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(** 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.
Set Implicit Arguments.
(** 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.
(** The languages are weakly typed, using only two types: [Tint] for
integers and pointers, and [Tfloat] for floating-point numbers. *)
Inductive typ : Set :=
| Tint : typ
| Tfloat : typ.
Definition typesize (ty: typ) : Z :=
match ty with Tint => 4 | Tfloat => 8 end.
(** Additionally, function definitions and function calls are annotated
by function signatures indicating the number and types of arguments,
as well as the type of the returned value if any. These signatures
are used in particular to determine appropriate calling conventions
for the function. *)
Record signature : Set := mksignature {
sig_args: list typ;
sig_res: option typ
}.
(** 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 : Set :=
| Mint8signed : memory_chunk (**r 8-bit signed integer *)
| Mint8unsigned : memory_chunk (**r 8-bit unsigned integer *)
| Mint16signed : memory_chunk (**r 16-bit signed integer *)
| Mint16unsigned : memory_chunk (**r 16-bit unsigned integer *)
| Mint32 : memory_chunk (**r 32-bit integer, or pointer *)
| Mfloat32 : memory_chunk (**r 32-bit single-precision float *)
| Mfloat64 : memory_chunk. (**r 64-bit double-precision float *)
(** Comparison instructions can perform one of the six following comparisons
between their two operands. *)
Inductive comparison : Set :=
| Ceq : comparison (**r same *)
| Cne : comparison (**r different *)
| Clt : comparison (**r less than *)
| Cle : comparison (**r less than or equal *)
| Cgt : comparison (**r greater than *)
| Cge : comparison. (**r greater than or equal *)
Definition negate_comparison (c: comparison): comparison :=
match c with
| Ceq => Cne
| Cne => Ceq
| Clt => Cge
| Cle => Cgt
| Cgt => Cle
| Cge => Clt
end.
Definition swap_comparison (c: comparison): comparison :=
match c with
| Ceq => Ceq
| Cne => Cne
| Clt => Cgt
| Cle => Cge
| Cgt => Clt
| Cge => Cle
end.
(** Whole programs consist of:
- a collection of function definitions (name and description);
- the name of the ``main'' function that serves as entry point in the program;
- a collection of global variable declarations (name and size in bytes).
The type of function descriptions varies among the various intermediate
languages and is taken as parameter to the [program] type. The other parts
of whole programs are common to all languages. *)
Record program (funct: Set) : Set := mkprogram {
prog_funct: list (ident * funct);
prog_main: ident;
prog_vars: list (ident * Z)
}.
(** 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: Set.
Variable transf: A -> B.
Fixpoint transf_program (l: list (ident * A)) : list (ident * B) :=
match l with
| nil => nil
| (id, fn) :: rem => (id, transf fn) :: transf_program rem
end.
Definition transform_program (p: program A) : program B :=
mkprogram
(transf_program p.(prog_funct))
p.(prog_main)
p.(prog_vars).
Remark transf_program_functions:
forall fl i tf,
In (i, tf) (transf_program fl) ->
exists f, In (i, f) fl /\ transf f = tf.
Proof.
induction fl; simpl.
tauto.
destruct a. simpl. intros.
elim H; intro. exists a. split. left. congruence. congruence.
generalize (IHfl _ _ H0). intros [f [IN TR]].
exists f. split. right. auto. auto.
Qed.
Lemma transform_program_function:
forall p i tf,
In (i, tf) (transform_program p).(prog_funct) ->
exists f, In (i, f) p.(prog_funct) /\ transf f = tf.
Proof.
simpl. intros. eapply transf_program_functions; eauto.
Qed.
End TRANSF_PROGRAM.
(** The following is a variant of [transform_program] where the
code transformation function can fail and therefore returns an
option type. *)
Section TRANSF_PARTIAL_PROGRAM.
Variable A B: Set.
Variable transf_partial: A -> option B.
Fixpoint transf_partial_program
(l: list (ident * A)) : option (list (ident * B)) :=
match l with
| nil => Some nil
| (id, fn) :: rem =>
match transf_partial fn with
| None => None
| Some fn' =>
match transf_partial_program rem with
| None => None
| Some res => Some ((id, fn') :: res)
end
end
end.
Definition transform_partial_program (p: program A) : option (program B) :=
match transf_partial_program p.(prog_funct) with
| None => None
| Some fl => Some (mkprogram fl p.(prog_main) p.(prog_vars))
end.
Remark transf_partial_program_functions:
forall fl tfl i tf,
transf_partial_program fl = Some tfl ->
In (i, tf) tfl ->
exists f, In (i, f) fl /\ transf_partial f = Some tf.
Proof.
induction fl; simpl.
intros; injection H; intro; subst tfl; contradiction.
case a; intros id fn. intros until tf.
caseEq (transf_partial fn).
intros tfn TFN.
caseEq (transf_partial_program fl).
intros tfl1 TFL1 EQ. injection EQ; intro; clear EQ; subst tfl.
simpl. intros [EQ1|IN1].
exists fn. intuition congruence.
generalize (IHfl _ _ _ TFL1 IN1).
intros [f [X Y]].
exists f. intuition congruence.
intros; discriminate.
intros; discriminate.
Qed.
Lemma transform_partial_program_function:
forall p tp i tf,
transform_partial_program p = Some tp ->
In (i, tf) tp.(prog_funct) ->
exists f, In (i, f) p.(prog_funct) /\ transf_partial f = Some tf.
Proof.
intros until tf.
unfold transform_partial_program.
caseEq (transf_partial_program (prog_funct p)).
intros. apply transf_partial_program_functions with l; auto.
injection H0; intros; subst tp. exact H1.
intros; discriminate.
Qed.
Lemma transform_partial_program_main:
forall p tp,
transform_partial_program p = Some tp ->
tp.(prog_main) = p.(prog_main).
Proof.
intros p tp. unfold transform_partial_program.
destruct (transf_partial_program (prog_funct p)).
intro EQ; injection EQ; intro EQ1; rewrite <- EQ1; reflexivity.
intro; discriminate.
Qed.
End TRANSF_PARTIAL_PROGRAM.
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