Error (MSG "In function " :: CTX id :: MSG ": " :: msg)
      | OK tf =>
          do tl' <- transf_globdefs l'; OK ((id, Gfun tf) :: tl')
      end
  | (id, Gvar v) :: l' =>
      match transf_globvar v with
      | Error msg => Error (MSG "In variable " :: CTX id :: MSG ": " :: msg)
      | OK tv =>
          do tl' <- transf_globdefs l'; OK ((id, Gvar tv) :: tl')
      end
  end.

Definition transform_partial_program2 (p: program A V) : res (program B W) :=
  do gl' <- transf_globdefs p.(prog_defs); OK(mkprogram gl' p.(prog_main)).

Lemma transform_partial_program2_function:
  forall p tp i tf,
  transform_partial_program2 p = OK tp ->
  In (i, Gfun tf) tp.(prog_defs) ->
  exists f, In (i, Gfun f) p.(prog_defs) /\ transf_fun f = OK tf.
Proof.
  intros. monadInv H. simpl in H0.
  revert x EQ H0. induction (prog_defs p); simpl; intros.
  inv EQ. contradiction.
  destruct a as [id [f|v]].
  destruct (transf_fun f) as [tf1|msg] eqn:?; monadInv EQ.
  simpl in H0; destruct H0. inv H. exists f; auto.
  exploit IHl; eauto. intros [f' [P Q]]; exists f'; auto.
  destruct (transf_globvar v) as [tv1|msg] eqn:?; monadInv EQ.
  simpl in H0; destruct H0. inv H.
  exploit IHl; eauto. intros [f' [P Q]]; exists f'; auto.
Qed.

Lemma transform_partial_program2_variable:
  forall p tp i tv,
  transform_partial_program2 p = OK tp ->
  In (i, Gvar tv) tp.(prog_defs) ->
  exists v,
     In (i, Gvar(mkglobvar v tv.(gvar_init) tv.(gvar_readonly) tv.(gvar_volatile))) p.(prog_defs)
  /\ transf_var v = OK tv.(gvar_info).
Proof.
  intros. monadInv H. simpl in H0.
  revert x EQ H0. induction (prog_defs p); simpl; intros.
  inv EQ. contradiction.
  destruct a as [id [f|v]].
  destruct (transf_fun f) as [tf1|msg] eqn:?; monadInv EQ.
  simpl in H0; destruct H0. inv H.
  exploit IHl; eauto. intros [v' [P Q]]; exists v'; auto.
  destruct (transf_globvar v) as [tv1|msg] eqn:?; monadInv EQ.
  simpl in H0; destruct H0. inv H.
  monadInv Heqr. simpl. exists (gvar_info v). split. left. destruct v; auto. auto.
  exploit IHl; eauto. intros [v' [P Q]]; exists v'; auto.
Qed.

Lemma transform_partial_program2_succeeds:
  forall p tp i g,
  transform_partial_program2 p = OK tp ->
  In (i, g) p.(prog_defs) ->
  match g with
  | Gfun fd => exists tfd, transf_fun fd = OK tfd
  | Gvar gv => exists tv, transf_var gv.(gvar_info) = OK tv
  end.
Proof.
  intros. monadInv H.
  revert x EQ H0. induction (prog_defs p); simpl; intros.
  contradiction.
  destruct a as [id1 g1]. destruct g1.
  destruct (transf_fun f) eqn:TF; try discriminate. monadInv EQ.
  destruct H0. inv H. econstructor; eauto. eapply IHl; eauto.
  destruct (transf_globvar v) eqn:TV; try discriminate. monadInv EQ.
  destruct H0. inv H. monadInv TV. econstructor; eauto. eapply IHl; eauto.
Qed.

Lemma transform_partial_program2_main:
  forall p tp,
  transform_partial_program2 p = OK tp ->
  tp.(prog_main) = p.(prog_main).
Proof.
  intros. monadInv H. reflexivity.
Qed.

Additionally, we can also "augment" the program with new global definitions and a different "main" function.

Section AUGMENT.

Variable new_globs: list(ident * globdef B W).
Variable new_main: ident.

Definition transform_partial_augment_program (p: program A V) : res (program B W) :=
  do gl' <- transf_globdefs p.(prog_defs);
  OK(mkprogram (gl' ++ new_globs) new_main).

Lemma transform_partial_augment_program_main:
  forall p tp,
  transform_partial_augment_program p = OK tp ->
  tp.(prog_main) = new_main.
Proof.
  intros. monadInv H. reflexivity.
Qed.

End AUGMENT.

Remark transform_partial_program2_augment:
  forall p,
  transform_partial_program2 p =
  transform_partial_augment_program nil p.(prog_main) p.
Proof.
  unfold transform_partial_program2, transform_partial_augment_program; intros.
  destruct (transf_globdefs (prog_defs p)); auto.
  simpl. f_equal. f_equal. rewrite <- app_nil_end. auto.
Qed.

End TRANSF_PROGRAM_GEN.

The following is a special case of transform_partial_program2, where only function definitions are transformed, but not variable definitions.

Section TRANSF_PARTIAL_PROGRAM.

Variable A B V: Type.
Variable transf_partial: A -> res B.

Definition transform_partial_program (p: program A V) : res (program B V) :=
  transform_partial_program2 transf_partial (fun v => OK v) p.

Lemma transform_partial_program_main:
  forall p tp,
  transform_partial_program p = OK tp ->
  tp.(prog_main) = p.(prog_main).
Proof.
  apply transform_partial_program2_main.
Qed.

Lemma transform_partial_program_function:
  forall p tp i tf,
  transform_partial_program p = OK tp ->
  In (i, Gfun tf) tp.(prog_defs) ->
  exists f, In (i, Gfun f) p.(prog_defs) /\ transf_partial f = OK tf.
Proof.
  apply transform_partial_program2_function.
Qed.

Lemma transform_partial_program_succeeds:
  forall p tp i fd,
  transform_partial_program p = OK tp ->
  In (i, Gfun fd) p.(prog_defs) ->
  exists tfd, transf_partial fd = OK tfd.
Proof.
  unfold transform_partial_program; intros.
  exploit transform_partial_program2_succeeds; eauto.
Qed.

End TRANSF_PARTIAL_PROGRAM.

Lemma transform_program_partial_program:
  forall (A B V: Type) (transf: A -> B) (p: program A V),
  transform_partial_program (fun f => OK(transf f)) p = OK(transform_program transf p).
Proof.
  intros.
  unfold transform_partial_program, transform_partial_program2, transform_program; intros.
  replace (transf_globdefs (fun f => OK (transf f)) (fun v => OK v) p.(prog_defs))
     with (OK (map (transform_program_globdef transf) p.(prog_defs))).
  auto.
  induction (prog_defs p); simpl.
  auto.
  destruct a as [id [f|v]]; rewrite <- IHl.
    auto.
    destruct v; auto.
Qed.

The following is a relational presentation of transform_partial_augment_preogram. Given relations between function definitions and between variable information, it defines a relation between programs stating that the two programs have appropriately related shapes (global names are preserved and possibly augmented, etc) and that identically-named function definitions and variable information are related.

Section MATCH_PROGRAM.

Variable A B V W: Type.
Variable match_fundef: A -> B -> Prop.
Variable match_varinfo: V -> W -> Prop.

Inductive match_globdef: ident * globdef A V -> ident * globdef B W -> Prop :=
  | match_glob_fun: forall id f1 f2,
      match_fundef f1 f2 ->
      match_globdef (id, Gfun f1) (id, Gfun f2)
  | match_glob_var: forall id init ro vo info1 info2,
      match_varinfo info1 info2 ->
      match_globdef (id, Gvar (mkglobvar info1 init ro vo)) (id, Gvar (mkglobvar info2 init ro vo)).

Definition match_program (new_globs : list (ident * globdef B W))
                         (new_main : ident)
                         (p1: program A V) (p2: program B W) : Prop :=
  (exists tglob, list_forall2 match_globdef p1.(prog_defs) tglob /\
                 p2.(prog_defs) = tglob ++ new_globs) /\
  p2.(prog_main) = new_main.

End MATCH_PROGRAM.

Lemma transform_partial_augment_program_match:
  forall (A B V W: Type)
         (transf_fun: A -> res B)
         (transf_var: V -> res W)
         (p: program A V)
         (new_globs : list (ident * globdef B W))
         (new_main : ident)
         (tp: program B W),
  transform_partial_augment_program transf_fun transf_var new_globs new_main p = OK tp ->
  match_program
    (fun fd tfd => transf_fun fd = OK tfd)
    (fun info tinfo => transf_var info = OK tinfo)
    new_globs new_main
    p tp.
Proof.
  unfold transform_partial_augment_program; intros. monadInv H.
  red; simpl. split; auto. exists x; split; auto.
  revert x EQ. generalize (prog_defs p). induction l; simpl; intros.
  monadInv EQ. constructor.
  destruct a as [id [f|v]].
  destruct (transf_fun f) as [tf|?] eqn:?; monadInv EQ.
  constructor; auto. constructor; auto.
  unfold transf_globvar in EQ.
  destruct (transf_var (gvar_info v)) as [tinfo|?] eqn:?; simpl in EQ; monadInv EQ.
  constructor; auto. destruct v; simpl in *. constructor; auto.
Qed.

External functions


For most languages, the functions composing the program are either internal functions, defined within the language, or external functions, defined outside. External functions include system calls but also compiler built-in functions. We define a type for external functions and associated operations.

Inductive external_function : Type :=
  | EF_external (name: ident) (sg: signature)
A system call or library function. Produces an event in the trace.
  | EF_builtin (name: ident) (sg: signature)
A compiler built-in function. Behaves like an external, but can be inlined by the compiler.
  | EF_vload (chunk: memory_chunk)
A volatile read operation. If the adress given as first argument points within a volatile global variable, generate an event and return the value found in this event. Otherwise, produce no event and behave like a regular memory load.
  | EF_vstore (chunk: memory_chunk)
A volatile store operation. If the adress given as first argument points within a volatile global variable, generate an event. Otherwise, produce no event and behave like a regular memory store.
  | EF_vload_global (chunk: memory_chunk) (id: ident) (ofs: int)
A volatile load operation from a global variable. Specialized version of EF_vload.
  | EF_vstore_global (chunk: memory_chunk) (id: ident) (ofs: int)
A volatile store operation in a global variable. Specialized version of EF_vstore.
  | EF_malloc
Dynamic memory allocation. Takes the requested size in bytes
  | EF_free
Dynamic memory deallocation. Takes a pointer to a block
  | EF_memcpy (sz: Z) (al: Z)
Block copy, of sz bytes, between addresses that are al-aligned.
  | EF_annot (text: ident) (targs: list annot_arg)
A programmer-supplied annotation. Takes zero, one or several arguments,
  | EF_annot_val (text: ident) (targ: typ)
Another form of annotation that takes one argument, produces
  | EF_inline_asm (text: ident)
Inline asm statements. Semantically, treated like an annotation with no parameters (EF_annot text nil). To be used with caution, as it can invalidate the semantic preservation theorem. Generated only if -finline-asm is given.

with annot_arg : Type :=
  | AA_arg (ty: typ)
  | AA_int (n: int)
  | AA_float (n: float).

The type signature of an external function.

Fixpoint annot_args_typ (targs: list annot_arg) : list typ :=
  match targs with
  | nil => nil
  | AA_arg ty :: targs' => ty :: annot_args_typ targs'
  | _ :: targs' => annot_args_typ targs'
  end.

Definition ef_sig (ef: external_function): signature :=
  match ef with
  | EF_external name sg => sg
  | EF_builtin name sg => sg
  | EF_vload chunk => mksignature (Tint :: nil) (Some (type_of_chunk chunk)) cc_default
  | EF_vstore chunk => mksignature (Tint :: type_of_chunk chunk :: nil) None cc_default
  | EF_vload_global chunk _ _ => mksignature nil (Some (type_of_chunk chunk)) cc_default
  | EF_vstore_global chunk _ _ => mksignature (type_of_chunk chunk :: nil) None cc_default
  | EF_memcpy sz al => mksignature (Tint :: Tint :: nil) None cc_default
  | EF_malloc => mksignature (Tint :: nil) (Some Tint) cc_default
  | EF_free => mksignature (Tint :: nil) None cc_default
  | EF_inline_asm text => mksignature nil None cc_default
  | EF_annot text targs => mksignature (annot_args_typ targs) None cc_default
  | EF_annot_val text targ => mksignature (targ :: nil) (Some targ) cc_default
  end.

Whether an external function should be inlined by the compiler.

Definition ef_inline (ef: external_function) : bool :=
  match ef with
  | EF_external name sg => false
  | EF_builtin name sg => true
  | EF_vload chunk => true
  | EF_vstore chunk => true
  | EF_vload_global chunk id ofs => true
  | EF_vstore_global chunk id ofs => true
  | EF_malloc => false
  | EF_free => false
  | EF_memcpy sz al => true
  | EF_annot text targs => true
  | EF_annot_val text targ => true
  | EF_inline_asm text => true
  end.

Whether an external function must reload its arguments.

Definition ef_reloads (ef: external_function) : bool :=
  match ef with
  | EF_annot text targs => false
  | _ => true
  end.

Equality between external functions. Used in module Allocation.

Definition external_function_eq: forall (ef1 ef2: external_function), {ef1=ef2} + {ef1<>ef2}.
Proof.
  generalize ident_eq signature_eq chunk_eq typ_eq zeq Int.eq_dec; intros.
  decide equality.
  decide equality. decide equality. apply Float.eq_dec.
Defined.
Global Opaque external_function_eq.

Function definitions are the union of internal and external functions.

Inductive fundef (F: Type): Type :=
  | Internal: F -> fundef F
  | External: external_function -> fundef F.

Implicit Arguments External [F].

Section TRANSF_FUNDEF.

Variable A B: Type.
Variable transf: A -> B.

Definition transf_fundef (fd: fundef A): fundef B :=
  match fd with
  | Internal f => Internal (transf f)
  | External ef => External ef
  end.

End TRANSF_FUNDEF.

Section TRANSF_PARTIAL_FUNDEF.

Variable A B: Type.
Variable transf_partial: A -> res B.

Definition transf_partial_fundef (fd: fundef A): res (fundef B) :=
  match fd with
  | Internal f => do f' <- transf_partial f; OK (Internal f')
  | External ef => OK (External ef)
  end.

End TRANSF_PARTIAL_FUNDEF.